2435:, called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space.
2542:
7798:
7203:
2526:
27:
4219:
2811:
2800:
2789:
2778:
2767:
2754:
2743:
2732:
2721:
2710:
2697:
2686:
2675:
2664:
2653:
2640:
2629:
2618:
2607:
2596:
15754:
7793:{\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},{\begin{pmatrix}\cos(2\pi /5)\\\sin(2\pi /5)\\\cos(4\pi /5)\\\sin(4\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(4\pi /5)\\\sin(4\pi /5)\\\cos(8\pi /5)\\\sin(8\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(6\pi /5)\\\sin(6\pi /5)\\\cos(2\pi /5)\\\sin(2\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(8\pi /5)\\\sin(8\pi /5)\\\cos(6\pi /5)\\\sin(6\pi /5)\end{pmatrix}},}
2829:
13956:, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using
7188:
3749:
5274:
forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is
8402:
6619:
11509:-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.
12463:
5227:
3061:
456:
8618:
9229:
5273:
is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices,
4589:
A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed,
4420:
6883:
4214:{\displaystyle {\begin{aligned}s_{0}&=0\\s_{1}&=s_{0}+t_{0}=t_{0}\\s_{2}&=s_{1}+t_{1}=t_{0}+t_{1}\\s_{3}&=s_{2}+t_{2}=t_{0}+t_{1}+t_{2}\\&\;\;\vdots \\s_{n}&=s_{n-1}+t_{n-1}=t_{0}+t_{1}+\cdots +t_{n-1}\\s_{n+1}&=s_{n}+t_{n}=t_{0}+t_{1}+\cdots +t_{n}=1\end{aligned}}}
8060:
5668:
14973:
8190:
6421:
15160:
10582:
677:
4677:
A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.
275:
9666:
13264:
6016:
13578:
15257:
Hence the components of these points with respect to each corresponding permuted basis are strictly ordered in the decreasing order. That explains why the simplexes are non-overlapping. The fact that the union of the simplexes is the whole unit
6837:
14646:
12284:
5055:
2873:
12267:
10005:
8458:
8168:
9086:
13068:
7183:{\displaystyle {\begin{pmatrix}\cos(2\pi /5)&-\sin(2\pi /5)&0&0\\\sin(2\pi /5)&\cos(2\pi /5)&0&0\\0&0&\cos(4\pi /5)&-\sin(4\pi /5)\\0&0&\sin(4\pi /5)&\cos(4\pi /5)\end{pmatrix}}.}
4238:
10265:
5523:
12567:, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.
7851:
5545:
14788:
13433:. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)
9799:
3504:
3312:
5539:-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
11992:
11147:
11013:
6335:
5780:
8998:
11209:
11075:
9808:
is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the
8397:{\displaystyle {\begin{pmatrix}1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\1\\-1/\surd 2\end{pmatrix}},{\begin{pmatrix}-1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\-1\\-1/\surd 2\end{pmatrix}},}
6614:{\displaystyle {\begin{pmatrix}\cos {\frac {2\pi \omega _{i}}{n+1}}&-\sin {\frac {2\pi \omega _{i}}{n+1}}\\\sin {\frac {2\pi \omega _{i}}{n+1}}&\cos {\frac {2\pi \omega _{i}}{n+1}}\end{pmatrix}},}
14781:-paths are isometric, and so is their convex hulls; this explains the congruence of the simplexes. To show the other assertions, it suffices to remark that the interior of the simplex determined by the
12563:. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite
539:
13771:
14978:
12891:
10706:
13652:
15255:
11390:
10093:
13367:
3754:
4930:
3557:
4842:
6122:
13460:
266:
10378:
15201:
14771:
10894:
9486:
9328:
13155:
2423:, requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron
4774:
2474:-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points:
14494:
13107:
5879:
964:
10367:
9894:
9498:
15282:
14733:
14704:
14675:
4491:
186:
13376:
is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the
11433:
9424:
12736:
10810:
10642:
10294:
10122:
9848:
9357:
9078:
9042:
4456:
534:
13167:
11256:
10765:
6172:
5890:
2411:, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle
13416:
4576:
11692:
6697:
6051:
5703:
5343:
14503:
2250:
2111:
1982:
1863:
1754:
1655:
1566:
1483:
1410:
1347:
12458:{\displaystyle \langle x,y\rangle ={\frac {1}{2D}}\sum _{i=1}^{D}\sum _{j=1}^{D}\log {\frac {x_{i}}{x_{j}}}\log {\frac {y_{i}}{y_{j}}}\qquad \forall x,y\in \Delta ^{D-1}}
5222:{\displaystyle \Delta _{c}^{n}=\left\{(t_{1},\ldots ,t_{n})\in \mathbf {R} ^{n}~{\Bigg |}~\sum _{i=1}^{n}t_{i}\leq 1{\text{ and }}t_{i}\geq 0{\text{ for all }}i\right\}.}
4732:
3056:{\displaystyle \Delta ^{n}=\left\{(t_{0},\dots ,t_{n})\in \mathbf {R} ^{n+1}~{\Bigg |}~\sum _{i=0}^{n}t_{i}=1{\text{ and }}t_{i}\geq 0{\text{ for }}i=0,\ldots ,n\right\}}
451:{\displaystyle C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\Bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for }}i=0,\dots ,k\right\}.}
5000:
2340:
2330:
2320:
2310:
2300:
2290:
2280:
2270:
2260:
2191:
2181:
2171:
2161:
2151:
2141:
2131:
2121:
2052:
2042:
2032:
2022:
2012:
2002:
1992:
1923:
1913:
1903:
1893:
1883:
1873:
1804:
1794:
1784:
1774:
1764:
1695:
1685:
1675:
1665:
1596:
1586:
1576:
1503:
1493:
1420:
1284:
13314:
5039:
12683:
8613:{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\left|\det {\begin{pmatrix}v_{1}-v_{0}&&v_{2}-v_{0}&&\cdots &&v_{n}-v_{0}\end{pmatrix}}\right|}
4527:
9380:
4951:
4865:
2335:
2325:
2315:
2305:
2295:
2285:
2275:
2265:
2255:
2186:
2176:
2166:
2156:
2146:
2136:
2126:
2116:
2047:
2037:
2027:
2017:
2007:
1997:
1987:
1918:
1908:
1898:
1888:
1878:
1868:
1799:
1789:
1779:
1769:
1759:
1690:
1680:
1670:
1660:
1591:
1581:
1571:
1498:
1488:
1415:
12009:
5300:
13294:
12713:
11460:
10321:
9906:
9706:
8685:
8084:
3597:
3393:
5049:
Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
13952:, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such
4635:
13889:
13853:
13798:
5823:
5803:
9224:{\displaystyle \mathrm {Volume} ={1 \over n!}\left|\det {\begin{pmatrix}v_{0}&v_{1}&\cdots &v_{n}\\1&1&\cdots &1\end{pmatrix}}\right|.}
15636:
4415:{\displaystyle \Delta _{*}^{n}=\left\{(s_{1},\ldots ,s_{n})\in \mathbf {R} ^{n}\mid 0=s_{0}\leq s_{1}\leq s_{2}\leq \dots \leq s_{n}\leq s_{n+1}=1\right\}.}
12902:
11266:
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent
10127:
5445:
8055:{\displaystyle \left\{\omega _{1},-\omega _{1},\dots ,\omega _{(n-1)/2},-\omega _{n-1)/2}\right\}=\left\{1,\dots ,(n-1)/2,(n+3)/2,\dots ,n\right\},}
5663:{\displaystyle {\frac {1}{\sqrt {2}}}\mathbf {e} _{i}-{\frac {1}{n{\sqrt {2}}}}{\bigg (}1\pm {\frac {1}{\sqrt {n+1}}}{\bigg )}\cdot (1,\dots ,1),}
14968:{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{\sigma (1)},\ v_{0}+e_{\sigma (1)}+e_{\sigma (2)}\ldots v_{0}+e_{\sigma (1)}+\cdots +e_{\sigma (n)}}
13657:
9737:
3413:
3221:
11707:
6260:
5711:
15685:
13457:-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is
11080:
10946:
8696:
14336:
14013:
12896:
It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
11152:
11018:
15155:{\displaystyle \scriptstyle v_{0}+(x_{1}+\cdots +x_{n})e_{\sigma (1)}+\cdots +(x_{n-1}+x_{n})e_{\sigma (n-1)}+x_{n}e_{\sigma (n)}}
16553:
5400:-simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form
15457:
14067:
of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a
12748:
2403:
as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point
10657:
15650:
15543:
15415:
14251:
14052:
13587:
3329:
2561:
In some conventions, the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if
15206:
13994:, strategies can be represented as points within a simplex. This representation simplifies the analysis of mixed strategies.
11300:
14385:
MacUlan, N.; De Paula, G. G. (1989). "A linear-time median-finding algorithm for projecting a vector on the simplex of n".
10017:
16580:
15988:
13322:
5396:-simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular
3510:
3520:
4870:
10577:{\displaystyle \det(v_{1}-v_{0},v_{2}-v_{0},\ldots ,v_{n}-v_{0})=\det(v_{1}-v_{0},v_{2}-v_{1},\ldots ,v_{n}-v_{n-1}).}
5884:
A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
4782:
15788:
15738:
15581:
15553:
15532:
15507:
15440:
6059:
672:{\displaystyle \left\{x\in \mathbf {R} ^{k}:x_{0}+\dots +x_{k-1}=1,x_{i}\geq 0{\text{ for }}i=0,\dots ,k-1\right\}.}
199:
13965:
15165:
5427:. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular
14738:
10815:
9731:, and these simplexes are congruent and pairwise non-overlapping. In particular, the volume of such a simplex is
9429:
9268:
13157:. In this case, both the summation convention for denoting the set, and the boundary operation commute with the
15678:
15567:
14111:
14083:
14060:
13384:
13119:
9235:
15284:". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes.
11270:
are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an
9661:{\displaystyle v_{1}=v_{0}+e_{\sigma (1)},\ v_{2}=v_{1}+e_{\sigma (2)},\ldots ,v_{n}=v_{n-1}+e_{\sigma (n)}}
4737:
2502:. A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point:
15361:
Donchian, P. S.; Coxeter, H. S. M. (July 1935). "1142. An n-dimensional extension of
Pythagoras' Theorem".
13957:
5358:
which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the
14477:
13076:
5831:
2427:, a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point
912:
15773:
10326:
9853:
694:
15265:
14709:
14680:
14651:
13259:{\displaystyle f\left(\sum \nolimits _{i}a_{i}\sigma _{i}\right)=\sum \nolimits _{i}a_{i}f(\sigma _{i})}
6643:
inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices
6011:{\displaystyle {\sqrt {1+n^{-1}}}\cdot \mathbf {e} _{i}-n^{-3/2}({\sqrt {n+1}}\pm 1)\cdot (1,\dots ,1),}
4645:. Alternatively, the volume can be computed by an iterated integral, whose successive integrands are 1,
4461:
2415:, a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point
145:
13930:
and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a
13573:{\displaystyle \Delta ^{n}:=\left\{x\in \mathbb {A} ^{n+1}~{\Bigg |}~\sum _{i=1}^{n+1}x_{i}=1\right\},}
12560:
12487:
11398:
9385:
9238:, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.
3604:
3357:
1154:
12718:
10774:
10603:
10270:
10098:
9824:
9333:
9054:
9018:
6832:{\displaystyle \{\omega _{1},n+1-\omega _{1},\dots ,\omega _{n/2},n+1-\omega _{n/2}\}=\{1,\dots ,n\},}
4432:
510:
15671:
14641:{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{1},\ v_{0}+e_{1}+e_{2},\ldots v_{0}+e_{1}+\cdots +e_{n}}
14146:
11214:
10732:
5431:-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a
13389:
6130:
4532:
16011:
15708:
15571:
14006:
11664:
11615:
possible outcomes. The correspondence is as follows: For each distribution described as an ordered
11591:
11560:
6024:
5676:
5305:
4493:(codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing,
1021:
846:
of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size
710:
17:
13949:
11623:-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose
15981:
15916:
15911:
15891:
15297:; Wills, Dean C. (October 2002). "An Elementary Calculation of the Dihedral Angle of the Regular
14044:
12498:
11624:
5302:; and the fact that the angle subtended through the center of the simplex by any two vertices is
4697:
4637:
mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume
14188:
4967:
15901:
15896:
15876:
14121:
13942:
13892:
13377:
13299:
12527:
12262:{\displaystyle \alpha \odot x=\left\qquad \forall x\in \Delta ^{D-1},\;\alpha \in \mathbb {R} }
5359:
5017:
722:
12604:
10000:{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {\det(e_{1},\ldots ,e_{n})}{n!}}.}
8163:{\displaystyle {\begin{pmatrix}0&-1&0\\1&0&0\\0&0&-1\\\end{pmatrix}}.}
4496:
875:-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the
16525:
16518:
16511:
15906:
15886:
15881:
14239:
13961:
11579:
9365:
9254:
4936:
4850:
3345:
16050:
16028:
16016:
15405:
15337:
5277:
16182:
16129:
14086:, simplices are used as building blocks of discretizations of spacetime; that is, to build
13581:
13272:
12691:
12491:
11438:
10299:
9684:
8663:
6673:
6404:
6233:
4691:
3647:
3570:
3366:
2530:
2498:. A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points:
2491:
989:
907:
794:
269:
189:
11608:-space form the space of possible probability distributions on a finite set consisting of
5014:
algorithms. Projecting onto the simplex is computationally similar to projecting onto the
8:
16537:
16436:
16186:
15783:
15778:
15520:
14204:
14131:
14087:
13972:
13941:, a simplex space is often used to represent the space of probability distributions. The
12583:
12531:
11658:
11470:
11466:
11275:
5011:
4579:
3672:
2569:
2511:
71:
15658:
11286:-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared
4617:
2541:
480:-simplex by connecting a new vertex to all original vertices by the common edge length.
16570:
16406:
16356:
16306:
16263:
16233:
16193:
16156:
15974:
15957:
15798:
15753:
15386:
15378:
15318:
14430:
14365:
14199:
14106:
13976:
13938:
13927:
13858:
13822:
13783:
13778:
13442:
12519:
12511:
11652:
11643:-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.
11564:
9818:
5808:
5788:
5432:
4687:
4591:
2483:
1115:
718:
690:
11211:
is perpendicular to the faces. So the vectors normal to the faces are permutations of
1150:
1048:
16575:
16545:
15793:
15646:
15614:
15577:
15549:
15528:
15503:
15468:
15436:
15411:
15390:
14398:
14247:
14193:
14151:
14126:
14002:
13980:
13816:
13419:
13063:{\displaystyle \partial ^{2}\sigma =\partial \left(\sum _{j=0}^{n}(-1)^{j}\right)=0.}
12515:
10372:
Finally, the formula at the beginning of this section is obtained by observing that
8637:
6197:
3072:
1273:
132:
5439:. Solving this equation shows that there are two choices for the additional vertex:
714:
16549:
16114:
16103:
16092:
16081:
16072:
16063:
16002:
15998:
15723:
15370:
15343:
15310:
14422:
14394:
14136:
14064:
13998:
11529:
9004:
3600:
3169:
745:
465:
15559:
13296:
are the integers denoting orientation and multiplicity. For the boundary operator
5251:
16139:
16124:
15768:
15713:
15640:
15632:
15294:
14075:
11267:
6225:
4599:
1336:
1099:
14331:"Sequence A135278 (Pascal's triangle with its left-hand edge removed)"
13895:
of schemes resp. rings, since the face and degeneracy maps are all polynomial).
10260:{\displaystyle e_{1}=v_{1}-v_{0},\ e_{2}=v_{2}-v_{1},\ldots e_{n}=v_{n}-v_{n-1}}
5518:{\displaystyle {\frac {1}{n}}\left(1\pm {\sqrt {n+1}}\right)\cdot (1,\dots ,1).}
2584:(skew orthogonal projections) show all the vertices of the regular simplex on a
2568:. This convention is more common in applications to algebraic topology (such as
16489:
15850:
15835:
14209:
14166:
14116:
14079:
13984:
10929:
5244:, which is based at the origin, and locally models a vertex on a polytope with
5241:
3740:
2581:
764:
749:
16564:
16506:
16394:
16387:
16380:
16344:
16337:
16330:
16294:
16287:
15840:
15617:
15339:
Connections between combinatorics of permutations and algorithms and geometry
14214:
14032:
14010:
12564:
12523:
12514:, simplices are used as building blocks to construct an interesting class of
11517:
11486:
5240:-cube, and is a standard orthogonal simplex. This is the simplex used in the
686:
15433:
Experiments with
Mixtures: Designs, Models, and the Analysis of Mixture Data
5528:
Either of these, together with the standard basis vectors, yields a regular
16446:
15860:
15825:
15718:
15495:
14354:, 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics)
14219:
14141:
13931:
10587:
From this formula, it follows immediately that the volume under a standard
6182:
4595:
3165:
2600:
2546:
504:
78:
15596:
p. 296, Table I (iii): Regular
Polytopes, three regular polytopes in
6246:-simplex. To carry this out, first observe that for any orthogonal matrix
2494:
is 3 ⋅ ( ) or {3}. A general 3-simplex is the join of 4 points:
713:, who wrote about these shapes in 1886 but called them "prime confines".
16455:
16416:
16366:
16316:
16273:
16243:
16175:
16161:
15945:
15728:
14326:
13991:
12542:
11435:
are facets being pairwise orthogonal to each other but not orthogonal to
9896:. As previously, this implies that the volume of a simplex coming from a
9360:
8633:
5421:
2622:
1472:
821:
122:
92:
55:
15262:-hypercube follows as well, replacing the strict inequalities above by "
13445:
allows one to talk about polynomial equations but not inequalities, the
12475:
Since all simplices are self-dual, they can form a series of compounds;
2525:
16441:
16425:
16375:
16325:
16282:
16252:
16166:
15940:
15820:
15382:
15322:
14434:
13923:
13910:
7807:
is odd, the condition means that exactly one of the diagonal blocks is
2815:
2804:
2793:
2782:
2771:
2758:
2747:
2736:
2725:
2714:
2701:
2240:
733:("simplest") and then with the same Latin adjective in the normal form
698:
59:
15347:
10943:
This can be seen by noting that the center of the standard simplex is
9794:{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {1}{n!}}.}
5785:
Note that there are two sets of vertices described here. One set uses
3499:{\displaystyle (t_{1},\ldots ,t_{n})\mapsto \sum _{i=1}^{n}t_{i}v_{i}}
3307:{\displaystyle (t_{0},\ldots ,t_{n})\mapsto \sum _{i=0}^{n}t_{i}v_{i}}
2478:. A general 2-simplex (scalene triangle) is the join of three points:
62:. The simplex is so-named because it represents the simplest possible
16497:
16411:
16361:
16311:
16268:
16238:
16207:
15921:
15830:
15743:
15694:
15622:
14455:
14176:
14171:
14020:
13158:
12598:-chain. Thus, if we denote one positively oriented affine simplex as
11694:. It defines the following operations on simplices and real numbers:
11541:
11502:
9234:
Another common way of computing the volume of the simplex is via the
6654:
form a basis for the non-trivial irreducible real representations of
3181:
2690:
2679:
2668:
2657:
2644:
2550:
2101:
1972:
1853:
1744:
1645:
1052:
825:
779:
15374:
15314:
14426:
11294:-dimensional volume of the facet opposite of the orthogonal corner.
9708:
does not depend on the permutation). The following assertions hold:
1020:
can have different meanings when describing types of simplices in a
16471:
16226:
16222:
16149:
15845:
15808:
15733:
14279:
14183:
14161:
14101:
14048:
14036:
13903:
13113:
12559:. The simplexes in a chain need not be unique; they may occur with
12526:
fashion. Simplicial complexes are used to define a certain kind of
12480:
11552:
3400:
2611:
2392:
1399:
1028:
682:
118:
85:
63:
51:
35:
14447:
14370:
11987:{\displaystyle x\oplus y=\left\qquad \forall x,y\in \Delta ^{D-1}}
11261:
11142:{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}
11008:{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}
6330:{\displaystyle Q=\operatorname {diag} (Q_{1},Q_{2},\dots ,Q_{k}),}
5775:{\displaystyle \pm {\frac {1}{\sqrt {2(n+1)}}}\cdot (1,\dots ,1).}
16450:
16217:
16212:
16203:
16144:
14450:
14068:
14024:
13953:
10369:, one sees that the previous formula is valid for every simplex.
8993:{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\det \left^{1/2}}
6666:, and the vector being rotated is not stabilized by any of them.
6636:
6371:
11204:{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}
11070:{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}
5348:
It is also possible to directly write down a particular regular
2810:
2799:
2788:
2777:
2766:
2753:
2742:
2731:
2720:
2709:
2696:
16420:
16370:
16320:
16277:
16247:
16198:
16134:
15515:(See chapter 10 for a simple review of topological properties.)
14056:
14040:
13112:
More generally, a simplex (and a chain) can be embedded into a
11505:'s edges, with the hypercube's vertices mapping to each of the
8421:
4606:-cube, meaning that the orbit of the ordered simplex under the
3701:
2685:
2674:
2663:
2652:
2639:
2633:
2628:
2617:
2606:
2595:
2585:
1555:
828:
99:
14364:
Yunmei Chen; Xiaojing Ye (2011). "Projection Onto A Simplex".
11015:, and the centers of its faces are coordinate permutations of
15812:
14295:
2828:
726:
30:
The four simplexes that can be fully represented in 3D space.
15663:
12522:. These spaces are built from simplices glued together in a
9011:-simplex's vertices are in a Euclidean space with more than
272:. Then, the simplex determined by them is the set of points
26:
16170:
14330:
14028:
13760:
13728:
13073:
Likewise, the boundary of the boundary of a chain is zero:
11469:
for triangles with a right angle and for a 3-simplex it is
10899:
3603:, or normalized exponential function; this generalizes the
3407:
vertices, given by the same equation (modifying indexing):
3363:
More generally, there is a canonical map from the standard
2529:
The numbers of faces in the above table are the same as in
13766:{\displaystyle R:=R\left/\left(1-\sum x_{i}\right)\right.}
10771:-simplex side length is 1), and normalizing by the length
3168:, with all coordinates as 0 or 1. It can also be seen one
12886:{\displaystyle \partial \sigma =\sum _{j=0}^{n}(-1)^{j}.}
2553:) shows the centroids of the 15 faces of the tetrahedron.
2446:-simplex can be constructed as a join (∨ operator) of an
15458:"Radial and Pruned Tetrahedral Interpolation Techniques"
10701:{\displaystyle {\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}}
3739:
An alternative coordinate system is given by taking the
2575:
14413:
Stein, P. (1966). "A Note on the Volume of a
Simplex".
14411:
A derivation of a very similar formula can be found in
14284:
Earliest Known Uses of Some of the Words of
Mathematics
14039:
bonds with one hydrogen atom and forms a line segment,
14027:
can resemble a simplex if one is to connect each atom.
13647:{\displaystyle \Delta _{n}(R)=\operatorname {Spec} (R)}
8065:
and each diagonal block acts upon a pair of entries of
4681:
4529:
here correspond to successive coordinates being equal,
721:
in 1900, called them "generalized tetrahedra". In 1902
697:, in which context the word "simplex" simply means any
15269:
15250:{\displaystyle \scriptstyle x_{1}+\cdots +x_{n}<1.}
15210:
15169:
14982:
14792:
14742:
14713:
14684:
14655:
14507:
14481:
14363:
14325:
14242:(2006) . "IV. five dimensional semiregular polytope".
11385:{\displaystyle \sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}}
11155:
11083:
11021:
10949:
10711:
as can be seen by multiplying the previous formula by
9141:
8882:
8752:
8513:
8346:
8296:
8246:
8199:
8093:
7660:
7525:
7390:
7255:
7212:
6892:
6430:
6133:
4873:
917:
693:. The associated combinatorial structure is called an
689:, it is common to "glue together" simplices to form a
413:
390:
15268:
15209:
15168:
14981:
14791:
14741:
14712:
14683:
14654:
14506:
14480:
14307:
13861:
13825:
13786:
13660:
13590:
13463:
13392:
13325:
13302:
13275:
13170:
13122:
13079:
12905:
12751:
12721:
12694:
12607:
12287:
12012:
11710:
11667:
11657:
Aitchinson geometry is a natural way to construct an
11462:, which is the facet opposite the orthogonal corner.
11441:
11401:
11303:
11217:
10818:
10777:
10735:
10660:
10606:
10591:-simplex (i.e. between the origin and the simplex in
10381:
10329:
10302:
10273:
10130:
10101:
10020:
9909:
9856:
9827:
9740:
9687:
9501:
9432:
9388:
9368:
9336:
9271:
9089:
9057:
9021:
8699:
8666:
8461:
8193:
8087:
7854:
7206:
6886:
6700:
6424:
6263:
6062:
6027:
5893:
5834:
5828:
This simplex is inscribed in a hypersphere of radius
5811:
5791:
5714:
5679:
5548:
5448:
5308:
5280:
5058:
5020:
4970:
4939:
4853:
4785:
4740:
4700:
4620:
4535:
4499:
4464:
4435:
4241:
3752:
3573:
3523:
3416:
3369:
3224:
2876:
915:
542:
513:
278:
202:
148:
14196:– an optimization method with inequality constraints
11476:
10088:{\displaystyle (v_{0},\ v_{1},\ v_{2},\ldots v_{n})}
7803:
each of which has distance √5 from the others. When
3336:-simplex. Such a general simplex is often called an
2419:
somewhere off the plane. The new shape, tetrahedron
13960:, and then a local maximum can be computed using a
13803:By using the same definitions as for the classical
13362:{\displaystyle \partial f(\rho )=f(\partial \rho )}
15276:
15249:
15195:
15154:
14967:
14765:
14727:
14698:
14669:
14640:
14488:
13883:
13847:
13792:
13765:
13646:
13572:
13410:
13361:
13308:
13288:
13258:
13149:
13101:
13062:
12885:
12730:
12707:
12677:
12457:
12261:
11986:
11686:
11596:In probability theory, the points of the standard
11454:
11427:
11384:
11250:
11203:
11141:
11069:
11007:
10888:
10804:
10759:
10700:
10636:
10576:
10361:
10315:
10288:
10259:
10116:
10087:
9999:
9888:
9842:
9793:
9700:
9660:
9480:
9418:
9374:
9351:
9322:
9257:. This can be understood as follows: Assume that
9223:
9072:
9036:
8992:
8679:
8612:
8396:
8162:
8054:
7792:
7182:
6831:
6613:
6329:
6166:
6116:
6045:
6010:
5873:
5817:
5797:
5774:
5697:
5662:
5517:
5337:
5294:
5221:
5033:
4994:
4945:
4925:{\textstyle \sum _{i}\max\{p_{i}+\Delta \,,0\}=1.}
4924:
4859:
4836:
4768:
4734:with possibly negative entries, the closest point
4726:
4629:
4570:
4521:
4485:
4458:(maximal dimension, codimension 0) rather than of
4450:
4414:
4213:
3591:
3552:{\displaystyle \Delta ^{n-1}\twoheadrightarrow P.}
3551:
3498:
3387:
3306:
3055:
958:
671:
528:
450:
260:
180:
13511:
11631:th vertex of the simplex is assigned to have the
11258:, from which the dihedral angles are calculated.
10725:from the origin, differentiating with respect to
10296:. Considering the parallelotope constructed from
9047:A more symmetric way to compute the volume of an
5628:
5597:
5137:
2956:
344:
16562:
15631:
15612:
14266:
11627:are precisely those probabilities. That is, the
10475:
10382:
9945:
9133:
8738:
8505:
4884:
4837:{\displaystyle t_{i}=\max\{p_{i}+\Delta \,,0\},}
4799:
4694:onto the standard simplex is of interest. Given
2407:somewhere off the line. The new shape, triangle
799:-dimensional space by infinitely many hypercubes
15360:
14384:
13773:the ring of regular functions on the algebraic
10928:-dimensional simplices, and they have the same
8071:which are not both zero. So, for example, when
6117:{\displaystyle \pm n^{-1/2}\cdot (1,\dots ,1).}
879:(singular: vertex), the 1-faces are called the
15336:Wills, Harold R.; Parks, Dean C. (June 2009).
11077:. Then, by symmetry, the vector pointing from
10721:-simplex as a function of its vertex distance
8407:each of which has distance 2 from the others.
6177:A highly symmetric way to construct a regular
4610:! elements of the symmetric group divides the
261:{\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}}
15982:
15679:
15430:
11473:for a tetrahedron with an orthogonal corner.
9723:-simplexes formed by the convex hull of each
2549:minus one. This figure (a projection of the
2431:somewhere outside the 3-space. The new shape
949:
920:
15590:pp. 120–121, §7.2. see illustration 7-2
15196:{\displaystyle \scriptstyle 0<x_{i}<1}
14244:The Semiregular Polytopes of the Hyperspaces
14031:does not react with hydrogen and as such is
13449:is commonly defined as the subset of affine
12300:
12288:
10920:-dimensional simplex are themselves regular
9413:
9389:
6823:
6805:
6799:
6701:
4913:
4887:
4828:
4802:
4224:This yields the alternative presentation by
3344:, to emphasize that the canonical map is an
2466:-simplex can be constructed as a join of an
499:-dimensional simplex whose vertices are the
23:Multi-dimensional generalization of triangle
14766:{\displaystyle \scriptstyle e_{\sigma (i)}}
11493:-simplex is isomorphic to the graph of the
10889:{\displaystyle (dx/(n+1),\ldots ,dx/(n+1))}
9481:{\displaystyle v_{0},\ v_{1},\ldots ,v_{n}}
9323:{\displaystyle (v_{0},e_{1},\ldots ,e_{n})}
8660:. This formula is particularly useful when
7821:; while the remaining diagonal blocks, say
6181:-simplex is to use a representation of the
3356:to emphasize that the canonical map may be
3187:There is a canonical map from the standard
2588:, and all vertex pairs connected by edges.
1342:{ } = ( ) ∨ ( ) = 2⋅( )
902:-simplex itself. In general, the number of
472:-simplex may be constructed from a regular
15989:
15975:
15686:
15672:
15335:
15293:
12247:
7197:results in the simplex whose vertices are
6870:acts are not both zero. For example, when
3995:
3994:
15519:
14369:
14337:On-Line Encyclopedia of Integer Sequences
13490:
12715:denoting the vertices, then the boundary
12255:
4906:
4821:
4582:corresponds to the inequalities becoming
3734:
50:) is a generalization of the notion of a
14005:, many methods first perform simplicial
13945:, for instance, is defined on a simplex.
9813:-parallelotope is the image of the unit
6370:, all of these matrices must have order
5805:in each calculation. The other set uses
4686:Especially in numerical applications of
2827:
2540:
2524:
2486:is the join of a 1-simplex and a point:
25:
16554:List of regular polytopes and compounds
15566:
15541:
15455:
14313:
14301:
14023:, the hydrides of most elements in the
13150:{\displaystyle f:\mathbf {R} ^{n}\to M}
13116:by means of smooth, differentiable map
9249:it is the formula for the volume of an
8410:
6242:will produce the vertices of a regular
2395:with the fewest vertices that requires
16563:
13855:assemble into one cosimplicial object
10124:, it can be supposed that the vectors
9265:-parallelotope constructed on a basis
6250:, there is a choice of basis in which
4953:can be easily calculated from sorting
4769:{\displaystyle \left(t_{i}\right)_{i}}
2545:The total number of faces is always a
709:The concept of a simplex was known to
15667:
15613:
15545:Non-Uniform Random Variate Generation
15494:
15407:Introduction to Topological Manifolds
14474:-path corresponding to a permutation
14448:
14412:
13436:
12505:
11646:
11262:Simplices with an "orthogonal corner"
8184:, the resulting simplex has vertices
5002:complexity, which can be improved to
3075:obtained by removing the restriction
2576:Symmetric graphs of regular simplices
1090:The number of 1-faces (edges) of the
725:described the concept first with the
66:in any given dimension. For example,
14489:{\displaystyle \scriptstyle \sigma }
14238:
13811:-simplices for different dimensions
13102:{\displaystyle \partial ^{2}\rho =0}
11465:For a 2-simplex, the theorem is the
7815:, and acts upon a non-zero entry of
5874:{\displaystyle {\sqrt {n/(2(n+1))}}}
5252:Cartesian coordinates for a regular
4682:Projection onto the standard simplex
3513:, and express every polytope as the
2399:dimensions. Consider a line segment
959:{\displaystyle {\tbinom {n+1}{m+1}}}
15500:Principles of Mathematical Analysis
15403:
14260:
14043:bonds with two hydrogen atoms in a
13218:
13180:
10362:{\displaystyle e_{1},\ldots ,e_{n}}
9889:{\displaystyle e_{1},\ldots ,e_{n}}
6127:The side length of this simplex is
3511:generalized barycentric coordinates
2823:
13:
15277:{\displaystyle \scriptstyle \leq }
14728:{\displaystyle \scriptstyle e_{i}}
14699:{\displaystyle \scriptstyle v_{0}}
14670:{\displaystyle \scriptstyle v_{0}}
14648:by the affine isometry that sends
14277:
13869:
13833:
13668:
13629:
13592:
13465:
13350:
13326:
13303:
13081:
12922:
12907:
12752:
12722:
12440:
12424:
12229:
12219:
11969:
11953:
11669:
10651:-simplex with unit side length is
9821:that sends the canonical basis of
9719:-hypercube, then the union of the
9106:
9103:
9100:
9097:
9094:
9091:
8716:
8713:
8710:
8707:
8704:
8701:
8478:
8475:
8472:
8469:
8466:
8463:
5060:
4940:
4903:
4854:
4818:
4590:the ordered simplex is a (closed)
4486:{\displaystyle \mathbf {R} ^{n+1}}
4243:
3525:
2878:
2572:) than to the study of polytopes.
924:
181:{\displaystyle u_{0},\dots ,u_{k}}
14:
16592:
15467:. HPL-98-95: 1–32. Archived from
13926:, simplices are sample spaces of
11428:{\displaystyle A_{1}\ldots A_{n}}
9419:{\displaystyle \{1,2,\ldots ,n\}}
8432:-dimensional space with vertices
5044:
3348:. It is also sometimes called an
974:-simplex may be found in column (
15752:
15342:(PhD). Oregon State University.
14706:, and whose linear part matches
14352:Combinatorial Algebraic Topology
13966:sequential quadratic programming
13909:and in the definition of higher
13131:
12731:{\displaystyle \partial \sigma }
12000:Powering (scalar multiplication)
10916:-dimensional faces of a regular
10805:{\displaystyle dx/{\sqrt {n+1}}}
10637:{\displaystyle {1 \over (n+1)!}}
10289:{\displaystyle \mathbf {R} ^{n}}
10276:
10117:{\displaystyle \mathbf {R} ^{n}}
10104:
9843:{\displaystyle \mathbf {R} ^{n}}
9830:
9352:{\displaystyle \mathbf {R} ^{n}}
9339:
9073:{\displaystyle \mathbf {R} ^{n}}
9060:
9037:{\displaystyle \mathbf {R} ^{3}}
9024:
9015:dimensions, e.g., a triangle in
5922:
5563:
5263:One way to write down a regular
5122:
4776:on the simplex has coordinates
4467:
4451:{\displaystyle \mathbf {R} ^{n}}
4438:
4305:
3567:to the interior of the standard
2935:
2809:
2798:
2787:
2776:
2765:
2752:
2741:
2730:
2719:
2708:
2695:
2684:
2673:
2662:
2651:
2638:
2627:
2616:
2605:
2594:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2134:
2129:
2124:
2119:
2114:
2109:
2050:
2045:
2040:
2035:
2030:
2025:
2020:
2015:
2010:
2005:
2000:
1995:
1990:
1985:
1980:
1921:
1916:
1911:
1906:
1901:
1896:
1891:
1886:
1881:
1876:
1871:
1866:
1861:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1693:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1594:
1589:
1584:
1579:
1574:
1569:
1564:
1501:
1496:
1491:
1486:
1481:
1418:
1413:
1408:
1345:
1282:
556:
529:{\displaystyle \mathbf {R} ^{k}}
516:
15527:(4th ed.). Prentice Hall.
15456:Vondran, Gary L. (April 1998).
15449:
15424:
15397:
15354:
15329:
15287:
14464:
14441:
14405:
14084:causal dynamical triangulations
14047:fashion resembling a triangle,
13916:
13425:is frequently referred to as a
12423:
12218:
11952:
11251:{\displaystyle (-n,1,\dots ,1)}
10900:Dihedral angles of the regular
10760:{\displaystyle x=1/{\sqrt {2}}}
7845:, there is an equality of sets
6691:, there is an equality of sets
6167:{\textstyle {\sqrt {2(n+1)/n}}}
1118:, the number of 3-faces of the
1102:, the number of 2-faces of the
15645:. Cambridge University Press.
15146:
15140:
15114:
15102:
15091:
15059:
15045:
15039:
15028:
14996:
14959:
14953:
14931:
14925:
14896:
14890:
14874:
14868:
14836:
14830:
14757:
14751:
14378:
14357:
14344:
14319:
14271:
14232:
14112:Causal dynamical triangulation
13979:problems can be solved by the
13902:-simplices are used in higher
13878:
13865:
13842:
13829:
13724:
13686:
13677:
13664:
13641:
13638:
13625:
13619:
13607:
13601:
13411:{\displaystyle f:\sigma \to X}
13402:
13356:
13347:
13338:
13332:
13253:
13240:
13141:
13046:
12970:
12961:
12951:
12877:
12801:
12792:
12782:
12672:
12614:
11585:
11372:
11356:
11342:
11326:
11245:
11218:
10883:
10880:
10868:
10845:
10833:
10819:
10717:, to get the volume under the
10625:
10613:
10568:
10478:
10469:
10385:
10082:
10021:
9980:
9948:
9925:
9919:
9756:
9750:
9653:
9647:
9593:
9587:
9542:
9536:
9317:
9272:
8377:
8324:
8274:
8224:
8021:
8009:
7995:
7983:
7945:
7912:
7900:
7776:
7759:
7746:
7729:
7716:
7699:
7686:
7669:
7641:
7624:
7611:
7594:
7581:
7564:
7551:
7534:
7506:
7489:
7476:
7459:
7446:
7429:
7416:
7399:
7371:
7354:
7341:
7324:
7311:
7294:
7281:
7264:
7166:
7149:
7138:
7121:
7098:
7081:
7067:
7050:
7017:
7000:
6989:
6972:
6949:
6932:
6918:
6901:
6321:
6276:
6151:
6139:
6108:
6090:
6002:
5984:
5978:
5956:
5866:
5863:
5851:
5845:
5766:
5748:
5739:
5727:
5654:
5636:
5509:
5491:
5332:
5315:
5114:
5082:
4989:
4974:
4715:
4701:
4571:{\displaystyle s_{i}=s_{i+1},}
4297:
4265:
3629:Δ is the line segment joining
3586:
3574:
3561:A commonly used function from
3540:
3452:
3449:
3417:
3382:
3370:
3260:
3257:
3225:
2927:
2895:
966:. Consequently, the number of
1:
15693:
15502:(3rd ed.). McGraw-Hill.
15488:
15303:American Mathematical Monthly
14415:American Mathematical Monthly
11687:{\displaystyle \Delta ^{D-1}}
6676:this means that every matrix
6046:{\displaystyle 1\leq i\leq n}
5698:{\displaystyle 1\leq i\leq n}
5338:{\displaystyle \arccos(-1/n)}
4964:. The sorting approach takes
1130:th 5-cell number, and so on.
744:family is the first of three
135:. More formally, suppose the
98:a 4-dimensional simplex is a
91:a 3-dimensional simplex is a
84:a 2-dimensional simplex is a
77:a 1-dimensional simplex is a
70:a 0-dimensional simplex is a
15410:. Springer. pp. 292–3.
14399:10.1016/0167-6377(89)90064-3
14267:Boyd & Vandenberghe 2004
13958:response surface methodology
13447:algebraic standard n-simplex
12470:
11274:-dimensional version of the
7193:Applying this to the vector
6232:is. Applying powers of this
5236:-simplex as a corner of the
2763:
2706:
2649:
2592:
2533:, without the left diagonal.
1035:-simplex can be computed by
464:is a simplex that is also a
7:
14451:"Cayley-Menger Determinant"
14387:Operations Research Letters
14094:
12570:Note that each facet of an
11547:
11528:-hypercube. It is also the
10896:, along the normal vector.
6395:matrix whose only entry is
6254:is a block diagonal matrix
4727:{\displaystyle (p_{i})_{i}}
3610:
2450:-simplex and a point,
1047:, like the coefficients of
815:
695:abstract simplicial complex
10:
16599:
16581:Multi-dimensional geometry
16543:
15970:
14327:Sloane, N. J. A.
13380:(by definition of a map).
12541:-simplexes embedded in an
12488:compound of two tetrahedra
11661:from the standard simplex
11650:
11589:
11489:of the face lattice of an
9426:, call a list of vertices
5388:. Begin with the standard
4995:{\displaystyle O(n\log n)}
4425:Geometrically, this is an
3605:standard logistic function
763:, the other two being the
704:
15:
15954:
15933:
15869:
15807:
15761:
15750:
15701:
15435:(third ed.). Wiley.
14304:, pp. 120–124, §7.2.
14147:List of regular polytopes
13309:{\displaystyle \partial }
11278:: The sum of the squared
9236:Cayley–Menger determinant
8623:where each column of the
8415:
6877:, one possible matrix is
6351:is orthogonal and either
6236:to an appropriate vector
5034:{\displaystyle \ell _{1}}
4232:-tuples between 0 and 1:
3191:-simplex to an arbitrary
3097:vertices of the standard
3087:in the above definition.
15363:The Mathematical Gazette
14246:. Simon & Schuster.
14226:
12678:{\displaystyle \sigma =}
11592:Categorical distribution
10647:The volume of a regular
9007:and works even when the
8640:that points from vertex
6669:In practical terms, for
6228:, but no lower power of
5256:-dimensional simplex in
4586:(increasing sequences).
4522:{\displaystyle t_{i}=0,}
4228:namely as nondecreasing
3195:-simplex with vertices (
3101:-simplex are the points
711:William Kingdon Clifford
18:Simplex (disambiguation)
15576:(3rd ed.). Dover.
14009:of the domain and then
13584:-theoretic description
12582:-simplex, and thus the
12499:compound of two 5-cells
11698:Perturbation (addition)
11625:barycentric coordinates
9375:{\displaystyle \sigma }
4946:{\displaystyle \Delta }
4860:{\displaystyle \Delta }
4429:-dimensional subset of
3330:barycentric coordinates
906:-faces is equal to the
890:)-faces are called the
860:defining points) is an
793:. A fourth family, the
192:, which means that the
15431:Cornell, John (2002).
15278:
15251:
15197:
15156:
14969:
14767:
14729:
14700:
14671:
14642:
14490:
14122:Delaunay triangulation
14074:In some approaches to
13943:Dirichlet distribution
13885:
13849:
13794:
13767:
13648:
13574:
13545:
13412:
13363:
13310:
13290:
13260:
13151:
13103:
13064:
12950:
12887:
12781:
12732:
12709:
12679:
12590:-simplex is an affine
12574:-simplex is an affine
12486:Two tetrahedra form a
12459:
12362:
12341:
12263:
12194:
12128:
12068:
11988:
11923:
11845:
11773:
11688:
11635:th probability of the
11456:
11429:
11386:
11324:
11252:
11205:
11143:
11071:
11009:
10890:
10806:
10761:
10702:
10638:
10578:
10363:
10317:
10290:
10261:
10118:
10089:
10001:
9890:
9844:
9795:
9702:
9662:
9482:
9420:
9376:
9353:
9324:
9225:
9074:
9038:
8994:
8681:
8614:
8398:
8164:
8056:
7794:
7184:
6833:
6615:
6331:
6168:
6118:
6047:
6012:
5875:
5819:
5799:
5776:
5699:
5664:
5519:
5339:
5296:
5295:{\displaystyle \pi /3}
5223:
5165:
5035:
4996:
4947:
4926:
4861:
4838:
4770:
4728:
4631:
4572:
4523:
4487:
4452:
4416:
4215:
3735:Increasing coordinates
3593:
3553:
3500:
3475:
3389:
3358:orientation preserving
3308:
3283:
3057:
2984:
2842:
2554:
2534:
960:
838:points that define an
723:Pieter Hendrik Schoute
673:
530:
452:
372:
262:
182:
31:
15542:Devroye, Luc (1986).
15404:Lee, John M. (2006).
15279:
15252:
15198:
15157:
14975:is the set of points
14970:
14768:
14730:
14701:
14672:
14643:
14491:
13962:nonlinear programming
13950:industrial statistics
13886:
13850:
13795:
13768:
13649:
13575:
13519:
13413:
13364:
13311:
13291:
13289:{\displaystyle a_{i}}
13261:
13152:
13104:
13065:
12930:
12888:
12761:
12733:
12710:
12708:{\displaystyle v_{j}}
12680:
12479:Two triangles form a
12460:
12342:
12321:
12264:
12174:
12108:
12048:
11989:
11903:
11825:
11753:
11689:
11580:manifold with corners
11516:-simplex is also the
11457:
11455:{\displaystyle A_{0}}
11430:
11387:
11304:
11253:
11206:
11144:
11072:
11010:
10891:
10807:
10762:
10703:
10639:
10579:
10364:
10318:
10316:{\displaystyle v_{0}}
10291:
10262:
10119:
10090:
10010:Conversely, given an
10002:
9891:
9845:
9796:
9703:
9701:{\displaystyle v_{n}}
9663:
9483:
9421:
9377:
9354:
9325:
9226:
9075:
9039:
8995:
8682:
8680:{\displaystyle v_{0}}
8615:
8399:
8165:
8057:
7795:
7185:
6834:
6616:
6332:
6169:
6119:
6048:
6013:
5876:
5825:in each calculation.
5820:
5800:
5777:
5700:
5665:
5520:
5340:
5297:
5224:
5145:
5036:
4997:
4948:
4927:
4862:
4839:
4771:
4729:
4632:
4573:
4524:
4488:
4453:
4417:
4216:
3594:
3592:{\displaystyle (n-1)}
3554:
3501:
3455:
3390:
3388:{\displaystyle (n-1)}
3346:affine transformation
3309:
3263:
3058:
2964:
2831:
2544:
2528:
2496:( ) ∨ ( ) ∨ ( ) ∨ ( )
961:
842:-simplex is called a
748:families, labeled by
674:
531:
453:
352:
263:
183:
29:
15870:Dimensions by number
15637:Vandenberghe, Lieven
15521:Tanenbaum, Andrew S.
15266:
15207:
15166:
14979:
14789:
14739:
14710:
14681:
14652:
14504:
14496:is the image of the
14478:
14189:Schläfli orthoscheme
14088:simplicial manifolds
13859:
13823:
13784:
13658:
13588:
13461:
13390:
13323:
13300:
13273:
13168:
13120:
13077:
12903:
12749:
12719:
12692:
12605:
12520:simplicial complexes
12285:
12010:
11708:
11665:
11439:
11399:
11301:
11215:
11153:
11081:
11019:
10947:
10816:
10775:
10733:
10658:
10604:
10379:
10327:
10300:
10271:
10128:
10099:
10018:
9907:
9854:
9825:
9738:
9685:
9499:
9430:
9386:
9366:
9334:
9269:
9087:
9055:
9019:
8697:
8664:
8459:
8411:Geometric properties
8191:
8085:
8078:, the matrix can be
7852:
7204:
6884:
6698:
6422:
6261:
6131:
6060:
6025:
5891:
5832:
5809:
5789:
5712:
5677:
5546:
5446:
5306:
5278:
5056:
5018:
4968:
4937:
4871:
4867:is chosen such that
4851:
4783:
4738:
4698:
4618:
4533:
4497:
4462:
4433:
4239:
3750:
3648:equilateral triangle
3571:
3521:
3414:
3367:
3222:
3138:= (0, 1, 0, ..., 0),
3127:= (1, 0, 0, ..., 0),
2874:
2518:or {3,3} and so on.
2492:equilateral triangle
1083:,8,28,56,70,56,28,8,
913:
908:binomial coefficient
864:-simplex, called an
540:
536:, or in other words
511:
276:
270:linearly independent
200:
190:affinely independent
146:
16:For other uses, see
16538:pentagonal polytope
16437:Uniform 10-polytope
15997:Fundamental convex
15642:Convex Optimization
15465:HP Technical Report
14205:Simplicial homology
14132:Geometric primitive
13973:operations research
12532:simplicial homology
12501:in four dimensions.
12497:Two 5-cells form a
12209:
12172:
12143:
12106:
12083:
12046:
11659:inner product space
11467:Pythagorean theorem
11276:Pythagorean theorem
8868:
8850:
8824:
8806:
8787:
8769:
6415:matrix of the form
6198:orthogonal matrices
5205: for all
5073:
4256:
3673:regular tetrahedron
3509:These are known as
3399:vertices) onto any
3164:is an example of a
3154:= (0, 0, 0, ..., 1)
2861:) is the subset of
2570:simplicial homology
2512:regular tetrahedron
2476:( ) ∨ ( ) = 2 ⋅ ( )
1138:
1049:polynomial products
898:-face is the whole
767:family, labeled as
489:probability simplex
16407:Uniform 9-polytope
16357:Uniform 8-polytope
16307:Uniform 7-polytope
16264:Uniform 6-polytope
16234:Uniform 5-polytope
16194:Uniform polychoron
16157:Uniform polyhedron
16005:in dimensions 2–10
15799:Degrees of freedom
15702:Dimensional spaces
15615:Weisstein, Eric W.
15523:(2003). "§2.5.3".
15274:
15273:
15247:
15246:
15193:
15192:
15152:
15151:
14965:
14964:
14777:. hence every two
14763:
14762:
14725:
14724:
14696:
14695:
14667:
14666:
14638:
14637:
14486:
14485:
14340:. OEIS Foundation.
14200:Simplicial complex
14107:Aitchison geometry
13977:linear programming
13939:probability theory
13928:compositional data
13881:
13845:
13819:, while the rings
13815:assemble into one
13790:
13777:-simplex (for any
13763:
13644:
13570:
13443:algebraic geometry
13437:Algebraic geometry
13408:
13359:
13306:
13286:
13256:
13147:
13099:
13060:
12883:
12728:
12705:
12675:
12516:topological spaces
12512:algebraic topology
12506:Algebraic topology
12455:
12259:
12195:
12158:
12129:
12092:
12069:
12032:
11984:
11684:
11653:Aitchison geometry
11647:Aitchison geometry
11452:
11425:
11382:
11248:
11201:
11139:
11067:
11005:
10886:
10812:of the increment,
10802:
10767: (where the
10757:
10698:
10634:
10574:
10359:
10313:
10286:
10257:
10114:
10085:
9997:
9886:
9840:
9819:linear isomorphism
9817:-hypercube by the
9791:
9698:
9658:
9478:
9416:
9372:
9349:
9320:
9221:
9207:
9070:
9034:
8990:
8965:
8871:
8854:
8836:
8810:
8792:
8773:
8755:
8677:
8649:to another vertex
8610:
8599:
8394:
8385:
8332:
8282:
8232:
8160:
8151:
8052:
7790:
7781:
7646:
7511:
7376:
7241:
7180:
7171:
6829:
6611:
6602:
6327:
6210:orthogonal matrix
6164:
6114:
6043:
6008:
5871:
5815:
5795:
5772:
5695:
5660:
5535:The above regular
5515:
5433:quadratic equation
5335:
5292:
5219:
5059:
5031:
4992:
4943:
4922:
4883:
4857:
4834:
4766:
4724:
4688:probability theory
4630:{\displaystyle n!}
4627:
4592:fundamental domain
4568:
4519:
4483:
4448:
4412:
4242:
4211:
4209:
3589:
3549:
3496:
3385:
3332:of a point in the
3304:
3053:
2843:
2555:
2535:
2484:isosceles triangle
2438:More formally, an
1478:{3,3} = 4⋅( )
1137:-Simplex elements
1133:
1116:tetrahedron number
1022:simplicial complex
956:
954:
719:algebraic topology
691:simplicial complex
669:
526:
448:
417:
394:
258:
178:
32:
16559:
16558:
16546:Polytope families
16003:uniform polytopes
15965:
15964:
15774:Lebesgue covering
15739:Algebraic variety
15652:978-1-107-39400-1
15593:
15573:Regular Polytopes
15525:Computer Networks
15417:978-0-387-22727-6
14846:
14808:
14552:
14523:
14449:Colins, Karen D.
14350:Kozlov, Dimitry,
14253:978-1-4181-7968-7
14194:Simplex algorithm
14127:Distance geometry
14051:reacts to form a
14011:fit interpolating
14003:computer graphics
13981:simplex algorithm
13884:{\displaystyle R}
13848:{\displaystyle R}
13817:simplicial object
13793:{\displaystyle R}
13580:which equals the
13518:
13508:
13420:topological space
12421:
12391:
12319:
12211:
12145:
12085:
11945:
11867:
11795:
11477:Relation to the (
11194:
11175:
11132:
11105:
11060:
11041:
10998:
10971:
10800:
10755:
10696:
10693:
10673:
10632:
10172:
10055:
10039:
9992:
9937:
9786:
9768:
9552:
9448:
9126:
8866:
8848:
8822:
8804:
8785:
8767:
8736:
8498:
6853:, the entries of
6639:between zero and
6598:
6557:
6514:
6470:
6380:. Therefore each
6162:
5970:
5915:
5869:
5818:{\displaystyle -}
5798:{\displaystyle +}
5743:
5742:
5624:
5623:
5593:
5590:
5559:
5558:
5481:
5457:
5206:
5185:
5144:
5134:
4874:
3700:Δ is the regular
3317:The coefficients
3073:affine hyperplane
3025:
3004:
2963:
2953:
2821:
2820:
2559:
2558:
2531:Pascal's triangle
2385:
2384:
2245:{3} = 11⋅( )
2106:{3} = 10⋅( )
1255:
1246:
1237:
1228:
1219:
1210:
1201:
1192:
1183:
1174:
1165:
1051:. For example, a
990:Pascal's triangle
947:
632:
416:
393:
351:
341:
16588:
16550:Regular polytope
16111:
16100:
16089:
16048:
15991:
15984:
15977:
15968:
15967:
15762:Other dimensions
15756:
15724:Projective space
15688:
15681:
15674:
15665:
15664:
15656:
15628:
15627:
15606:
15599:
15591:
15587:
15563:
15558:. Archived from
15538:
15513:
15483:
15482:
15480:
15479:
15473:
15462:
15453:
15447:
15446:
15428:
15422:
15421:
15401:
15395:
15394:
15358:
15352:
15351:
15333:
15327:
15326:
15300:
15295:Parks, Harold R.
15291:
15285:
15283:
15281:
15280:
15275:
15261:
15256:
15254:
15253:
15248:
15239:
15238:
15220:
15219:
15202:
15200:
15199:
15194:
15185:
15184:
15161:
15159:
15158:
15153:
15150:
15149:
15131:
15130:
15118:
15117:
15090:
15089:
15077:
15076:
15049:
15048:
15027:
15026:
15008:
15007:
14992:
14991:
14974:
14972:
14971:
14966:
14963:
14962:
14935:
14934:
14913:
14912:
14900:
14899:
14878:
14877:
14856:
14855:
14844:
14840:
14839:
14818:
14817:
14806:
14802:
14801:
14784:
14780:
14776:
14772:
14770:
14769:
14764:
14761:
14760:
14734:
14732:
14731:
14726:
14723:
14722:
14705:
14703:
14702:
14697:
14694:
14693:
14676:
14674:
14673:
14668:
14665:
14664:
14647:
14645:
14644:
14639:
14636:
14635:
14617:
14616:
14604:
14603:
14588:
14587:
14575:
14574:
14562:
14561:
14550:
14546:
14545:
14533:
14532:
14521:
14517:
14516:
14499:
14495:
14493:
14492:
14487:
14473:
14468:
14462:
14461:
14460:
14445:
14439:
14438:
14409:
14403:
14402:
14382:
14376:
14375:
14373:
14361:
14355:
14348:
14342:
14341:
14323:
14317:
14311:
14305:
14299:
14293:
14292:
14291:
14290:
14275:
14269:
14264:
14258:
14257:
14236:
14159:
14137:Hill tetrahedron
14065:Schlegel diagram
14016:to each simplex.
13999:geometric design
13964:method, such as
13901:
13890:
13888:
13887:
13882:
13877:
13876:
13854:
13852:
13851:
13846:
13841:
13840:
13814:
13810:
13806:
13799:
13797:
13796:
13791:
13776:
13772:
13770:
13769:
13764:
13762:
13759:
13755:
13754:
13753:
13723:
13722:
13698:
13697:
13676:
13675:
13653:
13651:
13650:
13645:
13637:
13636:
13600:
13599:
13579:
13577:
13576:
13571:
13566:
13562:
13555:
13554:
13544:
13533:
13516:
13515:
13514:
13506:
13505:
13504:
13493:
13473:
13472:
13456:
13441:Since classical
13430:
13424:
13417:
13415:
13414:
13409:
13375:
13368:
13366:
13365:
13360:
13315:
13313:
13312:
13307:
13295:
13293:
13292:
13287:
13285:
13284:
13265:
13263:
13262:
13257:
13252:
13251:
13236:
13235:
13226:
13225:
13213:
13209:
13208:
13207:
13198:
13197:
13188:
13187:
13156:
13154:
13153:
13148:
13140:
13139:
13134:
13108:
13106:
13105:
13100:
13089:
13088:
13069:
13067:
13066:
13061:
13053:
13049:
13045:
13044:
13026:
13025:
13007:
13006:
12982:
12981:
12969:
12968:
12949:
12944:
12915:
12914:
12892:
12890:
12889:
12884:
12876:
12875:
12857:
12856:
12838:
12837:
12813:
12812:
12800:
12799:
12780:
12775:
12741:
12737:
12735:
12734:
12729:
12714:
12712:
12711:
12706:
12704:
12703:
12684:
12682:
12681:
12676:
12671:
12670:
12652:
12651:
12639:
12638:
12626:
12625:
12597:
12589:
12581:
12573:
12556:
12550:
12540:
12537:A finite set of
12492:stella octangula
12464:
12462:
12461:
12456:
12454:
12453:
12422:
12420:
12419:
12410:
12409:
12400:
12392:
12390:
12389:
12380:
12379:
12370:
12361:
12356:
12340:
12335:
12320:
12318:
12307:
12268:
12266:
12265:
12260:
12258:
12243:
12242:
12217:
12213:
12212:
12210:
12208:
12203:
12193:
12188:
12171:
12166:
12157:
12146:
12144:
12142:
12137:
12127:
12122:
12105:
12100:
12091:
12086:
12084:
12082:
12077:
12067:
12062:
12045:
12040:
12031:
11993:
11991:
11990:
11985:
11983:
11982:
11951:
11947:
11946:
11944:
11943:
11942:
11933:
11932:
11922:
11917:
11901:
11900:
11899:
11890:
11889:
11879:
11868:
11866:
11865:
11864:
11855:
11854:
11844:
11839:
11823:
11822:
11821:
11812:
11811:
11801:
11796:
11794:
11793:
11792:
11783:
11782:
11772:
11767:
11751:
11750:
11749:
11740:
11739:
11729:
11693:
11691:
11690:
11685:
11683:
11682:
11642:
11634:
11630:
11622:
11614:
11607:
11599:
11577:
11573:
11567:
11558:
11539:
11527:
11515:
11508:
11500:
11492:
11471:de Gua's theorem
11461:
11459:
11458:
11453:
11451:
11450:
11434:
11432:
11431:
11426:
11424:
11423:
11411:
11410:
11391:
11389:
11388:
11383:
11381:
11380:
11375:
11369:
11368:
11359:
11351:
11350:
11345:
11339:
11338:
11329:
11323:
11318:
11293:
11285:
11273:
11257:
11255:
11254:
11249:
11210:
11208:
11207:
11202:
11200:
11196:
11195:
11187:
11176:
11168:
11148:
11146:
11145:
11140:
11138:
11134:
11133:
11131:
11117:
11106:
11104:
11090:
11076:
11074:
11073:
11068:
11066:
11062:
11061:
11053:
11042:
11034:
11014:
11012:
11011:
11006:
11004:
11000:
10999:
10997:
10983:
10972:
10970:
10956:
10939:
10927:
10919:
10915:
10895:
10893:
10892:
10887:
10867:
10832:
10811:
10809:
10808:
10803:
10801:
10790:
10788:
10770:
10766:
10764:
10763:
10758:
10756:
10751:
10749:
10728:
10724:
10720:
10716:
10707:
10705:
10704:
10699:
10697:
10695:
10694:
10692:
10691:
10682:
10663:
10662:
10650:
10643:
10641:
10640:
10635:
10633:
10631:
10608:
10596:
10590:
10583:
10581:
10580:
10575:
10567:
10566:
10548:
10547:
10529:
10528:
10516:
10515:
10503:
10502:
10490:
10489:
10468:
10467:
10455:
10454:
10436:
10435:
10423:
10422:
10410:
10409:
10397:
10396:
10368:
10366:
10365:
10360:
10358:
10357:
10339:
10338:
10322:
10320:
10319:
10314:
10312:
10311:
10295:
10293:
10292:
10287:
10285:
10284:
10279:
10267:form a basis of
10266:
10264:
10263:
10258:
10256:
10255:
10237:
10236:
10224:
10223:
10208:
10207:
10195:
10194:
10182:
10181:
10170:
10166:
10165:
10153:
10152:
10140:
10139:
10123:
10121:
10120:
10115:
10113:
10112:
10107:
10094:
10092:
10091:
10086:
10081:
10080:
10065:
10064:
10053:
10049:
10048:
10037:
10033:
10032:
10013:
10006:
10004:
10003:
9998:
9993:
9991:
9983:
9979:
9978:
9960:
9959:
9943:
9938:
9936:
9928:
9911:
9899:
9895:
9893:
9892:
9887:
9885:
9884:
9866:
9865:
9849:
9847:
9846:
9841:
9839:
9838:
9833:
9816:
9812:
9807:
9800:
9798:
9797:
9792:
9787:
9785:
9774:
9769:
9767:
9759:
9742:
9730:
9726:
9722:
9718:
9714:
9707:
9705:
9704:
9699:
9697:
9696:
9680:
9677:
9667:
9665:
9664:
9659:
9657:
9656:
9635:
9634:
9616:
9615:
9597:
9596:
9575:
9574:
9562:
9561:
9550:
9546:
9545:
9524:
9523:
9511:
9510:
9491:
9487:
9485:
9484:
9479:
9477:
9476:
9458:
9457:
9446:
9442:
9441:
9425:
9423:
9422:
9417:
9381:
9379:
9378:
9373:
9358:
9356:
9355:
9350:
9348:
9347:
9342:
9329:
9327:
9326:
9321:
9316:
9315:
9297:
9296:
9284:
9283:
9264:
9260:
9252:
9248:
9230:
9228:
9227:
9222:
9217:
9213:
9212:
9211:
9182:
9181:
9165:
9164:
9153:
9152:
9127:
9125:
9114:
9109:
9079:
9077:
9076:
9071:
9069:
9068:
9063:
9050:
9043:
9041:
9040:
9035:
9033:
9032:
9027:
9014:
9010:
9005:Gram determinant
8999:
8997:
8996:
8991:
8989:
8988:
8984:
8975:
8971:
8970:
8969:
8962:
8961:
8949:
8948:
8932:
8931:
8919:
8918:
8907:
8906:
8894:
8893:
8876:
8875:
8867:
8864:
8862:
8849:
8846:
8844:
8823:
8820:
8818:
8805:
8802:
8800:
8786:
8783:
8781:
8768:
8765:
8763:
8737:
8735:
8724:
8719:
8686:
8684:
8683:
8678:
8676:
8675:
8659:
8648:
8632:
8619:
8617:
8616:
8611:
8609:
8605:
8604:
8603:
8596:
8595:
8583:
8582:
8572:
8566:
8564:
8563:
8551:
8550:
8540:
8538:
8537:
8525:
8524:
8499:
8497:
8486:
8481:
8451:
8431:
8427:
8403:
8401:
8400:
8395:
8390:
8389:
8376:
8337:
8336:
8323:
8287:
8286:
8273:
8237:
8236:
8223:
8183:
8181:
8180:
8169:
8167:
8166:
8161:
8156:
8155:
8077:
8070:
8061:
8059:
8058:
8053:
8048:
8044:
8028:
8002:
7962:
7958:
7957:
7956:
7952:
7924:
7923:
7919:
7885:
7884:
7869:
7868:
7844:
7840:
7820:
7814:
7810:
7806:
7799:
7797:
7796:
7791:
7786:
7785:
7772:
7742:
7712:
7682:
7651:
7650:
7637:
7607:
7577:
7547:
7516:
7515:
7502:
7472:
7442:
7412:
7381:
7380:
7367:
7337:
7307:
7277:
7246:
7245:
7196:
7189:
7187:
7186:
7181:
7176:
7175:
7162:
7134:
7094:
7063:
7013:
6985:
6945:
6914:
6876:
6869:
6858:
6852:
6838:
6836:
6835:
6830:
6798:
6797:
6793:
6765:
6764:
6760:
6738:
6737:
6713:
6712:
6690:
6686:
6672:
6665:
6653:
6642:
6634:
6620:
6618:
6617:
6612:
6607:
6606:
6599:
6597:
6586:
6585:
6584:
6568:
6558:
6556:
6545:
6544:
6543:
6527:
6515:
6513:
6502:
6501:
6500:
6484:
6471:
6469:
6458:
6457:
6456:
6440:
6414:
6410:
6402:
6398:
6394:
6390:
6379:
6369:
6362:
6358:
6354:
6350:
6336:
6334:
6333:
6328:
6320:
6319:
6301:
6300:
6288:
6287:
6253:
6249:
6245:
6241:
6231:
6223:
6213:
6209:
6195:
6180:
6173:
6171:
6170:
6165:
6163:
6158:
6135:
6123:
6121:
6120:
6115:
6086:
6085:
6081:
6052:
6050:
6049:
6044:
6017:
6015:
6014:
6009:
5971:
5960:
5955:
5954:
5950:
5931:
5930:
5925:
5916:
5914:
5913:
5895:
5880:
5878:
5877:
5872:
5870:
5844:
5836:
5824:
5822:
5821:
5816:
5804:
5802:
5801:
5796:
5781:
5779:
5778:
5773:
5744:
5723:
5719:
5704:
5702:
5701:
5696:
5669:
5667:
5666:
5661:
5632:
5631:
5625:
5613:
5609:
5601:
5600:
5594:
5592:
5591:
5586:
5577:
5572:
5571:
5566:
5560:
5554:
5550:
5538:
5531:
5524:
5522:
5521:
5516:
5487:
5483:
5482:
5471:
5458:
5450:
5438:
5430:
5426:
5419:
5399:
5395:
5387:
5376:
5367:
5357:
5351:
5344:
5342:
5341:
5336:
5328:
5301:
5299:
5298:
5293:
5288:
5272:
5266:
5255:
5247:
5239:
5235:
5228:
5226:
5225:
5220:
5215:
5211:
5207:
5204:
5196:
5195:
5186:
5183:
5175:
5174:
5164:
5159:
5142:
5141:
5140:
5132:
5131:
5130:
5125:
5113:
5112:
5094:
5093:
5072:
5067:
5040:
5038:
5037:
5032:
5030:
5029:
5009:
5001:
4999:
4998:
4993:
4963:
4952:
4950:
4949:
4944:
4931:
4929:
4928:
4923:
4899:
4898:
4882:
4866:
4864:
4863:
4858:
4843:
4841:
4840:
4835:
4814:
4813:
4795:
4794:
4775:
4773:
4772:
4767:
4765:
4764:
4759:
4755:
4754:
4733:
4731:
4730:
4725:
4723:
4722:
4713:
4712:
4673:
4662:
4655:
4648:
4644:
4636:
4634:
4633:
4628:
4613:
4609:
4605:
4577:
4575:
4574:
4569:
4564:
4563:
4545:
4544:
4528:
4526:
4525:
4520:
4509:
4508:
4492:
4490:
4489:
4484:
4482:
4481:
4470:
4457:
4455:
4454:
4449:
4447:
4446:
4441:
4428:
4421:
4419:
4418:
4413:
4408:
4404:
4397:
4396:
4378:
4377:
4359:
4358:
4346:
4345:
4333:
4332:
4314:
4313:
4308:
4296:
4295:
4277:
4276:
4255:
4250:
4231:
4220:
4218:
4217:
4212:
4210:
4200:
4199:
4181:
4180:
4168:
4167:
4155:
4154:
4142:
4141:
4125:
4124:
4105:
4104:
4080:
4079:
4067:
4066:
4054:
4053:
4035:
4034:
4012:
4011:
3990:
3986:
3985:
3973:
3972:
3960:
3959:
3947:
3946:
3934:
3933:
3917:
3916:
3903:
3902:
3890:
3889:
3877:
3876:
3864:
3863:
3847:
3846:
3833:
3832:
3820:
3819:
3807:
3806:
3790:
3789:
3766:
3765:
3729:
3723:
3719:
3715:
3711:
3707:
3696:
3690:
3686:
3682:
3678:
3667:
3661:
3657:
3653:
3642:
3636:
3632:
3625:
3619:
3601:softmax function
3599:-simplex is the
3598:
3596:
3595:
3590:
3566:
3558:
3556:
3555:
3550:
3539:
3538:
3505:
3503:
3502:
3497:
3495:
3494:
3485:
3484:
3474:
3469:
3448:
3447:
3429:
3428:
3406:
3398:
3394:
3392:
3391:
3386:
3353:
3350:oriented affine
3341:
3335:
3327:
3313:
3311:
3310:
3305:
3303:
3302:
3293:
3292:
3282:
3277:
3256:
3255:
3237:
3236:
3214:
3203:
3194:
3190:
3179:
3162:standard simplex
3155:
3139:
3128:
3115:
3100:
3096:
3086:
3070:
3062:
3060:
3059:
3054:
3052:
3048:
3026:
3023:
3015:
3014:
3005:
3002:
2994:
2993:
2983:
2978:
2961:
2960:
2959:
2951:
2950:
2949:
2938:
2926:
2925:
2907:
2906:
2886:
2885:
2866:
2858:
2850:
2841:
2835:
2824:Standard simplex
2813:
2802:
2791:
2780:
2769:
2756:
2745:
2734:
2723:
2712:
2699:
2688:
2677:
2666:
2655:
2642:
2631:
2620:
2609:
2598:
2591:
2590:
2567:
2521:
2520:
2517:
2509:
2505:
2501:
2497:
2489:
2481:
2477:
2473:
2470:-simplex and an
2469:
2465:
2453:
2449:
2445:
2430:
2418:
2406:
2398:
2391:-simplex is the
2390:
2343:
2342:
2341:
2337:
2336:
2332:
2331:
2327:
2326:
2322:
2321:
2317:
2316:
2312:
2311:
2307:
2306:
2302:
2301:
2297:
2296:
2292:
2291:
2287:
2286:
2282:
2281:
2277:
2276:
2272:
2271:
2267:
2266:
2262:
2261:
2257:
2256:
2252:
2251:
2194:
2193:
2192:
2188:
2187:
2183:
2182:
2178:
2177:
2173:
2172:
2168:
2167:
2163:
2162:
2158:
2157:
2153:
2152:
2148:
2147:
2143:
2142:
2138:
2137:
2133:
2132:
2128:
2127:
2123:
2122:
2118:
2117:
2113:
2112:
2055:
2054:
2053:
2049:
2048:
2044:
2043:
2039:
2038:
2034:
2033:
2029:
2028:
2024:
2023:
2019:
2018:
2014:
2013:
2009:
2008:
2004:
2003:
1999:
1998:
1994:
1993:
1989:
1988:
1984:
1983:
1977:{3} = 9⋅( )
1926:
1925:
1924:
1920:
1919:
1915:
1914:
1910:
1909:
1905:
1904:
1900:
1899:
1895:
1894:
1890:
1889:
1885:
1884:
1880:
1879:
1875:
1874:
1870:
1869:
1865:
1864:
1858:{3} = 8⋅( )
1807:
1806:
1805:
1801:
1800:
1796:
1795:
1791:
1790:
1786:
1785:
1781:
1780:
1776:
1775:
1771:
1770:
1766:
1765:
1761:
1760:
1756:
1755:
1749:{3} = 7⋅( )
1698:
1697:
1696:
1692:
1691:
1687:
1686:
1682:
1681:
1677:
1676:
1672:
1671:
1667:
1666:
1662:
1661:
1657:
1656:
1650:{3} = 6⋅( )
1599:
1598:
1597:
1593:
1592:
1588:
1587:
1583:
1582:
1578:
1577:
1573:
1572:
1568:
1567:
1561:{3} = 5⋅( )
1506:
1505:
1504:
1500:
1499:
1495:
1494:
1490:
1489:
1485:
1484:
1423:
1422:
1421:
1417:
1416:
1412:
1411:
1405:{3} = 3⋅( )
1350:
1349:
1348:
1287:
1286:
1285:
1253:
1244:
1235:
1226:
1217:
1208:
1199:
1190:
1181:
1172:
1163:
1144:
1139:
1136:
1132:
1129:
1122:-simplex is the
1121:
1113:
1106:-simplex is the
1105:
1097:
1094:-simplex is the
1093:
1046:
1034:
1011:
1007:
1003:
995:
987:
980:
973:
969:
965:
963:
962:
957:
955:
953:
952:
946:
935:
923:
905:
901:
897:
889:
874:
868:
863:
859:
852:
841:
837:
811:
801:, he labeled as
798:
795:tessellation of
792:
777:
762:
746:regular polytope
717:, writing about
678:
676:
675:
670:
665:
661:
633:
630:
622:
621:
603:
602:
578:
577:
565:
564:
559:
535:
533:
532:
527:
525:
524:
519:
502:
498:
485:standard simplex
479:
471:
466:regular polytope
457:
455:
454:
449:
444:
440:
418:
414:
405:
404:
395:
391:
382:
381:
371:
366:
349:
348:
347:
339:
338:
337:
328:
327:
309:
308:
299:
298:
267:
265:
264:
259:
257:
256:
244:
243:
225:
224:
212:
211:
195:
187:
185:
184:
179:
177:
176:
158:
157:
141:
131:
116:
110:
106:Specifically, a
16598:
16597:
16591:
16590:
16589:
16587:
16586:
16585:
16561:
16560:
16529:
16522:
16515:
16398:
16391:
16384:
16348:
16341:
16334:
16298:
16291:
16125:Regular polygon
16118:
16109:
16102:
16098:
16091:
16087:
16078:
16069:
16062:
16058:
16046:
16040:
16036:
16024:
16006:
15995:
15966:
15961:
15950:
15929:
15865:
15803:
15757:
15748:
15714:Euclidean space
15697:
15692:
15653:
15601:
15597:
15584:
15568:Coxeter, H.S.M.
15556:
15535:
15510:
15491:
15486:
15477:
15475:
15471:
15460:
15454:
15450:
15443:
15429:
15425:
15418:
15402:
15398:
15375:10.2307/3605876
15359:
15355:
15334:
15330:
15315:10.2307/3072403
15298:
15292:
15288:
15267:
15264:
15263:
15259:
15234:
15230:
15215:
15211:
15208:
15205:
15204:
15180:
15176:
15167:
15164:
15163:
15136:
15132:
15126:
15122:
15098:
15094:
15085:
15081:
15066:
15062:
15035:
15031:
15022:
15018:
15003:
14999:
14987:
14983:
14980:
14977:
14976:
14949:
14945:
14921:
14917:
14908:
14904:
14886:
14882:
14864:
14860:
14851:
14847:
14826:
14822:
14813:
14809:
14797:
14793:
14790:
14787:
14786:
14782:
14778:
14774:
14747:
14743:
14740:
14737:
14736:
14718:
14714:
14711:
14708:
14707:
14689:
14685:
14682:
14679:
14678:
14660:
14656:
14653:
14650:
14649:
14631:
14627:
14612:
14608:
14599:
14595:
14583:
14579:
14570:
14566:
14557:
14553:
14541:
14537:
14528:
14524:
14512:
14508:
14505:
14502:
14501:
14497:
14479:
14476:
14475:
14471:
14469:
14465:
14446:
14442:
14427:10.2307/2315353
14410:
14406:
14383:
14379:
14362:
14358:
14349:
14345:
14324:
14320:
14312:
14308:
14300:
14296:
14288:
14286:
14276:
14272:
14265:
14261:
14254:
14237:
14233:
14229:
14224:
14157:
14097:
14076:quantum gravity
13919:
13899:
13872:
13868:
13860:
13857:
13856:
13836:
13832:
13824:
13821:
13820:
13812:
13808:
13804:
13785:
13782:
13781:
13774:
13749:
13745:
13735:
13731:
13727:
13712:
13708:
13693:
13689:
13671:
13667:
13659:
13656:
13655:
13632:
13628:
13595:
13591:
13589:
13586:
13585:
13550:
13546:
13534:
13523:
13510:
13509:
13494:
13489:
13488:
13481:
13477:
13468:
13464:
13462:
13459:
13458:
13450:
13439:
13428:
13422:
13391:
13388:
13387:
13373:
13324:
13321:
13320:
13301:
13298:
13297:
13280:
13276:
13274:
13271:
13270:
13247:
13243:
13231:
13227:
13221:
13217:
13203:
13199:
13193:
13189:
13183:
13179:
13178:
13174:
13169:
13166:
13165:
13135:
13130:
13129:
13121:
13118:
13117:
13084:
13080:
13078:
13075:
13074:
13040:
13036:
13015:
13011:
12996:
12992:
12977:
12973:
12964:
12960:
12945:
12934:
12929:
12925:
12910:
12906:
12904:
12901:
12900:
12871:
12867:
12846:
12842:
12827:
12823:
12808:
12804:
12795:
12791:
12776:
12765:
12750:
12747:
12746:
12739:
12720:
12717:
12716:
12699:
12695:
12693:
12690:
12689:
12666:
12662:
12647:
12643:
12634:
12630:
12621:
12617:
12606:
12603:
12602:
12591:
12587:
12575:
12571:
12554:
12546:
12538:
12508:
12473:
12443:
12439:
12415:
12411:
12405:
12401:
12399:
12385:
12381:
12375:
12371:
12369:
12357:
12346:
12336:
12325:
12311:
12306:
12286:
12283:
12282:
12254:
12232:
12228:
12204:
12199:
12189:
12178:
12173:
12167:
12162:
12156:
12138:
12133:
12123:
12112:
12107:
12101:
12096:
12090:
12078:
12073:
12063:
12052:
12047:
12041:
12036:
12030:
12029:
12025:
12011:
12008:
12007:
11972:
11968:
11938:
11934:
11928:
11924:
11918:
11907:
11902:
11895:
11891:
11885:
11881:
11880:
11878:
11860:
11856:
11850:
11846:
11840:
11829:
11824:
11817:
11813:
11807:
11803:
11802:
11800:
11788:
11784:
11778:
11774:
11768:
11757:
11752:
11745:
11741:
11735:
11731:
11730:
11728:
11727:
11723:
11709:
11706:
11705:
11672:
11668:
11666:
11663:
11662:
11655:
11649:
11636:
11632:
11628:
11616:
11609:
11601:
11597:
11594:
11588:
11575:
11574:-simplex is an
11571:
11565:
11556:
11550:
11533:
11521:
11513:
11506:
11494:
11490:
11483:
11446:
11442:
11440:
11437:
11436:
11419:
11415:
11406:
11402:
11400:
11397:
11396:
11376:
11371:
11370:
11364:
11360:
11355:
11346:
11341:
11340:
11334:
11330:
11325:
11319:
11308:
11302:
11299:
11298:
11287:
11279:
11271:
11264:
11216:
11213:
11212:
11186:
11167:
11160:
11156:
11154:
11151:
11150:
11121:
11116:
11094:
11089:
11088:
11084:
11082:
11079:
11078:
11052:
11033:
11026:
11022:
11020:
11017:
11016:
10987:
10982:
10960:
10955:
10954:
10950:
10948:
10945:
10944:
10933:
10921:
10917:
10909:
10906:
10863:
10828:
10817:
10814:
10813:
10789:
10784:
10776:
10773:
10772:
10768:
10750:
10745:
10734:
10731:
10730:
10726:
10722:
10718:
10712:
10687:
10683:
10681:
10674:
10661:
10659:
10656:
10655:
10648:
10612:
10607:
10605:
10602:
10601:
10592:
10588:
10556:
10552:
10543:
10539:
10524:
10520:
10511:
10507:
10498:
10494:
10485:
10481:
10463:
10459:
10450:
10446:
10431:
10427:
10418:
10414:
10405:
10401:
10392:
10388:
10380:
10377:
10376:
10353:
10349:
10334:
10330:
10328:
10325:
10324:
10307:
10303:
10301:
10298:
10297:
10280:
10275:
10274:
10272:
10269:
10268:
10245:
10241:
10232:
10228:
10219:
10215:
10203:
10199:
10190:
10186:
10177:
10173:
10161:
10157:
10148:
10144:
10135:
10131:
10129:
10126:
10125:
10108:
10103:
10102:
10100:
10097:
10096:
10076:
10072:
10060:
10056:
10044:
10040:
10028:
10024:
10019:
10016:
10015:
10011:
9984:
9974:
9970:
9955:
9951:
9944:
9942:
9929:
9912:
9910:
9908:
9905:
9904:
9897:
9880:
9876:
9861:
9857:
9855:
9852:
9851:
9834:
9829:
9828:
9826:
9823:
9822:
9814:
9810:
9805:
9778:
9773:
9760:
9743:
9741:
9739:
9736:
9735:
9728:
9724:
9720:
9716:
9712:
9692:
9688:
9686:
9683:
9682:
9678:
9672:
9643:
9639:
9624:
9620:
9611:
9607:
9583:
9579:
9570:
9566:
9557:
9553:
9532:
9528:
9519:
9515:
9506:
9502:
9500:
9497:
9496:
9489:
9472:
9468:
9453:
9449:
9437:
9433:
9431:
9428:
9427:
9387:
9384:
9383:
9367:
9364:
9363:
9343:
9338:
9337:
9335:
9332:
9331:
9311:
9307:
9292:
9288:
9279:
9275:
9270:
9267:
9266:
9262:
9258:
9250:
9242:
9206:
9205:
9200:
9195:
9190:
9184:
9183:
9177:
9173:
9171:
9166:
9160:
9156:
9154:
9148:
9144:
9137:
9136:
9132:
9128:
9118:
9113:
9090:
9088:
9085:
9084:
9064:
9059:
9058:
9056:
9053:
9052:
9048:
9028:
9023:
9022:
9020:
9017:
9016:
9012:
9008:
8980:
8976:
8964:
8963:
8957:
8953:
8944:
8940:
8938:
8933:
8927:
8923:
8914:
8910:
8908:
8902:
8898:
8889:
8885:
8878:
8877:
8870:
8869:
8863:
8858:
8845:
8840:
8833:
8832:
8826:
8825:
8819:
8814:
8801:
8796:
8789:
8788:
8782:
8777:
8764:
8759:
8748:
8747:
8746:
8742:
8741:
8728:
8723:
8700:
8698:
8695:
8694:
8690:The expression
8687:is the origin.
8671:
8667:
8665:
8662:
8661:
8658:
8650:
8647:
8641:
8624:
8598:
8597:
8591:
8587:
8578:
8574:
8571:
8565:
8559:
8555:
8546:
8542:
8539:
8533:
8529:
8520:
8516:
8509:
8508:
8504:
8500:
8490:
8485:
8462:
8460:
8457:
8456:
8449:
8440:
8433:
8429:
8425:
8418:
8413:
8384:
8383:
8372:
8363:
8362:
8353:
8352:
8342:
8341:
8331:
8330:
8319:
8313:
8312:
8306:
8305:
8292:
8291:
8281:
8280:
8269:
8260:
8259:
8253:
8252:
8242:
8241:
8231:
8230:
8219:
8213:
8212:
8206:
8205:
8195:
8194:
8192:
8189:
8188:
8178:
8176:
8174:
8173:For the vector
8150:
8149:
8141:
8136:
8130:
8129:
8124:
8119:
8113:
8112:
8107:
8099:
8089:
8088:
8086:
8083:
8082:
8072:
8066:
8024:
7998:
7970:
7966:
7948:
7935:
7931:
7915:
7899:
7895:
7880:
7876:
7864:
7860:
7859:
7855:
7853:
7850:
7849:
7842:
7839:
7828:
7822:
7816:
7812:
7808:
7804:
7780:
7779:
7768:
7750:
7749:
7738:
7720:
7719:
7708:
7690:
7689:
7678:
7656:
7655:
7645:
7644:
7633:
7615:
7614:
7603:
7585:
7584:
7573:
7555:
7554:
7543:
7521:
7520:
7510:
7509:
7498:
7480:
7479:
7468:
7450:
7449:
7438:
7420:
7419:
7408:
7386:
7385:
7375:
7374:
7363:
7345:
7344:
7333:
7315:
7314:
7303:
7285:
7284:
7273:
7251:
7250:
7240:
7239:
7233:
7232:
7226:
7225:
7219:
7218:
7208:
7207:
7205:
7202:
7201:
7194:
7170:
7169:
7158:
7141:
7130:
7113:
7108:
7102:
7101:
7090:
7070:
7059:
7042:
7037:
7031:
7030:
7025:
7020:
7009:
6992:
6981:
6963:
6962:
6957:
6952:
6941:
6921:
6910:
6888:
6887:
6885:
6882:
6881:
6871:
6868:
6860:
6854:
6851:
6843:
6842:and, for every
6789:
6785:
6781:
6756:
6752:
6748:
6733:
6729:
6708:
6704:
6699:
6696:
6695:
6688:
6685:
6677:
6670:
6664:
6655:
6652:
6644:
6640:
6633:
6625:
6601:
6600:
6587:
6580:
6576:
6569:
6567:
6559:
6546:
6539:
6535:
6528:
6526:
6517:
6516:
6503:
6496:
6492:
6485:
6483:
6472:
6459:
6452:
6448:
6441:
6439:
6426:
6425:
6423:
6420:
6419:
6412:
6408:
6400:
6396:
6392:
6389:
6381:
6374:
6364:
6360:
6359:. In order for
6356:
6352:
6349:
6341:
6315:
6311:
6296:
6292:
6283:
6279:
6262:
6259:
6258:
6251:
6247:
6243:
6237:
6229:
6226:identity matrix
6215:
6211:
6201:
6194:
6185:
6178:
6154:
6134:
6132:
6129:
6128:
6077:
6070:
6066:
6061:
6058:
6057:
6026:
6023:
6022:
5959:
5946:
5939:
5935:
5926:
5921:
5920:
5906:
5902:
5894:
5892:
5889:
5888:
5840:
5835:
5833:
5830:
5829:
5810:
5807:
5806:
5790:
5787:
5786:
5718:
5713:
5710:
5709:
5678:
5675:
5674:
5627:
5626:
5608:
5596:
5595:
5585:
5581:
5576:
5567:
5562:
5561:
5549:
5547:
5544:
5543:
5536:
5529:
5470:
5463:
5459:
5449:
5447:
5444:
5443:
5436:
5428:
5424:
5401:
5397:
5389:
5386:
5378:
5375:
5369:
5363:
5353:
5349:
5324:
5307:
5304:
5303:
5284:
5279:
5276:
5275:
5268:
5264:
5261:
5253:
5245:
5237:
5233:
5232:This yields an
5203:
5191:
5187:
5184: and
5182:
5170:
5166:
5160:
5149:
5136:
5135:
5126:
5121:
5120:
5108:
5104:
5089:
5085:
5081:
5077:
5068:
5063:
5057:
5054:
5053:
5047:
5025:
5021:
5019:
5016:
5015:
5010:complexity via
5003:
4969:
4966:
4965:
4962:
4954:
4938:
4935:
4934:
4894:
4890:
4878:
4872:
4869:
4868:
4852:
4849:
4848:
4809:
4805:
4790:
4786:
4784:
4781:
4780:
4760:
4750:
4746:
4742:
4741:
4739:
4736:
4735:
4718:
4714:
4708:
4704:
4699:
4696:
4695:
4684:
4664:
4657:
4650:
4646:
4638:
4619:
4616:
4615:
4611:
4607:
4603:
4600:symmetric group
4553:
4549:
4540:
4536:
4534:
4531:
4530:
4504:
4500:
4498:
4495:
4494:
4471:
4466:
4465:
4463:
4460:
4459:
4442:
4437:
4436:
4434:
4431:
4430:
4426:
4386:
4382:
4373:
4369:
4354:
4350:
4341:
4337:
4328:
4324:
4309:
4304:
4303:
4291:
4287:
4272:
4268:
4264:
4260:
4251:
4246:
4240:
4237:
4236:
4229:
4208:
4207:
4195:
4191:
4176:
4172:
4163:
4159:
4150:
4146:
4137:
4133:
4126:
4114:
4110:
4107:
4106:
4094:
4090:
4075:
4071:
4062:
4058:
4043:
4039:
4024:
4020:
4013:
4007:
4003:
4000:
3999:
3988:
3987:
3981:
3977:
3968:
3964:
3955:
3951:
3942:
3938:
3929:
3925:
3918:
3912:
3908:
3905:
3904:
3898:
3894:
3885:
3881:
3872:
3868:
3859:
3855:
3848:
3842:
3838:
3835:
3834:
3828:
3824:
3815:
3811:
3802:
3798:
3791:
3785:
3781:
3778:
3777:
3767:
3761:
3757:
3753:
3751:
3748:
3747:
3737:
3725:
3722:(0, 0, 0, 0, 1)
3721:
3718:(0, 0, 0, 1, 0)
3717:
3714:(0, 0, 1, 0, 0)
3713:
3710:(0, 1, 0, 0, 0)
3709:
3706:(1, 0, 0, 0, 0)
3705:
3692:
3688:
3684:
3680:
3676:
3663:
3659:
3655:
3651:
3638:
3634:
3630:
3621:
3617:
3616:Δ is the point
3613:
3572:
3569:
3568:
3562:
3528:
3524:
3522:
3519:
3518:
3490:
3486:
3480:
3476:
3470:
3459:
3443:
3439:
3424:
3420:
3415:
3412:
3411:
3404:
3396:
3395:-simplex (with
3368:
3365:
3364:
3351:
3339:
3333:
3328:are called the
3326:
3318:
3298:
3294:
3288:
3284:
3278:
3267:
3251:
3247:
3232:
3228:
3223:
3220:
3219:
3213:
3205:
3202:
3196:
3192:
3188:
3173:
3153:
3145:
3137:
3131:
3126:
3120:
3110:
3102:
3098:
3091:
3084:
3076:
3068:
3024: for
3022:
3010:
3006:
3003: and
3001:
2989:
2985:
2979:
2968:
2955:
2954:
2939:
2934:
2933:
2921:
2917:
2902:
2898:
2894:
2890:
2881:
2877:
2875:
2872:
2871:
2862:
2856:
2848:
2837:
2833:
2826:
2814:
2803:
2792:
2781:
2770:
2757:
2746:
2735:
2724:
2713:
2700:
2689:
2678:
2667:
2656:
2643:
2632:
2621:
2610:
2599:
2582:Petrie polygons
2578:
2562:
2515:
2507:
2503:
2500:{ } ∨ ( ) ∨ ( )
2499:
2495:
2487:
2480:( ) ∨ ( ) ∨ ( )
2479:
2475:
2471:
2467:
2455:
2451:
2447:
2439:
2428:
2416:
2404:
2396:
2388:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2247:
2246:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2108:
2107:
2051:
2046:
2041:
2036:
2031:
2026:
2021:
2016:
2011:
2006:
2001:
1996:
1991:
1986:
1981:
1979:
1978:
1922:
1917:
1912:
1907:
1902:
1897:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1860:
1859:
1803:
1798:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1751:
1750:
1694:
1689:
1684:
1679:
1674:
1669:
1664:
1659:
1654:
1652:
1651:
1595:
1590:
1585:
1580:
1575:
1570:
1565:
1563:
1562:
1553:
1502:
1497:
1492:
1487:
1482:
1480:
1479:
1470:
1419:
1414:
1409:
1407:
1406:
1397:
1346:
1344:
1343:
1334:
1283:
1281:
1280:
1271:
1261:
1252:
1250:
1243:
1241:
1234:
1232:
1225:
1223:
1216:
1214:
1207:
1205:
1198:
1196:
1189:
1187:
1180:
1178:
1171:
1169:
1162:
1160:
1153:
1142:
1134:
1123:
1119:
1107:
1103:
1100:triangle number
1095:
1091:
1036:
1032:
1009:
1005:
1001:
993:
982:
975:
971:
967:
948:
936:
925:
919:
918:
916:
914:
911:
910:
903:
899:
895:
894:, and the sole
884:
872:
866:
861:
854:
847:
839:
832:
818:
810:
802:
796:
791:
783:
776:
768:
761:
753:
742:regular simplex
707:
631: for
629:
617:
613:
592:
588:
573:
569:
560:
555:
554:
547:
543:
541:
538:
537:
520:
515:
514:
512:
509:
508:
500:
492:
473:
469:
462:regular simplex
415: for
412:
400:
396:
392: and
389:
377:
373:
367:
356:
343:
342:
333:
329:
323:
319:
304:
300:
294:
290:
289:
285:
277:
274:
273:
252:
248:
239:
235:
220:
216:
207:
203:
201:
198:
197:
193:
172:
168:
153:
149:
147:
144:
143:
136:
126:
114:
108:
24:
21:
12:
11:
5:
16596:
16595:
16584:
16583:
16578:
16573:
16557:
16556:
16541:
16540:
16531:
16527:
16520:
16513:
16509:
16500:
16483:
16474:
16463:
16462:
16460:
16458:
16453:
16444:
16439:
16433:
16432:
16430:
16428:
16423:
16414:
16409:
16403:
16402:
16400:
16396:
16389:
16382:
16378:
16373:
16364:
16359:
16353:
16352:
16350:
16346:
16339:
16332:
16328:
16323:
16314:
16309:
16303:
16302:
16300:
16296:
16289:
16285:
16280:
16271:
16266:
16260:
16259:
16257:
16255:
16250:
16241:
16236:
16230:
16229:
16220:
16215:
16210:
16201:
16196:
16190:
16189:
16180:
16178:
16173:
16164:
16159:
16153:
16152:
16147:
16142:
16137:
16132:
16127:
16121:
16120:
16116:
16112:
16107:
16096:
16085:
16076:
16067:
16060:
16054:
16044:
16038:
16032:
16026:
16020:
16014:
16008:
16007:
15996:
15994:
15993:
15986:
15979:
15971:
15963:
15962:
15955:
15952:
15951:
15949:
15948:
15943:
15937:
15935:
15931:
15930:
15928:
15927:
15919:
15914:
15909:
15904:
15899:
15894:
15889:
15884:
15879:
15873:
15871:
15867:
15866:
15864:
15863:
15858:
15853:
15851:Cross-polytope
15848:
15843:
15838:
15836:Hyperrectangle
15833:
15828:
15823:
15817:
15815:
15805:
15804:
15802:
15801:
15796:
15791:
15786:
15781:
15776:
15771:
15765:
15763:
15759:
15758:
15751:
15749:
15747:
15746:
15741:
15736:
15731:
15726:
15721:
15716:
15711:
15705:
15703:
15699:
15698:
15691:
15690:
15683:
15676:
15668:
15662:
15661:
15651:
15629:
15610:
15609:
15608:
15594:
15582:
15564:
15562:on 2009-05-05.
15554:
15539:
15533:
15517:
15508:
15490:
15487:
15485:
15484:
15448:
15441:
15423:
15416:
15396:
15353:
15328:
15286:
15272:
15245:
15242:
15237:
15233:
15229:
15226:
15223:
15218:
15214:
15191:
15188:
15183:
15179:
15175:
15172:
15148:
15145:
15142:
15139:
15135:
15129:
15125:
15121:
15116:
15113:
15110:
15107:
15104:
15101:
15097:
15093:
15088:
15084:
15080:
15075:
15072:
15069:
15065:
15061:
15058:
15055:
15052:
15047:
15044:
15041:
15038:
15034:
15030:
15025:
15021:
15017:
15014:
15011:
15006:
15002:
14998:
14995:
14990:
14986:
14961:
14958:
14955:
14952:
14948:
14944:
14941:
14938:
14933:
14930:
14927:
14924:
14920:
14916:
14911:
14907:
14903:
14898:
14895:
14892:
14889:
14885:
14881:
14876:
14873:
14870:
14867:
14863:
14859:
14854:
14850:
14843:
14838:
14835:
14832:
14829:
14825:
14821:
14816:
14812:
14805:
14800:
14796:
14759:
14756:
14753:
14750:
14746:
14721:
14717:
14692:
14688:
14663:
14659:
14634:
14630:
14626:
14623:
14620:
14615:
14611:
14607:
14602:
14598:
14594:
14591:
14586:
14582:
14578:
14573:
14569:
14565:
14560:
14556:
14549:
14544:
14540:
14536:
14531:
14527:
14520:
14515:
14511:
14484:
14463:
14440:
14421:(3): 299–301.
14404:
14377:
14356:
14343:
14318:
14316:, p. 120.
14306:
14294:
14278:Miller, Jeff,
14270:
14259:
14252:
14230:
14228:
14225:
14223:
14222:
14217:
14212:
14210:Simplicial set
14207:
14202:
14197:
14191:
14186:
14181:
14180:
14179:
14174:
14169:
14167:Cross-polytope
14156:Other regular
14154:
14152:Metcalfe's law
14149:
14144:
14139:
14134:
14129:
14124:
14119:
14117:Complete graph
14114:
14109:
14104:
14098:
14096:
14093:
14092:
14091:
14080:Regge calculus
14072:
14017:
14007:triangulations
13995:
13988:
13985:George Dantzig
13969:
13946:
13935:
13918:
13915:
13898:The algebraic
13880:
13875:
13871:
13867:
13864:
13844:
13839:
13835:
13831:
13828:
13807:-simplex, the
13789:
13761:
13758:
13752:
13748:
13744:
13741:
13738:
13734:
13730:
13726:
13721:
13718:
13715:
13711:
13707:
13704:
13701:
13696:
13692:
13688:
13685:
13682:
13679:
13674:
13670:
13666:
13663:
13643:
13640:
13635:
13631:
13627:
13624:
13621:
13618:
13615:
13612:
13609:
13606:
13603:
13598:
13594:
13569:
13565:
13561:
13558:
13553:
13549:
13543:
13540:
13537:
13532:
13529:
13526:
13522:
13513:
13503:
13500:
13497:
13492:
13487:
13484:
13480:
13476:
13471:
13467:
13438:
13435:
13407:
13404:
13401:
13398:
13395:
13385:continuous map
13370:
13369:
13358:
13355:
13352:
13349:
13346:
13343:
13340:
13337:
13334:
13331:
13328:
13305:
13283:
13279:
13267:
13266:
13255:
13250:
13246:
13242:
13239:
13234:
13230:
13224:
13220:
13216:
13212:
13206:
13202:
13196:
13192:
13186:
13182:
13177:
13173:
13146:
13143:
13138:
13133:
13128:
13125:
13098:
13095:
13092:
13087:
13083:
13071:
13070:
13059:
13056:
13052:
13048:
13043:
13039:
13035:
13032:
13029:
13024:
13021:
13018:
13014:
13010:
13005:
13002:
12999:
12995:
12991:
12988:
12985:
12980:
12976:
12972:
12967:
12963:
12959:
12956:
12953:
12948:
12943:
12940:
12937:
12933:
12928:
12924:
12921:
12918:
12913:
12909:
12894:
12893:
12882:
12879:
12874:
12870:
12866:
12863:
12860:
12855:
12852:
12849:
12845:
12841:
12836:
12833:
12830:
12826:
12822:
12819:
12816:
12811:
12807:
12803:
12798:
12794:
12790:
12787:
12784:
12779:
12774:
12771:
12768:
12764:
12760:
12757:
12754:
12727:
12724:
12702:
12698:
12686:
12685:
12674:
12669:
12665:
12661:
12658:
12655:
12650:
12646:
12642:
12637:
12633:
12629:
12624:
12620:
12616:
12613:
12610:
12507:
12504:
12503:
12502:
12495:
12484:
12472:
12469:
12468:
12467:
12466:
12465:
12452:
12449:
12446:
12442:
12438:
12435:
12432:
12429:
12426:
12418:
12414:
12408:
12404:
12398:
12395:
12388:
12384:
12378:
12374:
12368:
12365:
12360:
12355:
12352:
12349:
12345:
12339:
12334:
12331:
12328:
12324:
12317:
12314:
12310:
12305:
12302:
12299:
12296:
12293:
12290:
12277:
12276:
12272:
12271:
12270:
12269:
12257:
12253:
12250:
12246:
12241:
12238:
12235:
12231:
12227:
12224:
12221:
12216:
12207:
12202:
12198:
12192:
12187:
12184:
12181:
12177:
12170:
12165:
12161:
12155:
12152:
12149:
12141:
12136:
12132:
12126:
12121:
12118:
12115:
12111:
12104:
12099:
12095:
12089:
12081:
12076:
12072:
12066:
12061:
12058:
12055:
12051:
12044:
12039:
12035:
12028:
12024:
12021:
12018:
12015:
12002:
12001:
11997:
11996:
11995:
11994:
11981:
11978:
11975:
11971:
11967:
11964:
11961:
11958:
11955:
11950:
11941:
11937:
11931:
11927:
11921:
11916:
11913:
11910:
11906:
11898:
11894:
11888:
11884:
11877:
11874:
11871:
11863:
11859:
11853:
11849:
11843:
11838:
11835:
11832:
11828:
11820:
11816:
11810:
11806:
11799:
11791:
11787:
11781:
11777:
11771:
11766:
11763:
11760:
11756:
11748:
11744:
11738:
11734:
11726:
11722:
11719:
11716:
11713:
11700:
11699:
11681:
11678:
11675:
11671:
11651:Main article:
11648:
11645:
11590:Main article:
11587:
11584:
11549:
11546:
11482:
11481:+ 1)-hypercube
11475:
11449:
11445:
11422:
11418:
11414:
11409:
11405:
11393:
11392:
11379:
11374:
11367:
11363:
11358:
11354:
11349:
11344:
11337:
11333:
11328:
11322:
11317:
11314:
11311:
11307:
11263:
11260:
11247:
11244:
11241:
11238:
11235:
11232:
11229:
11226:
11223:
11220:
11199:
11193:
11190:
11185:
11182:
11179:
11174:
11171:
11166:
11163:
11159:
11137:
11130:
11127:
11124:
11120:
11115:
11112:
11109:
11103:
11100:
11097:
11093:
11087:
11065:
11059:
11056:
11051:
11048:
11045:
11040:
11037:
11032:
11029:
11025:
11003:
10996:
10993:
10990:
10986:
10981:
10978:
10975:
10969:
10966:
10963:
10959:
10953:
10930:dihedral angle
10905:
10898:
10885:
10882:
10879:
10876:
10873:
10870:
10866:
10862:
10859:
10856:
10853:
10850:
10847:
10844:
10841:
10838:
10835:
10831:
10827:
10824:
10821:
10799:
10796:
10793:
10787:
10783:
10780:
10754:
10748:
10744:
10741:
10738:
10709:
10708:
10690:
10686:
10680:
10677:
10672:
10669:
10666:
10645:
10644:
10630:
10627:
10624:
10621:
10618:
10615:
10611:
10585:
10584:
10573:
10570:
10565:
10562:
10559:
10555:
10551:
10546:
10542:
10538:
10535:
10532:
10527:
10523:
10519:
10514:
10510:
10506:
10501:
10497:
10493:
10488:
10484:
10480:
10477:
10474:
10471:
10466:
10462:
10458:
10453:
10449:
10445:
10442:
10439:
10434:
10430:
10426:
10421:
10417:
10413:
10408:
10404:
10400:
10395:
10391:
10387:
10384:
10356:
10352:
10348:
10345:
10342:
10337:
10333:
10310:
10306:
10283:
10278:
10254:
10251:
10248:
10244:
10240:
10235:
10231:
10227:
10222:
10218:
10214:
10211:
10206:
10202:
10198:
10193:
10189:
10185:
10180:
10176:
10169:
10164:
10160:
10156:
10151:
10147:
10143:
10138:
10134:
10111:
10106:
10084:
10079:
10075:
10071:
10068:
10063:
10059:
10052:
10047:
10043:
10036:
10031:
10027:
10023:
10008:
10007:
9996:
9990:
9987:
9982:
9977:
9973:
9969:
9966:
9963:
9958:
9954:
9950:
9947:
9941:
9935:
9932:
9927:
9924:
9921:
9918:
9915:
9883:
9879:
9875:
9872:
9869:
9864:
9860:
9837:
9832:
9802:
9801:
9790:
9784:
9781:
9777:
9772:
9766:
9763:
9758:
9755:
9752:
9749:
9746:
9695:
9691:
9671:(so there are
9669:
9668:
9655:
9652:
9649:
9646:
9642:
9638:
9633:
9630:
9627:
9623:
9619:
9614:
9610:
9606:
9603:
9600:
9595:
9592:
9589:
9586:
9582:
9578:
9573:
9569:
9565:
9560:
9556:
9549:
9544:
9541:
9538:
9535:
9531:
9527:
9522:
9518:
9514:
9509:
9505:
9475:
9471:
9467:
9464:
9461:
9456:
9452:
9445:
9440:
9436:
9415:
9412:
9409:
9406:
9403:
9400:
9397:
9394:
9391:
9371:
9346:
9341:
9319:
9314:
9310:
9306:
9303:
9300:
9295:
9291:
9287:
9282:
9278:
9274:
9232:
9231:
9220:
9216:
9210:
9204:
9201:
9199:
9196:
9194:
9191:
9189:
9186:
9185:
9180:
9176:
9172:
9170:
9167:
9163:
9159:
9155:
9151:
9147:
9143:
9142:
9140:
9135:
9131:
9124:
9121:
9117:
9112:
9108:
9105:
9102:
9099:
9096:
9093:
9067:
9062:
9031:
9026:
9001:
9000:
8987:
8983:
8979:
8974:
8968:
8960:
8956:
8952:
8947:
8943:
8939:
8937:
8934:
8930:
8926:
8922:
8917:
8913:
8909:
8905:
8901:
8897:
8892:
8888:
8884:
8883:
8881:
8874:
8861:
8857:
8853:
8843:
8839:
8835:
8834:
8831:
8828:
8827:
8817:
8813:
8809:
8799:
8795:
8791:
8790:
8780:
8776:
8772:
8762:
8758:
8754:
8753:
8751:
8745:
8740:
8734:
8731:
8727:
8722:
8718:
8715:
8712:
8709:
8706:
8703:
8674:
8670:
8654:
8645:
8621:
8620:
8608:
8602:
8594:
8590:
8586:
8581:
8577:
8573:
8570:
8567:
8562:
8558:
8554:
8549:
8545:
8541:
8536:
8532:
8528:
8523:
8519:
8515:
8514:
8512:
8507:
8503:
8496:
8493:
8489:
8484:
8480:
8477:
8474:
8471:
8468:
8465:
8445:
8438:
8417:
8414:
8412:
8409:
8405:
8404:
8393:
8388:
8382:
8379:
8375:
8371:
8368:
8365:
8364:
8361:
8358:
8355:
8354:
8351:
8348:
8347:
8345:
8340:
8335:
8329:
8326:
8322:
8318:
8315:
8314:
8311:
8308:
8307:
8304:
8301:
8298:
8297:
8295:
8290:
8285:
8279:
8276:
8272:
8268:
8265:
8262:
8261:
8258:
8255:
8254:
8251:
8248:
8247:
8245:
8240:
8235:
8229:
8226:
8222:
8218:
8215:
8214:
8211:
8208:
8207:
8204:
8201:
8200:
8198:
8171:
8170:
8159:
8154:
8148:
8145:
8142:
8140:
8137:
8135:
8132:
8131:
8128:
8125:
8123:
8120:
8118:
8115:
8114:
8111:
8108:
8106:
8103:
8100:
8098:
8095:
8094:
8092:
8063:
8062:
8051:
8047:
8043:
8040:
8037:
8034:
8031:
8027:
8023:
8020:
8017:
8014:
8011:
8008:
8005:
8001:
7997:
7994:
7991:
7988:
7985:
7982:
7979:
7976:
7973:
7969:
7965:
7961:
7955:
7951:
7947:
7944:
7941:
7938:
7934:
7930:
7927:
7922:
7918:
7914:
7911:
7908:
7905:
7902:
7898:
7894:
7891:
7888:
7883:
7879:
7875:
7872:
7867:
7863:
7858:
7833:
7826:
7801:
7800:
7789:
7784:
7778:
7775:
7771:
7767:
7764:
7761:
7758:
7755:
7752:
7751:
7748:
7745:
7741:
7737:
7734:
7731:
7728:
7725:
7722:
7721:
7718:
7715:
7711:
7707:
7704:
7701:
7698:
7695:
7692:
7691:
7688:
7685:
7681:
7677:
7674:
7671:
7668:
7665:
7662:
7661:
7659:
7654:
7649:
7643:
7640:
7636:
7632:
7629:
7626:
7623:
7620:
7617:
7616:
7613:
7610:
7606:
7602:
7599:
7596:
7593:
7590:
7587:
7586:
7583:
7580:
7576:
7572:
7569:
7566:
7563:
7560:
7557:
7556:
7553:
7550:
7546:
7542:
7539:
7536:
7533:
7530:
7527:
7526:
7524:
7519:
7514:
7508:
7505:
7501:
7497:
7494:
7491:
7488:
7485:
7482:
7481:
7478:
7475:
7471:
7467:
7464:
7461:
7458:
7455:
7452:
7451:
7448:
7445:
7441:
7437:
7434:
7431:
7428:
7425:
7422:
7421:
7418:
7415:
7411:
7407:
7404:
7401:
7398:
7395:
7392:
7391:
7389:
7384:
7379:
7373:
7370:
7366:
7362:
7359:
7356:
7353:
7350:
7347:
7346:
7343:
7340:
7336:
7332:
7329:
7326:
7323:
7320:
7317:
7316:
7313:
7310:
7306:
7302:
7299:
7296:
7293:
7290:
7287:
7286:
7283:
7280:
7276:
7272:
7269:
7266:
7263:
7260:
7257:
7256:
7254:
7249:
7244:
7238:
7235:
7234:
7231:
7228:
7227:
7224:
7221:
7220:
7217:
7214:
7213:
7211:
7191:
7190:
7179:
7174:
7168:
7165:
7161:
7157:
7154:
7151:
7148:
7145:
7142:
7140:
7137:
7133:
7129:
7126:
7123:
7120:
7117:
7114:
7112:
7109:
7107:
7104:
7103:
7100:
7097:
7093:
7089:
7086:
7083:
7080:
7077:
7074:
7071:
7069:
7066:
7062:
7058:
7055:
7052:
7049:
7046:
7043:
7041:
7038:
7036:
7033:
7032:
7029:
7026:
7024:
7021:
7019:
7016:
7012:
7008:
7005:
7002:
6999:
6996:
6993:
6991:
6988:
6984:
6980:
6977:
6974:
6971:
6968:
6965:
6964:
6961:
6958:
6956:
6953:
6951:
6948:
6944:
6940:
6937:
6934:
6931:
6928:
6925:
6922:
6920:
6917:
6913:
6909:
6906:
6903:
6900:
6897:
6894:
6893:
6891:
6864:
6847:
6840:
6839:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6807:
6804:
6801:
6796:
6792:
6788:
6784:
6780:
6777:
6774:
6771:
6768:
6763:
6759:
6755:
6751:
6747:
6744:
6741:
6736:
6732:
6728:
6725:
6722:
6719:
6716:
6711:
6707:
6703:
6681:
6659:
6648:
6629:
6622:
6621:
6610:
6605:
6596:
6593:
6590:
6583:
6579:
6575:
6572:
6566:
6563:
6560:
6555:
6552:
6549:
6542:
6538:
6534:
6531:
6525:
6522:
6519:
6518:
6512:
6509:
6506:
6499:
6495:
6491:
6488:
6482:
6479:
6476:
6473:
6468:
6465:
6462:
6455:
6451:
6447:
6444:
6438:
6435:
6432:
6431:
6429:
6385:
6363:to have order
6345:
6338:
6337:
6326:
6323:
6318:
6314:
6310:
6307:
6304:
6299:
6295:
6291:
6286:
6282:
6278:
6275:
6272:
6269:
6266:
6189:
6161:
6157:
6153:
6150:
6147:
6144:
6141:
6138:
6125:
6124:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6084:
6080:
6076:
6073:
6069:
6065:
6042:
6039:
6036:
6033:
6030:
6019:
6018:
6007:
6004:
6001:
5998:
5995:
5992:
5989:
5986:
5983:
5980:
5977:
5974:
5969:
5966:
5963:
5958:
5953:
5949:
5945:
5942:
5938:
5934:
5929:
5924:
5919:
5912:
5909:
5905:
5901:
5898:
5868:
5865:
5862:
5859:
5856:
5853:
5850:
5847:
5843:
5839:
5814:
5794:
5783:
5782:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5741:
5738:
5735:
5732:
5729:
5726:
5722:
5717:
5694:
5691:
5688:
5685:
5682:
5671:
5670:
5659:
5656:
5653:
5650:
5647:
5644:
5641:
5638:
5635:
5630:
5622:
5619:
5616:
5612:
5607:
5604:
5599:
5589:
5584:
5580:
5575:
5570:
5565:
5557:
5553:
5526:
5525:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5493:
5490:
5486:
5480:
5477:
5474:
5469:
5466:
5462:
5456:
5453:
5382:
5373:
5334:
5331:
5327:
5323:
5320:
5317:
5314:
5311:
5291:
5287:
5283:
5260:
5250:
5242:simplex method
5230:
5229:
5218:
5214:
5210:
5202:
5199:
5194:
5190:
5181:
5178:
5173:
5169:
5163:
5158:
5155:
5152:
5148:
5139:
5129:
5124:
5119:
5116:
5111:
5107:
5103:
5100:
5097:
5092:
5088:
5084:
5080:
5076:
5071:
5066:
5062:
5046:
5045:Corner of cube
5043:
5028:
5024:
5012:median-finding
4991:
4988:
4985:
4982:
4979:
4976:
4973:
4958:
4942:
4921:
4918:
4915:
4912:
4909:
4905:
4902:
4897:
4893:
4889:
4886:
4881:
4877:
4856:
4845:
4844:
4833:
4830:
4827:
4824:
4820:
4817:
4812:
4808:
4804:
4801:
4798:
4793:
4789:
4763:
4758:
4753:
4749:
4745:
4721:
4717:
4711:
4707:
4703:
4683:
4680:
4626:
4623:
4567:
4562:
4559:
4556:
4552:
4548:
4543:
4539:
4518:
4515:
4512:
4507:
4503:
4480:
4477:
4474:
4469:
4445:
4440:
4423:
4422:
4411:
4407:
4403:
4400:
4395:
4392:
4389:
4385:
4381:
4376:
4372:
4368:
4365:
4362:
4357:
4353:
4349:
4344:
4340:
4336:
4331:
4327:
4323:
4320:
4317:
4312:
4307:
4302:
4299:
4294:
4290:
4286:
4283:
4280:
4275:
4271:
4267:
4263:
4259:
4254:
4249:
4245:
4222:
4221:
4206:
4203:
4198:
4194:
4190:
4187:
4184:
4179:
4175:
4171:
4166:
4162:
4158:
4153:
4149:
4145:
4140:
4136:
4132:
4129:
4127:
4123:
4120:
4117:
4113:
4109:
4108:
4103:
4100:
4097:
4093:
4089:
4086:
4083:
4078:
4074:
4070:
4065:
4061:
4057:
4052:
4049:
4046:
4042:
4038:
4033:
4030:
4027:
4023:
4019:
4016:
4014:
4010:
4006:
4002:
4001:
3998:
3993:
3991:
3989:
3984:
3980:
3976:
3971:
3967:
3963:
3958:
3954:
3950:
3945:
3941:
3937:
3932:
3928:
3924:
3921:
3919:
3915:
3911:
3907:
3906:
3901:
3897:
3893:
3888:
3884:
3880:
3875:
3871:
3867:
3862:
3858:
3854:
3851:
3849:
3845:
3841:
3837:
3836:
3831:
3827:
3823:
3818:
3814:
3810:
3805:
3801:
3797:
3794:
3792:
3788:
3784:
3780:
3779:
3776:
3773:
3770:
3768:
3764:
3760:
3756:
3755:
3741:indefinite sum
3736:
3733:
3732:
3731:
3704:with vertices
3698:
3675:with vertices
3669:
3650:with vertices
3644:
3627:
3612:
3609:
3588:
3585:
3582:
3579:
3576:
3548:
3545:
3542:
3537:
3534:
3531:
3527:
3517:of a simplex:
3507:
3506:
3493:
3489:
3483:
3479:
3473:
3468:
3465:
3462:
3458:
3454:
3451:
3446:
3442:
3438:
3435:
3432:
3427:
3423:
3419:
3384:
3381:
3378:
3375:
3372:
3360:or reversing.
3322:
3315:
3314:
3301:
3297:
3291:
3287:
3281:
3276:
3273:
3270:
3266:
3262:
3259:
3254:
3250:
3246:
3243:
3240:
3235:
3231:
3227:
3209:
3200:
3158:
3157:
3149:
3143:
3140:
3135:
3129:
3124:
3106:
3080:
3065:
3064:
3051:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3021:
3018:
3013:
3009:
3000:
2997:
2992:
2988:
2982:
2977:
2974:
2971:
2967:
2958:
2948:
2945:
2942:
2937:
2932:
2929:
2924:
2920:
2916:
2913:
2910:
2905:
2901:
2897:
2893:
2889:
2884:
2880:
2825:
2822:
2819:
2818:
2807:
2796:
2785:
2774:
2762:
2761:
2750:
2739:
2728:
2717:
2705:
2704:
2693:
2682:
2671:
2660:
2648:
2647:
2636:
2625:
2614:
2603:
2577:
2574:
2557:
2556:
2537:
2536:
2383:
2382:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2243:
2238:
2234:
2233:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2104:
2099:
2095:
2094:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
1975:
1970:
1966:
1965:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1856:
1851:
1847:
1846:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1747:
1742:
1738:
1737:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1648:
1643:
1639:
1638:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1559:
1550:
1546:
1545:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1476:
1467:
1463:
1462:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1403:
1394:
1390:
1389:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1340:
1331:
1327:
1326:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1277:
1268:
1264:
1263:
1256:
1247:
1238:
1229:
1220:
1211:
1202:
1193:
1184:
1175:
1166:
1157:
1148:
1145:
951:
945:
942:
939:
934:
931:
928:
922:
817:
814:
806:
787:
772:
765:cross-polytope
757:
750:Donald Coxeter
731:simplicissimum
715:Henri Poincaré
706:
703:
668:
664:
660:
657:
654:
651:
648:
645:
642:
639:
636:
628:
625:
620:
616:
612:
609:
606:
601:
598:
595:
591:
587:
584:
581:
576:
572:
568:
563:
558:
553:
550:
546:
523:
518:
447:
443:
439:
436:
433:
430:
427:
424:
421:
411:
408:
403:
399:
388:
385:
380:
376:
370:
365:
362:
359:
355:
346:
336:
332:
326:
322:
318:
315:
312:
307:
303:
297:
293:
288:
284:
281:
255:
251:
247:
242:
238:
234:
231:
228:
223:
219:
215:
210:
206:
175:
171:
167:
164:
161:
156:
152:
104:
103:
96:
89:
82:
75:
22:
9:
6:
4:
3:
2:
16594:
16593:
16582:
16579:
16577:
16574:
16572:
16569:
16568:
16566:
16555:
16551:
16547:
16542:
16539:
16535:
16532:
16530:
16523:
16516:
16510:
16508:
16504:
16501:
16499:
16495:
16491:
16487:
16484:
16482:
16478:
16475:
16473:
16469:
16465:
16464:
16461:
16459:
16457:
16454:
16452:
16448:
16445:
16443:
16440:
16438:
16435:
16434:
16431:
16429:
16427:
16424:
16422:
16418:
16415:
16413:
16410:
16408:
16405:
16404:
16401:
16399:
16392:
16385:
16379:
16377:
16374:
16372:
16368:
16365:
16363:
16360:
16358:
16355:
16354:
16351:
16349:
16342:
16335:
16329:
16327:
16324:
16322:
16318:
16315:
16313:
16310:
16308:
16305:
16304:
16301:
16299:
16292:
16286:
16284:
16281:
16279:
16275:
16272:
16270:
16267:
16265:
16262:
16261:
16258:
16256:
16254:
16251:
16249:
16245:
16242:
16240:
16237:
16235:
16232:
16231:
16228:
16224:
16221:
16219:
16216:
16214:
16213:Demitesseract
16211:
16209:
16205:
16202:
16200:
16197:
16195:
16192:
16191:
16188:
16184:
16181:
16179:
16177:
16174:
16172:
16168:
16165:
16163:
16160:
16158:
16155:
16154:
16151:
16148:
16146:
16143:
16141:
16138:
16136:
16133:
16131:
16128:
16126:
16123:
16122:
16119:
16113:
16110:
16106:
16099:
16095:
16088:
16084:
16079:
16075:
16070:
16066:
16061:
16059:
16057:
16053:
16043:
16039:
16037:
16035:
16031:
16027:
16025:
16023:
16019:
16015:
16013:
16010:
16009:
16004:
16000:
15992:
15987:
15985:
15980:
15978:
15973:
15972:
15969:
15960:
15959:
15953:
15947:
15944:
15942:
15939:
15938:
15936:
15932:
15926:
15924:
15920:
15918:
15915:
15913:
15910:
15908:
15905:
15903:
15900:
15898:
15895:
15893:
15890:
15888:
15885:
15883:
15880:
15878:
15875:
15874:
15872:
15868:
15862:
15859:
15857:
15854:
15852:
15849:
15847:
15844:
15842:
15841:Demihypercube
15839:
15837:
15834:
15832:
15829:
15827:
15824:
15822:
15819:
15818:
15816:
15814:
15810:
15806:
15800:
15797:
15795:
15792:
15790:
15787:
15785:
15782:
15780:
15777:
15775:
15772:
15770:
15767:
15766:
15764:
15760:
15755:
15745:
15742:
15740:
15737:
15735:
15732:
15730:
15727:
15725:
15722:
15720:
15717:
15715:
15712:
15710:
15707:
15706:
15704:
15700:
15696:
15689:
15684:
15682:
15677:
15675:
15670:
15669:
15666:
15660:
15654:
15648:
15644:
15643:
15638:
15634:
15633:Boyd, Stephen
15630:
15625:
15624:
15619:
15616:
15611:
15604:
15595:
15589:
15588:
15585:
15583:0-486-61480-8
15579:
15575:
15574:
15569:
15565:
15561:
15557:
15555:0-387-96305-7
15551:
15547:
15546:
15540:
15536:
15534:0-13-066102-3
15530:
15526:
15522:
15518:
15516:
15511:
15509:0-07-054235-X
15505:
15501:
15497:
15496:Rudin, Walter
15493:
15492:
15474:on 2011-06-07
15470:
15466:
15459:
15452:
15444:
15442:0-471-07916-2
15438:
15434:
15427:
15419:
15413:
15409:
15408:
15400:
15392:
15388:
15384:
15380:
15376:
15372:
15368:
15364:
15357:
15349:
15345:
15341:
15340:
15332:
15324:
15320:
15316:
15312:
15308:
15304:
15296:
15290:
15270:
15243:
15240:
15235:
15231:
15227:
15224:
15221:
15216:
15212:
15189:
15186:
15181:
15177:
15173:
15170:
15143:
15137:
15133:
15127:
15123:
15119:
15111:
15108:
15105:
15099:
15095:
15086:
15082:
15078:
15073:
15070:
15067:
15063:
15056:
15053:
15050:
15042:
15036:
15032:
15023:
15019:
15015:
15012:
15009:
15004:
15000:
14993:
14988:
14984:
14956:
14950:
14946:
14942:
14939:
14936:
14928:
14922:
14918:
14914:
14909:
14905:
14901:
14893:
14887:
14883:
14879:
14871:
14865:
14861:
14857:
14852:
14848:
14841:
14833:
14827:
14823:
14819:
14814:
14810:
14803:
14798:
14794:
14773:for all
14754:
14748:
14744:
14719:
14715:
14690:
14686:
14661:
14657:
14632:
14628:
14624:
14621:
14618:
14613:
14609:
14605:
14600:
14596:
14592:
14589:
14584:
14580:
14576:
14571:
14567:
14563:
14558:
14554:
14547:
14542:
14538:
14534:
14529:
14525:
14518:
14513:
14509:
14482:
14467:
14458:
14457:
14452:
14444:
14436:
14432:
14428:
14424:
14420:
14416:
14408:
14400:
14396:
14392:
14388:
14381:
14372:
14367:
14360:
14353:
14347:
14339:
14338:
14332:
14328:
14322:
14315:
14310:
14303:
14298:
14285:
14281:
14274:
14268:
14263:
14255:
14249:
14245:
14241:
14235:
14231:
14221:
14218:
14216:
14215:Spectrahedron
14213:
14211:
14208:
14206:
14203:
14201:
14198:
14195:
14192:
14190:
14187:
14185:
14182:
14178:
14175:
14173:
14170:
14168:
14165:
14164:
14163:
14155:
14153:
14150:
14148:
14145:
14143:
14140:
14138:
14135:
14133:
14130:
14128:
14125:
14123:
14120:
14118:
14115:
14113:
14110:
14108:
14105:
14103:
14100:
14099:
14089:
14085:
14081:
14077:
14073:
14070:
14066:
14063:resembling a
14062:
14058:
14054:
14050:
14046:
14042:
14038:
14034:
14030:
14026:
14022:
14018:
14015:
14012:
14008:
14004:
14000:
13996:
13993:
13989:
13986:
13982:
13978:
13974:
13970:
13967:
13963:
13959:
13955:
13951:
13947:
13944:
13940:
13936:
13933:
13929:
13925:
13921:
13920:
13914:
13912:
13908:
13906:
13896:
13894:
13873:
13862:
13837:
13826:
13818:
13801:
13787:
13780:
13756:
13750:
13746:
13742:
13739:
13736:
13732:
13719:
13716:
13713:
13709:
13705:
13702:
13699:
13694:
13690:
13683:
13680:
13672:
13661:
13633:
13622:
13616:
13613:
13610:
13604:
13596:
13583:
13567:
13563:
13559:
13556:
13551:
13547:
13541:
13538:
13535:
13530:
13527:
13524:
13520:
13501:
13498:
13495:
13485:
13482:
13478:
13474:
13469:
13454:
13448:
13444:
13434:
13432:
13421:
13405:
13399:
13396:
13393:
13386:
13381:
13379:
13378:map operation
13353:
13344:
13341:
13335:
13329:
13319:
13318:
13317:
13281:
13277:
13248:
13244:
13237:
13232:
13228:
13222:
13214:
13210:
13204:
13200:
13194:
13190:
13184:
13175:
13171:
13164:
13163:
13162:
13160:
13144:
13136:
13126:
13123:
13115:
13110:
13096:
13093:
13090:
13085:
13057:
13054:
13050:
13041:
13037:
13033:
13030:
13027:
13022:
13019:
13016:
13012:
13008:
13003:
13000:
12997:
12993:
12989:
12986:
12983:
12978:
12974:
12965:
12957:
12954:
12946:
12941:
12938:
12935:
12931:
12926:
12919:
12916:
12911:
12899:
12898:
12897:
12880:
12872:
12868:
12864:
12861:
12858:
12853:
12850:
12847:
12843:
12839:
12834:
12831:
12828:
12824:
12820:
12817:
12814:
12809:
12805:
12796:
12788:
12785:
12777:
12772:
12769:
12766:
12762:
12758:
12755:
12745:
12744:
12743:
12742:is the chain
12725:
12700:
12696:
12667:
12663:
12659:
12656:
12653:
12648:
12644:
12640:
12635:
12631:
12627:
12622:
12618:
12611:
12608:
12601:
12600:
12599:
12595:
12585:
12579:
12568:
12566:
12562:
12558:
12551:is called an
12549:
12544:
12535:
12533:
12529:
12525:
12524:combinatorial
12521:
12517:
12513:
12500:
12496:
12493:
12489:
12485:
12482:
12478:
12477:
12476:
12450:
12447:
12444:
12436:
12433:
12430:
12427:
12416:
12412:
12406:
12402:
12396:
12393:
12386:
12382:
12376:
12372:
12366:
12363:
12358:
12353:
12350:
12347:
12343:
12337:
12332:
12329:
12326:
12322:
12315:
12312:
12308:
12303:
12297:
12294:
12291:
12281:
12280:
12279:
12278:
12275:Inner product
12274:
12273:
12251:
12248:
12244:
12239:
12236:
12233:
12225:
12222:
12214:
12205:
12200:
12196:
12190:
12185:
12182:
12179:
12175:
12168:
12163:
12159:
12153:
12150:
12147:
12139:
12134:
12130:
12124:
12119:
12116:
12113:
12109:
12102:
12097:
12093:
12087:
12079:
12074:
12070:
12064:
12059:
12056:
12053:
12049:
12042:
12037:
12033:
12026:
12022:
12019:
12016:
12013:
12006:
12005:
12004:
12003:
11999:
11998:
11979:
11976:
11973:
11965:
11962:
11959:
11956:
11948:
11939:
11935:
11929:
11925:
11919:
11914:
11911:
11908:
11904:
11896:
11892:
11886:
11882:
11875:
11872:
11869:
11861:
11857:
11851:
11847:
11841:
11836:
11833:
11830:
11826:
11818:
11814:
11808:
11804:
11797:
11789:
11785:
11779:
11775:
11769:
11764:
11761:
11758:
11754:
11746:
11742:
11736:
11732:
11724:
11720:
11717:
11714:
11711:
11704:
11703:
11702:
11701:
11697:
11696:
11695:
11679:
11676:
11673:
11660:
11654:
11644:
11640:
11626:
11620:
11612:
11605:
11593:
11583:
11581:
11578:-dimensional
11569:
11562:
11554:
11553:Topologically
11545:
11543:
11537:
11531:
11525:
11519:
11518:vertex figure
11510:
11504:
11498:
11488:
11487:Hasse diagram
11480:
11474:
11472:
11468:
11463:
11447:
11443:
11420:
11416:
11412:
11407:
11403:
11377:
11365:
11361:
11352:
11347:
11335:
11331:
11320:
11315:
11312:
11309:
11305:
11297:
11296:
11295:
11291:
11283:
11277:
11269:
11259:
11242:
11239:
11236:
11233:
11230:
11227:
11224:
11221:
11197:
11191:
11188:
11183:
11180:
11177:
11172:
11169:
11164:
11161:
11157:
11135:
11128:
11125:
11122:
11118:
11113:
11110:
11107:
11101:
11098:
11095:
11091:
11085:
11063:
11057:
11054:
11049:
11046:
11043:
11038:
11035:
11030:
11027:
11023:
11001:
10994:
10991:
10988:
10984:
10979:
10976:
10973:
10967:
10964:
10961:
10957:
10951:
10941:
10937:
10931:
10925:
10913:
10903:
10897:
10877:
10874:
10871:
10864:
10860:
10857:
10854:
10851:
10848:
10842:
10839:
10836:
10829:
10825:
10822:
10797:
10794:
10791:
10785:
10781:
10778:
10752:
10746:
10742:
10739:
10736:
10715:
10688:
10684:
10678:
10675:
10670:
10667:
10664:
10654:
10653:
10652:
10628:
10622:
10619:
10616:
10609:
10600:
10599:
10598:
10595:
10571:
10563:
10560:
10557:
10553:
10549:
10544:
10540:
10536:
10533:
10530:
10525:
10521:
10517:
10512:
10508:
10504:
10499:
10495:
10491:
10486:
10482:
10472:
10464:
10460:
10456:
10451:
10447:
10443:
10440:
10437:
10432:
10428:
10424:
10419:
10415:
10411:
10406:
10402:
10398:
10393:
10389:
10375:
10374:
10373:
10370:
10354:
10350:
10346:
10343:
10340:
10335:
10331:
10308:
10304:
10281:
10252:
10249:
10246:
10242:
10238:
10233:
10229:
10225:
10220:
10216:
10212:
10209:
10204:
10200:
10196:
10191:
10187:
10183:
10178:
10174:
10167:
10162:
10158:
10154:
10149:
10145:
10141:
10136:
10132:
10109:
10077:
10073:
10069:
10066:
10061:
10057:
10050:
10045:
10041:
10034:
10029:
10025:
9994:
9988:
9985:
9975:
9971:
9967:
9964:
9961:
9956:
9952:
9939:
9933:
9930:
9922:
9916:
9913:
9903:
9902:
9901:
9881:
9877:
9873:
9870:
9867:
9862:
9858:
9835:
9820:
9788:
9782:
9779:
9775:
9770:
9764:
9761:
9753:
9747:
9744:
9734:
9733:
9732:
9709:
9693:
9689:
9675:
9650:
9644:
9640:
9636:
9631:
9628:
9625:
9621:
9617:
9612:
9608:
9604:
9601:
9598:
9590:
9584:
9580:
9576:
9571:
9567:
9563:
9558:
9554:
9547:
9539:
9533:
9529:
9525:
9520:
9516:
9512:
9507:
9503:
9495:
9494:
9493:
9473:
9469:
9465:
9462:
9459:
9454:
9450:
9443:
9438:
9434:
9410:
9407:
9404:
9401:
9398:
9395:
9392:
9369:
9362:
9344:
9312:
9308:
9304:
9301:
9298:
9293:
9289:
9285:
9280:
9276:
9256:
9255:parallelotope
9246:
9239:
9237:
9218:
9214:
9208:
9202:
9197:
9192:
9187:
9178:
9174:
9168:
9161:
9157:
9149:
9145:
9138:
9129:
9122:
9119:
9115:
9110:
9083:
9082:
9081:
9065:
9045:
9029:
9006:
8985:
8981:
8977:
8972:
8966:
8958:
8954:
8950:
8945:
8941:
8935:
8928:
8924:
8920:
8915:
8911:
8903:
8899:
8895:
8890:
8886:
8879:
8872:
8859:
8855:
8851:
8841:
8837:
8829:
8815:
8811:
8807:
8797:
8793:
8778:
8774:
8770:
8760:
8756:
8749:
8743:
8732:
8729:
8725:
8720:
8693:
8692:
8691:
8688:
8672:
8668:
8657:
8653:
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7364:
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7215:
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6878:
6874:
6867:
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6811:
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6757:
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6717:
6714:
6709:
6705:
6694:
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6684:
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6638:
6632:
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6608:
6603:
6594:
6591:
6588:
6581:
6577:
6573:
6570:
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6553:
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6520:
6510:
6507:
6504:
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6463:
6460:
6453:
6449:
6445:
6442:
6436:
6433:
6427:
6418:
6417:
6416:
6411:; or it is a
6406:
6388:
6384:
6377:
6373:
6367:
6348:
6344:
6324:
6316:
6312:
6308:
6305:
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6289:
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6267:
6264:
6257:
6256:
6255:
6240:
6235:
6227:
6222:
6218:
6208:
6204:
6200:. This is an
6199:
6192:
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6184:
6175:
6159:
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6145:
6142:
6136:
6111:
6105:
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5440:
5434:
5423:
5417:
5413:
5409:
5405:
5393:
5385:
5381:
5372:
5366:
5361:
5360:basis vectors
5356:
5346:
5329:
5325:
5321:
5318:
5312:
5309:
5289:
5285:
5281:
5271:
5259:
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5212:
5208:
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5127:
5117:
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5013:
5007:
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4443:
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4398:
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4387:
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4379:
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4370:
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4360:
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3965:
3961:
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3930:
3926:
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3920:
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3909:
3899:
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3891:
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3882:
3878:
3873:
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3865:
3860:
3856:
3852:
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3843:
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3829:
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3816:
3812:
3808:
3803:
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3786:
3782:
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3771:
3769:
3762:
3758:
3746:
3745:
3744:
3742:
3728:
3703:
3699:
3695:
3674:
3670:
3666:
3649:
3645:
3641:
3628:
3624:
3615:
3614:
3608:
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3602:
3583:
3580:
3577:
3565:
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3535:
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3516:
3512:
3491:
3487:
3481:
3477:
3471:
3466:
3463:
3460:
3456:
3444:
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3436:
3433:
3430:
3425:
3421:
3410:
3409:
3408:
3402:
3379:
3376:
3373:
3361:
3359:
3355:
3347:
3343:
3331:
3325:
3321:
3299:
3295:
3289:
3285:
3279:
3274:
3271:
3268:
3264:
3252:
3248:
3244:
3241:
3238:
3233:
3229:
3218:
3217:
3216:
3212:
3208:
3199:
3185:
3183:
3177:
3172:of a regular
3171:
3167:
3163:
3152:
3148:
3144:
3141:
3134:
3130:
3123:
3119:
3118:
3117:
3114:
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3105:
3094:
3088:
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3079:
3074:
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3033:
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3027:
3019:
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3011:
3007:
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2986:
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2946:
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2940:
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2922:
2918:
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2911:
2908:
2903:
2899:
2891:
2887:
2882:
2870:
2869:
2868:
2865:
2860:
2852:
2840:
2832:The standard
2830:
2817:
2812:
2808:
2806:
2801:
2797:
2795:
2790:
2786:
2784:
2779:
2775:
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2768:
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2698:
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2654:
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2646:
2641:
2637:
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2619:
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2608:
2604:
2602:
2597:
2593:
2589:
2587:
2583:
2573:
2571:
2565:
2552:
2548:
2543:
2539:
2538:
2532:
2527:
2523:
2522:
2519:
2513:
2493:
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2381:
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2345:
2244:
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2239:
2236:
2235:
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2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2105:
2103:
2100:
2097:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
1976:
1974:
1971:
1968:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1857:
1855:
1852:
1849:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1748:
1746:
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1740:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1649:
1647:
1644:
1641:
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1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1560:
1557:
1551:
1548:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1477:
1474:
1468:
1465:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1404:
1401:
1395:
1392:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1341:
1338:
1332:
1329:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1278:
1275:
1269:
1266:
1265:
1262:= 2 − 1
1260:
1257:
1248:
1239:
1230:
1221:
1212:
1203:
1194:
1185:
1176:
1167:
1158:
1156:
1152:
1149:
1146:
1141:
1140:
1131:
1127:
1117:
1111:
1101:
1088:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1044:
1040:
1030:
1027:The extended
1025:
1023:
1019:
1015:
1008:is a face of
1000:of a simplex
999:
991:
985:
978:
970:-faces of an
943:
940:
937:
932:
929:
926:
909:
893:
887:
882:
878:
870:
857:
850:
845:
835:
830:
827:
823:
813:
809:
805:
800:
790:
786:
782:, labeled as
781:
775:
771:
766:
760:
756:
751:
747:
743:
738:
736:
732:
728:
724:
720:
716:
712:
702:
701:of vertices.
700:
696:
692:
688:
687:combinatorics
684:
679:
666:
662:
658:
655:
652:
649:
646:
643:
640:
637:
634:
626:
623:
618:
614:
610:
607:
604:
599:
596:
593:
589:
585:
582:
579:
574:
570:
566:
561:
551:
548:
544:
521:
506:
496:
490:
486:
481:
477:
467:
463:
458:
445:
441:
437:
434:
431:
428:
425:
422:
419:
409:
406:
401:
397:
386:
383:
378:
374:
368:
363:
360:
357:
353:
334:
330:
324:
320:
316:
313:
310:
305:
301:
295:
291:
286:
282:
279:
271:
253:
249:
245:
240:
236:
232:
229:
226:
221:
217:
213:
208:
204:
191:
173:
169:
165:
162:
159:
154:
150:
139:
134:
129:
124:
120:
117:-dimensional
112:
101:
97:
94:
90:
87:
83:
80:
76:
73:
69:
68:
67:
65:
61:
58:to arbitrary
57:
53:
49:
45:
41:
37:
28:
19:
16533:
16502:
16493:
16485:
16480:
16476:
16467:
16447:10-orthoplex
16183:Dodecahedron
16104:
16093:
16082:
16073:
16064:
16055:
16051:
16041:
16033:
16029:
16021:
16017:
15956:
15922:
15861:Hyperpyramid
15855:
15826:Hypersurface
15719:Affine space
15709:Vector space
15641:
15621:
15602:
15600:dimensions (
15572:
15560:the original
15548:. Springer.
15544:
15524:
15514:
15499:
15476:. Retrieved
15469:the original
15464:
15451:
15432:
15426:
15406:
15399:
15369:(234): 206.
15366:
15362:
15356:
15338:
15331:
15309:(8): 756–8.
15306:
15302:
15289:
14466:
14454:
14443:
14418:
14414:
14407:
14390:
14386:
14380:
14359:
14351:
14346:
14334:
14321:
14314:Coxeter 1973
14309:
14302:Coxeter 1973
14297:
14287:, retrieved
14283:
14273:
14262:
14243:
14234:
14220:Ternary plot
14142:Hypersimplex
13932:ternary plot
13917:Applications
13904:
13897:
13802:
13452:
13446:
13440:
13426:
13382:
13371:
13268:
13111:
13072:
12895:
12687:
12593:
12577:
12569:
12561:multiplicity
12552:
12547:
12536:
12509:
12474:
11656:
11638:
11618:
11610:
11603:
11600:-simplex in
11595:
11559:-simplex is
11551:
11535:
11523:
11511:
11496:
11484:
11478:
11464:
11394:
11289:
11281:
11265:
10942:
10935:
10923:
10911:
10907:
10901:
10713:
10710:
10646:
10593:
10586:
10371:
10009:
9803:
9715:is the unit
9710:
9673:
9670:
9244:
9241:Without the
9240:
9233:
9051:-simplex in
9046:
9002:
8689:
8655:
8651:
8642:
8629:
8625:
8622:
8446:
8442:
8435:
8428:-simplex in
8419:
8406:
8172:
8073:
8067:
8064:
7835:
7830:
7823:
7817:
7802:
7195:(1, 0, 1, 0)
7192:
6872:
6865:
6861:
6855:
6848:
6844:
6841:
6682:
6678:
6668:
6660:
6656:
6649:
6645:
6630:
6626:
6623:
6391:is either a
6386:
6382:
6375:
6365:
6346:
6342:
6339:
6238:
6220:
6216:
6206:
6202:
6190:
6186:
6183:cyclic group
6176:
6126:
6020:
5883:
5827:
5784:
5672:
5534:
5527:
5415:
5411:
5407:
5403:
5391:
5383:
5379:
5370:
5364:
5354:
5352:-simplex in
5347:
5269:
5267:-simplex in
5262:
5257:
5231:
5048:
5005:
4959:
4955:
4933:
4846:
4685:
4676:
4669:
4665:
4658:
4651:
4640:
4588:
4583:
4424:
4225:
4223:
3738:
3726:
3693:
3689:(0, 0, 0, 1)
3685:(0, 0, 1, 0)
3681:(0, 1, 0, 0)
3677:(1, 0, 0, 0)
3664:
3639:
3622:
3563:
3560:
3514:
3508:
3362:
3349:
3337:
3323:
3319:
3316:
3210:
3206:
3197:
3186:
3175:
3166:0/1-polytope
3161:
3159:
3150:
3146:
3132:
3121:
3112:
3107:
3103:
3092:
3089:
3081:
3077:
3071:lies in the
3067:The simplex
3066:
2863:
2854:
2846:
2844:
2838:
2579:
2563:
2560:
2547:power of two
2461:
2457:
2441:
2437:
2432:
2424:
2420:
2412:
2408:
2400:
2386:
2379:
2230:
2091:
1962:
1843:
1734:
1635:
1542:
1459:
1386:
1337:line segment
1323:
1258:
1125:
1109:
1089:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1042:
1038:
1026:
1017:
1013:
997:
992:. A simplex
983:
976:
891:
885:
880:
876:
865:
855:
848:
843:
833:
819:
807:
803:
788:
784:
773:
769:
758:
754:
741:
739:
737:("simple").
734:
730:
729:superlative
708:
680:
505:unit vectors
494:
488:
484:
482:
475:
468:. A regular
461:
459:
137:
127:
121:that is the
107:
105:
79:line segment
47:
43:
39:
33:
16456:10-demicube
16417:9-orthoplex
16367:8-orthoplex
16317:7-orthoplex
16274:6-orthoplex
16244:5-orthoplex
16199:Pentachoron
16187:Icosahedron
16162:Tetrahedron
15946:Codimension
15925:-dimensions
15846:Hypersphere
15729:Free module
15301:-Simplex".
14061:a structure
14053:tetrahedron
14014:polynomials
13992:game theory
13911:Chow groups
13316:, one has:
13161:. That is,
12565:orientation
12543:open subset
11586:Probability
9681:-paths and
9361:permutation
8634:determinant
7811:, equal to
6859:upon which
6624:where each
6340:where each
5422:real number
4614:-cube into
3215:) given by
1473:tetrahedron
822:convex hull
123:convex hull
93:tetrahedron
56:tetrahedron
16565:Categories
16442:10-simplex
16426:9-demicube
16376:8-demicube
16326:7-demicube
16283:6-demicube
16253:5-demicube
16167:Octahedron
15941:Hyperspace
15821:Hyperplane
15489:References
15478:2009-11-11
15348:1957/11929
14393:(4): 219.
14289:2018-01-08
14240:Elte, E.L.
14078:, such as
13924:statistics
13269:where the
11561:equivalent
9900:-path is:
9492:-path if
9359:. Given a
9003:employs a
6214:such that
5532:-simplex.
4692:projection
4578:while the
2241:10-simplex
1164:(vertices)
981:) of row (
780:hypercubes
778:, and the
699:finite set
60:dimensions
16571:Polytopes
16490:orthoplex
16412:9-simplex
16362:8-simplex
16312:7-simplex
16269:6-simplex
16239:5-simplex
16208:Tesseract
15831:Hypercube
15809:Polytopes
15789:Minkowski
15784:Hausdorff
15779:Inductive
15744:Spacetime
15695:Dimension
15623:MathWorld
15618:"Simplex"
15391:125391795
15271:≤
15225:⋯
15138:σ
15109:−
15100:σ
15071:−
15054:⋯
15037:σ
15013:⋯
14951:σ
14940:⋯
14923:σ
14902:…
14888:σ
14866:σ
14828:σ
14749:σ
14622:⋯
14593:…
14483:σ
14456:MathWorld
14371:1101.6081
14280:"Simplex"
14177:Tesseract
14172:Hypercube
14162:polytopes
14021:chemistry
13874:∙
13870:Δ
13834:Δ
13743:∑
13740:−
13703:…
13669:Δ
13630:Δ
13617:
13593:Δ
13521:∑
13486:∈
13466:Δ
13427:singular
13403:→
13400:σ
13354:ρ
13351:∂
13336:ρ
13327:∂
13304:∂
13245:σ
13219:∑
13201:σ
13181:∑
13159:embedding
13142:→
13091:ρ
13082:∂
13031:…
13001:−
12987:…
12955:−
12932:∑
12923:∂
12917:σ
12908:∂
12862:…
12832:−
12818:…
12786:−
12763:∑
12756:σ
12753:∂
12726:σ
12723:∂
12688:with the
12657:…
12609:σ
12471:Compounds
12448:−
12441:Δ
12437:∈
12425:∀
12397:
12367:
12344:∑
12323:∑
12301:⟩
12289:⟨
12252:∈
12249:α
12237:−
12230:Δ
12226:∈
12220:∀
12206:α
12176:∑
12169:α
12151:…
12140:α
12110:∑
12103:α
12080:α
12050:∑
12043:α
12017:⊙
12014:α
11977:−
11970:Δ
11966:∈
11954:∀
11905:∑
11873:…
11827:∑
11755:∑
11715:⊕
11677:−
11670:Δ
11542:orthoplex
11503:hypercube
11413:…
11306:∑
11237:…
11222:−
11181:…
11111:…
11047:…
10977:…
10852:…
10561:−
10550:−
10534:…
10518:−
10492:−
10457:−
10441:…
10425:−
10399:−
10344:…
10250:−
10239:−
10213:…
10197:−
10155:−
10070:…
10014:-simplex
9965:…
9917:
9871:…
9748:
9727:-path is
9645:σ
9629:−
9602:…
9585:σ
9534:σ
9463:…
9405:…
9370:σ
9302:…
9198:⋯
9169:⋯
8951:−
8936:⋯
8921:−
8896:−
8852:−
8830:⋮
8808:−
8771:−
8585:−
8569:⋯
8553:−
8527:−
8378:√
8367:−
8357:−
8325:√
8300:−
8275:√
8264:−
8225:√
8175:(1, 0, 1/
8144:−
8102:−
8036:…
7990:−
7978:…
7940:−
7933:ω
7929:−
7907:−
7897:ω
7890:…
7878:ω
7874:−
7862:ω
7766:π
7757:
7736:π
7727:
7706:π
7697:
7676:π
7667:
7631:π
7622:
7601:π
7592:
7571:π
7562:
7541:π
7532:
7496:π
7487:
7466:π
7457:
7436:π
7427:
7406:π
7397:
7361:π
7352:
7331:π
7322:
7301:π
7292:
7271:π
7262:
7156:π
7147:
7128:π
7119:
7088:π
7079:
7073:−
7057:π
7048:
7007:π
6998:
6979:π
6970:
6939:π
6930:
6924:−
6908:π
6899:
6815:…
6783:ω
6779:−
6750:ω
6743:…
6731:ω
6727:−
6706:ω
6578:ω
6574:π
6565:
6537:ω
6533:π
6524:
6494:ω
6490:π
6481:
6475:−
6450:ω
6446:π
6437:
6306:…
6274:
6100:…
6088:⋅
6072:−
6064:±
6038:≤
6032:≤
5994:…
5982:⋅
5973:±
5941:−
5933:−
5918:⋅
5908:−
5813:−
5758:…
5746:⋅
5716:±
5690:≤
5684:≤
5646:…
5634:⋅
5606:±
5574:−
5501:…
5489:⋅
5468:±
5420:for some
5319:−
5313:
5282:π
5198:≥
5177:≤
5147:∑
5118:∈
5099:…
5061:Δ
5023:ℓ
4984:
4941:Δ
4904:Δ
4876:∑
4855:Δ
4819:Δ
4380:≤
4367:≤
4364:⋯
4361:≤
4348:≤
4335:≤
4316:∣
4301:∈
4282:…
4248:∗
4244:Δ
4186:⋯
4099:−
4085:⋯
4048:−
4029:−
3997:⋮
3671:Δ is the
3660:(0, 0, 1)
3656:(0, 1, 0)
3652:(1, 0, 0)
3646:Δ is the
3581:−
3541:↠
3533:−
3526:Δ
3457:∑
3453:↦
3434:…
3377:−
3265:∑
3261:↦
3242:…
3182:orthoplex
3040:…
3017:≥
2966:∑
2931:∈
2912:…
2879:Δ
2867:given by
2847:standard
2834:2-simplex
2551:tesseract
2504:3.( )∨( )
2488:{ } ∨ ( )
2102:9-simplex
1973:8-simplex
1854:7-simplex
1745:6-simplex
1646:5-simplex
1552:4-simplex
1469:3-simplex
1396:2-simplex
1333:1-simplex
1270:0-simplex
1053:7-simplex
656:−
647:…
624:≥
597:−
583:⋯
552:∈
503:standard
432:…
407:≥
398:θ
375:θ
354:∑
321:θ
314:⋯
292:θ
246:−
230:…
214:−
163:…
48:simplices
44:simplexes
42:(plural:
16576:Topology
16544:Topics:
16507:demicube
16472:polytope
16466:Uniform
16227:600-cell
16223:120-cell
16176:Demicube
16150:Pentagon
16130:Triangle
15958:Category
15934:See also
15734:Manifold
15639:(2004).
15570:(1973).
15498:(1976).
14184:Polytope
14102:3-sphere
14095:See also
14049:nitrogen
14037:fluorine
13954:mixtures
13893:category
13891:(in the
13431:-simplex
13114:manifold
12584:boundary
12528:homology
12481:hexagram
11570:. Every
11548:Topology
10908:Any two
10904:-simplex
7838:− 1) / 2
6372:dividing
5377:through
5248:facets.
4594:for the
4580:interior
3611:Examples
3401:polytope
3354:-simplex
3342:-simplex
3116:, where
2859:-simplex
2851:-simplex
2393:polytope
1400:triangle
1151:Schläfli
1029:f-vector
877:vertices
853:(of the
826:nonempty
816:Elements
683:topology
196:vectors
133:vertices
119:polytope
111:-simplex
86:triangle
64:polytope
52:triangle
36:geometry
16481:simplex
16451:10-cube
16218:24-cell
16204:16-cell
16145:Hexagon
15999:regular
15856:Simplex
15794:Fractal
15383:3605876
15323:3072403
15162:, with
14435:2315353
14329:(ed.).
14069:halogen
14033:a point
14025:p-block
13907:-theory
12553:affine
12530:called
12518:called
11532:of the
11520:of the
8441:, ...,
8177:√
7829:, ...,
6637:integer
6399:or, if
6224:is the
5410:, ...,
4663:, ...,
4602:on the
4598:of the
3338:affine
3204:, ...,
2516:4 ⋅ ( )
2508:{3}∨( )
2227:
2088:
2085:
1959:
1956:
1953:
1840:
1837:
1834:
1831:
1731:
1728:
1725:
1722:
1719:
1632:
1629:
1626:
1623:
1620:
1617:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1191:(cells)
1182:(faces)
1173:(edges)
1155:Coxeter
1075:,4,6,4,
1031:for an
883:, the (
871:of the
831:of the
824:of any
735:simplex
705:History
491:is the
142:points
125:of its
40:simplex
16421:9-cube
16371:8-cube
16321:7-cube
16278:6-cube
16248:5-cube
16135:Square
16012:Family
15813:shapes
15649:
15580:
15552:
15531:
15506:
15439:
15414:
15389:
15381:
15321:
14845:
14807:
14785:-path
14551:
14522:
14500:-path
14470:Every
14433:
14250:
14059:forms
14057:carbon
14055:, and
14041:oxygen
13582:scheme
13517:
13507:
13372:where
12586:of an
12557:-chain
12483:{6/2}.
11563:to an
11395:where
10934:cos(1/
10171:
10054:
10038:
9551:
9447:
9261:is an
8638:vector
8424:of an
8422:volume
8416:Volume
7841:, are
6635:is an
6234:matrix
6053:, and
6021:where
5705:, and
5310:arccos
5143:
5133:
5041:ball.
4847:where
4596:action
4584:strict
4226:order,
3702:5-cell
3635:(0, 1)
3631:(1, 0)
2962:
2952:
2586:circle
2580:These
1556:5-cell
1254:
1245:
1236:
1227:
1218:
1209:
1200:
998:coface
892:facets
829:subset
350:
340:
100:5-cell
16140:p-gon
15917:Eight
15912:Seven
15892:Three
15769:Krull
15472:(PDF)
15461:(PDF)
15387:S2CID
15379:JSTOR
15319:JSTOR
14431:JSTOR
14366:arXiv
14227:Notes
14071:atom.
13654:with
13418:to a
11568:-ball
11555:, an
11530:facet
11268:faces
10729:, at
10597:) is
8636:is a
7843:2 × 2
7809:1 × 1
6689:2 × 2
6413:2 × 2
6393:1 × 1
6357:1 × 1
6353:2 × 2
3515:image
3403:with
3170:facet
2855:unit
2490:. An
2482:. An
2454:. An
2433:ABCDE
1274:point
1251:faces
1242:faces
1233:faces
1224:faces
1215:faces
1206:faces
1197:faces
1188:faces
1179:faces
1170:faces
1161:faces
1147:Name
1079:) = (
1071:) = (
1063:) = (
1018:facet
996:is a
988:) of
881:edges
869:-face
727:Latin
113:is a
95:, and
72:point
16498:cube
16171:Cube
16001:and
15902:Five
15897:Four
15877:Zero
15811:and
15647:ISBN
15578:ISBN
15550:ISBN
15529:ISBN
15504:ISBN
15437:ISBN
15412:ISBN
15241:<
15203:and
15187:<
15174:<
14335:The
14248:ISBN
14082:and
14045:bent
14029:Neon
14001:and
13779:ring
13614:Spec
13455:+ 1)
12596:− 1)
12580:− 1)
11641:+ 1)
11621:+ 1)
11606:+ 1)
11538:+ 1)
11526:+ 1)
11512:The
11499:+ 1)
11485:The
11292:− 1)
11284:− 1)
10926:− 1)
10914:− 1)
10323:and
8420:The
6674:even
6271:diag
5673:for
5435:for
5394:− 1)
3720:and
3687:and
3658:and
3633:and
3178:+ 1)
3090:The
2853:(or
2845:The
2566:= −1
2510:. A
2464:+ 1)
2444:+ 1)
2425:ABCD
2421:ABCD
2380:2047
2367:165
2364:330
2361:462
2358:462
2355:330
2352:165
2231:1023
2215:120
2212:210
2209:252
2206:210
2203:120
2070:126
2067:126
1128:− 2)
1112:− 1)
1098:-th
1055:is (
1016:and
1014:Face
844:face
820:The
740:The
685:and
497:− 1)
483:The
478:− 1)
268:are
188:are
38:, a
16047:(p)
15907:Six
15887:Two
15882:One
15659:PDF
15657:As
15605:≥ 5
15371:doi
15344:hdl
15311:doi
15307:109
14735:to
14677:to
14423:doi
14395:doi
14019:In
13997:In
13990:In
13983:of
13971:In
13948:In
13937:In
13922:In
13800:).
12738:of
12545:of
12510:In
12490:or
12394:log
12364:log
11613:+ 1
11149:to
10940:.
10932:of
10476:det
10383:det
10095:of
9946:det
9914:Vol
9850:to
9804:If
9745:Vol
9711:If
9382:of
9330:of
9134:det
9080:is
9044:.
8739:det
8506:det
8452:is
8076:= 3
7754:sin
7724:cos
7694:sin
7664:cos
7619:sin
7589:cos
7559:sin
7529:cos
7484:sin
7454:cos
7424:sin
7394:cos
7349:sin
7319:cos
7289:sin
7259:cos
7144:cos
7116:sin
7076:sin
7045:cos
6995:cos
6967:sin
6927:sin
6896:cos
6875:= 4
6687:is
6562:cos
6521:sin
6478:sin
6434:cos
6405:odd
6403:is
6378:+ 1
6368:+ 1
6355:or
6196:by
5368:by
5362:of
4981:log
4885:max
4800:max
4661:/3!
3724:in
3691:in
3662:in
3637:in
3620:in
3095:+ 1
3085:≥ 0
2836:in
2514:is
2506:or
2452:( )
2413:ABC
2409:ABC
2387:An
2373:11
2370:55
2349:55
2346:11
2221:10
2218:45
2200:45
2197:10
2092:511
2076:36
2073:84
2064:84
2061:36
1963:255
1944:28
1941:56
1938:70
1935:56
1932:28
1844:127
1822:21
1819:35
1816:35
1813:21
1710:15
1707:20
1704:15
1608:10
1605:10
1279:( )
1259:Sum
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8873:)
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