1285:
564:
limit positions (in the elastic-edge transformation they are rigid). The passage of the Jessen's icosahedron point (important in the elastic-edge transformation because the short edges reach their minimum size and begin to lengthen again) is not visible at all. The helical contraction of the cuboctahedron into successively smaller-radius polyhedra is visible as it occurs in the rigid-edge cuboctahedron transformation (and in the elastic-edge transformation it would be quite similar).
1421:
643:
149:
2135:, since there are two choices of diagonal on which to begin folding the square faces. The red-and-yellow animation in this article cycles endlessly through the same chiral form of the rigid-edge cuboctahedron transformation. It could (but does not) cycle through both chiral forms of the rigid-edge transformation alternately, departing from the cuboctahedron by folding the opposite set of square face diagonals each time. This would take it all the way around the topological
2082:. He discovered the symmetry transformations of the cuboctahedron, understood their relationship to the tensegrity icosahedron, and even gave demonstrations of the rigid-edge cuboctahedron transformation before audiences (in the days before computer-rendered animations). His demonstration with commentary of the "vector equilibrium", as he called the cuboctahedron, is still far more illuminating than the animations in this article.
1407:
629:
135:
1414:
636:
142:
1428:
650:
156:
2576:, p. 203; "As Clinton observed in his paper on expanding rigid structures, each triangle is subject to a translation-rotation along its symmetry axis. When starting from the position in the octahedron, these axes are the four triangular symmetry axes of the octahedron. When describing cylinders about the triangles along the axes, each vertex common to two triangles moves along the intersecting
2440:; that kinematics is beyond the scope of this article, except to note that it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations described in this article): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the
1383:
its edges are elastic (able to lengthen under tension). Forcing the polyhedron away from its stable resting shape (in either direction) involves stretching its 24 short edges slightly and equally. Force applied to any pair of parallel long edges, to move them closer together or farther apart, is transferred automatically to stretch
2432:, pp. 50–52, §3.7: Coordinates for the vertices of the regular and quasi-regular solids; describes the cuboctahedron transformation in Euclidean 3-space; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the
2712:
inscribed in the cube) at all times; actually, Fuller could have rotated any of the kinematic polyhedra in an inscribing cube in this way: the entire cuboctahedron transformation cycle takes place inside an inscribing cube of varying edge length, with the 12 vertices always on the surface of the cube.
2711:
transformation (which he observed in the tensegrity icosahedron), but he does show how a rigid regular icosahedron can be rotated inside an inscribing "vector edge cube" (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron
2032:
Finally, both transformations are pure abstractions, the two limit cases of an infinite family of cuboctahedron transformations in which there are two elasticity parameters and no requirement that one of them be 0. Neither limit case is apt to apply perfectly to most real tensegrity structures, which
563:
of the elastic-edge transformation: it does not illustrate the Jessen's icosahedron's long edges at all, and the short edges do not lengthen (as they do by 15% at the elastic-edge transformation's cuboctahedron and octahedron limit positions); instead, the (invisible) long edges shorten by 15% at the
2020:
cuboctahedron transformations. In particular, the vertices always move in helices toward the center as the cuboctahedron transforms into the octahedron, and the Jessen's icosahedron (with 90° dihedral angles and three invariant orthogonal planes) is always the median point, stable to the extent that
2143:
chiral forms. But instead, it reverses at the limit cases (cuboctahedron and octahedron), and endlessly shuffles back and forth along the same half of the Mobius loop, never setting foot on the other half of it. Watch the blue-and-white animation carefully and see if you can tell if it is doing the
2015:
The rigid-edge and elastic-edge cuboctahedron transformations differ only in having reciprocal parameters: in the elastic-edge transformation the Jessen's icosahedron's short edges stretch and its long edges are rigid, and in the rigid-edge transformation its long edges compress and its short edges
1382:
Although the transformation is described above as a contraction of the cuboctahedron, the stable equilibrium point of the tensegrity is the Jessen's icosahedron; the tensegrity icosahedron resists being deformed from that shape and can only be forced to expand or contract from it to the extent that
558:
Continuous transformation between the cuboctahedron and the octahedron pausing at the vertex position of the regular icosahedron. The vertex position of the Jessen's icosahedron lies between the regular icosahedron and the octahedron (but the animation does not pause there). This is an animation of
2037:
the cables and the struts, giving their actual behavior metrics which are non-trivial to calculate. In engineering practice, only a tiny amount of elasticity is required to allow a significant degree of motion, so most tensegrity structures are constructed to be "drum-tight" using nearly inelastic
2028:
because it comes closest to modeling how most actual tensegrity icosahedron structures behave. However, one could certainly construct a tensegrity icosahedron in which the short edges (cables) were perfectly inelastic, and the long edges (struts) were compressable springs. Such a tensegrity would
1387:
the short edges uniformly, shrinking the polyhedron from its medium-size Jessen's icosahedron toward the smaller octahedron, or expanding it toward the larger regular icosahedron and still larger cuboctahedron, respectively. Releasing the force causes the polyhedron to spring back to its Jessen's
2202:
which lie in concave troughs). The 6 long edges occur as 3 orthogonal pairs of parallel edges on opposite sides of the polyhedron, and in their resting state each parallel pair is exactly as far apart as 1/2 their length; each resting long edge lies 1/4 its length from the center, defined as the
1391:
In the elastic-edge transformation the cuboctahedron edges are not rigid (though the Jessen's icosahedron's 6 long edges are). What the cuboctahedron actually transforms into is a regular icosahedron of shorter radius and shorter edge length, a Jessen's icosahedron of still shorter radius and
537:
contraction. The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles. The 12 vertices of the cuboctahedron spiral inward (toward the center) and move closer together until they reach the points where they form a regular
110:
Adding a central vertex, connected by rigid edges to all the other vertices, subdivides the cuboctahedron into square pyramids and tetrahedra, meeting at the central vertex. Unlike the cuboctahedron itself, the resulting system of edges and joints is rigid, and forms part of the infinite
40:
1182:
1392:(minimum) edge length, and finally an octahedron of still shorter radius but the same (maximum) edge length as the cuboctahedron (but only after the edges have shortened and lengthened again, and come together in coincident pairs).
1946:
545:
The general cuboctahedron transformation can be parameterized along a continuum of special-case transformations with two limit-cases: one in which the edges of the cuboctahedron are rigid, and one in which they are elastic.
893:
2694:
555:
2518:
Uberti, R.; Janse van
Rensburg, E. J.; Orlandini, E.; Tesi, M. C.; Whittington, S. G. (1998), "Minimal links in the cubic lattice", in Whittington, Stuart G.; Sumners, Witt De; Lodge, Timothy (eds.),
1787:
1543:
1030:
1281:
be deformed into symmetrical polyhedra along that spectrum from cuboctahedron to octahedron. It is called the tensegrity icosahedron because its median stable form is the Jessen's icosahedron.
2321:
the polyhedron, forcing all 3 pairs toward each other similarly until they coincide and become the 3 orthogonal axes of the regular octahedron. At that point (the limit of contraction) the
2593:
There is a continuous motion of Q (the cuboctahedron without square faces) shown in Fig. 5a onto the octahedron W satisfying the following conditions for each face F of Q, e.g. F = 𝚫a
1874:
1612:
1493:
952:
828:
1979:
1576:
1248:
1215:
1114:
990:
2355:. In a Jessen's icosahedron of unit short radius one set of these three rectangles (the set in which the Jessen's icosahedron's long edges are the rectangles' long edges) measures
2198:
has only 8 of the regular icosahedron's 20 equilateral triangle faces and 24 of its 30 edges, but it also has 12 isosceles triangle faces that meet in pairs at 6 longer edges (its
2263:, not tension load like the short edges. Unlike the elastic short edges which stretch and lengthen slightly, the long edges must resist compression perfectly and not shorten.
263:
282:
224:
205:
787:
760:
733:
706:
2508:
does not occur, often because the cuboctahedron limit position is never actually reached; even when it is, the structure is constrained by the struts to one chiral form.
302:
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253:
244:
234:
215:
195:
2379:
297:
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268:
258:
239:
229:
210:
200:
1291:. All dihedral angles are 90°. The vertices of the inscribed cube are the centers of the equilateral triangle faces. The polyhedron is a construct of the lengths
99:
When interpreted as a framework of rigid flat faces, connected along the edges by hinges, the cuboctahedron is a rigid structure, as are all convex polyhedra, by
2002:
1833:
1810:
1737:
1709:
1686:
1663:
1640:
1122:
1076:
1053:
916:
2408:, §3. Pyritohedral Symmetry; "The pyritohedral 3D symmetry group is the unique polyhedral point group that is neither a rotation group nor a reflection group."
2185:
at both the cuboctahedron limit (where they are the diagonals of the square faces) and at the regular octahedron limit (where they are the long diameter).
1882:
836:
602:
What the cuboctahedron with rigid edges actually can transform into (and through) is a regular icosahedron from which 6 edges are missing (a
71:
and consequently its vertices can be repositioned by folding (changing the dihedral angle) at edges and face diagonals. The cuboctahedron's
2786:
591:
those other polyhedra, and they cannot transform into each other (if they have rigid edges), because unlike the cuboctahedron they
17:
103:. However, when the faces are removed, leaving only rigid edges connected by flexible joints at the vertices, the result is not a
2423:
2447:
1745:
1501:
2867:
2535:
2276:
the polyhedron, forcing all 3 pairs apart similarly until they become chords of the regular icosahedron, at which point the
2066:
for example) he did not have any mathematics to give. But he was the first to stress the importance of the cuboctahedron's
998:
67:, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have
2770:
2691:
and gives a continuous flattening motion for its 2-skeleton , which is related to the
Jitterbug by Buckminster Fuller."
1329:
2471:
2411:
2207:(so the long edge length is 4, and their distance apart at rest is 2). In the resting state the short edge length is
614:
coincident rigid edges connecting each pair of vertices (formed by making pairs of cuboctahedron vertices coincide).
2847:
2703:, Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire
2055:
542:; and they continue to spiral toward each other until they coincide in pairs as the 6 vertices of the octahedron.
2941:
2381:. These three rectangles are the shortest possible representation of the Borromean rings using only edges of the
2727:
1846:
1584:
1465:
924:
800:
107:(unlike polyhedra whose faces are all triangles, to which Cauchy's theorem applies despite the missing faces).
100:
2582:
2308:≈ 2.828 the long edges will become the square face diagonals of the cuboctahedron (at the limit of expansion).
2161:
is their mid-edge-to-center distance. In the rigid-edge cuboctahedron transformation the rigid-edge length is
1954:
1551:
1223:
1190:
1089:
965:
529:
In their spatial relationships the cuboctahedron, icosahedron, Jessen's icosahedron, and octahedron nest like
2946:
2492:, multiple kinematics can happen: the polyhedron can depart the cuboctahedron in either direction along the
584:, in the sense that the polyhedron's vertices take on the vertex positions of those polyhedra successively.
2936:
2246:
at the cuboctahedron limit of expansion, where the cuboctahedron's radius is also its edge length (it is
2067:
2522:, IMA Volumes in Mathematics and its Applications, vol. 103, New York: Springer, pp. 89–100,
2505:
506:
The cuboctahedron can be transformed cyclically through four polyhedra, repeating the cycle endlessly.
2493:
607:
2441:
2731:
2195:
1447:
1288:
768:
741:
714:
687:
669:
577:
539:
515:
175:
80:
2489:
2437:
2358:
187:
88:
2919:
2851:
2025:
1270:
2809:
2559:
Clinton, J.D. (1971). "A geometric transformation concept for expanding rigid structures".
2545:
2247:
2075:
310:
1177:{\displaystyle \left({\tfrac {\sqrt {10+2{\sqrt {5}}}}{4}}\right){\sqrt {6}}\approx 2.330}
8:
2823:
2046:
is a kinematic cuboctahedron transformation with reciprocal small elasticity parameters.
2024:
The elastic-edge cuboctahedron transformation is usually given as the mathematics of the
1442:
1284:
664:
596:
573:
170:
76:
68:
60:
2759:
2741:
2481:
2239:: in the elastic-edge transformation from 2 at the octahedron limit of contraction to 2
2063:
2059:
2054:
The twisting, expansive-contractive transformations between these polyhedra were named
1987:
1818:
1795:
1722:
1694:
1671:
1648:
1625:
1452:
1061:
1038:
901:
674:
180:
2902:
2885:
2855:
2144:
same thing, or if it is performing the entire rigid-edge cuboctahedron transformation.
75:
is noteworthy in that its vertices can be repositioned to the vertex positions of the
52:. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.
2863:
2800:
2781:
2766:
2531:
2344:
The vertices of the regular icosahedron form five sets of three concentric, mutually
603:
45:
2328:≈ 2.449 short edges have also stretched (not contracted!) to their limit length of 2
606:), a Jessen's icosahedron in which the 6 reflex edges are missing or elastic, and a
2897:
2832:
2795:
2523:
2348:
2301:≈ 2.472. If the short edges are elastic enough that they can be stretched ~15% to 2
554:
2805:
2609:
F is rotated and moved toward along the line l joining the centroids of F and 𝚫v
2541:
2382:
2352:
1941:{\displaystyle {\tfrac {2{\sqrt {10+2{\sqrt {5}}}}}{1+{\sqrt {5}}}}\approx 2.351}
2642:
2527:
2459:
2399:
888:{\displaystyle \left({\tfrac {1+{\sqrt {5}}}{2}}\right){\sqrt {6}}\approx 3.963}
2837:
2818:
27:
Symmetrical transformations of the cuboctahedron into related uniform polyhedra
2745:
2501:
2099:
2016:
are rigid. Everything in the descriptions above except the metrics applies to
511:
2930:
2884:
Koca, Nazife; Al-Mukhaini, Aida; Koca, Mehmet; Al Qanobi, Amal (2016-12-01).
2874:
2782:"The complete set of Jitterbug transformers and the analysis of their motion"
2335:≈ 2.828, and now coincide in pairs as the 12 edges of the regular octahedron.
2062:. Fuller did not give any mathematics; like many great geometers before him (
2010:
1437:
659:
530:
165:
64:
2480:, cuboctahedron as vector equilibrium, a completely unstable condition; as
104:
2819:"Continuous flattening of the 2-dimensional skeleton of a regular 24-cell"
2625:
F always touches the cylinder T(F), that is, F is always orthogonal to l.
2136:
2071:
1420:
642:
148:
112:
2420:, p. 145, 4. Pyritohedral Group and Related Polyhedra; see Table 1.
2345:
1262:
581:
84:
72:
49:
2517:
2079:
2496:, folding on either set of square diagonals to select either of two
2591:, p. 13, §4. From the 24-cell onto an octahedron; "Lemma 4.2.
2488:
equilibrium point of the cycle. At that point, as at an airplane's
2433:
2103:
507:
2860:
Bridges
Leeuwarden: Mathematics, Music, Art, Architecture, Culture
538:
icosahedron; they move slightly closer together until they form a
2688:
519:
2707:
transformation cycle; in this film, he does not demonstrate the
1406:
628:
134:
2654:
2630:
2497:
2260:
2132:
523:
2228:≈ 1.414, and the long radius of the polyhedron expands by the
1413:
635:
141:
2577:
2484:
was the first to realize, the cuboctahedron is the antipodal
2444:(a smaller regular icosahedron nested inside the octahedron).
1427:
649:
534:
155:
2883:
2678:
2456:, 4.1 Construction of the vertices of the pseudoicosahedron.
2453:
2417:
2259:
The force places the long edges under compression load like
2221:≈ 2.236. The altitude of the isosceles triangles at rest is
2283:≈ 2.449 short edges have stretched only ~1% to a length of
1265:
polyhedron which embodies and enforces the closely related
39:
2563:. Vol. CR-1735. Washington, D.C.: Southern Ill. Univ.
2317:
Forcing any pair of parallel long edges toward each other
2153:
The Jessen's icosahedron's long reflex edges have length 4
2011:
Duality of the rigid-edge and elastic-edge transformations
2561:
NASA Report: Advanced structural geometry studies, Part 2
2074:, intuiting that it plays a fundamental role not only in
118:
1782:{\displaystyle {\tfrac {4}{1+{\sqrt {5}}}}\approx 1.236}
1538:{\displaystyle {\tfrac {8}{1+{\sqrt {5}}}}\approx 2.472}
2567:
2272:
Forcing any pair of parallel long edges slightly apart
2021:
there is resistance to the stretching or compressing.
1887:
1750:
1506:
1131:
1003:
845:
2852:"The Borromean Rings: A video about the New IMU logo"
2361:
2029:
perform the rigid-edge cuboctahedron transformation.
1990:
1957:
1885:
1849:
1821:
1798:
1748:
1725:
1697:
1674:
1651:
1628:
1587:
1554:
1504:
1468:
1226:
1193:
1125:
1092:
1064:
1041:
1001:
968:
927:
904:
839:
803:
771:
744:
717:
690:
2666:
2070:
which he applied structurally (and patented) as the
1025:{\displaystyle {\tfrac {\sqrt {6}}{2}}\approx 1.225}
2214:≈ 2.449, and the long radius (center to vertex) is
94:
2758:
2373:
1996:
1973:
1940:
1868:
1827:
1804:
1781:
1731:
1703:
1680:
1657:
1634:
1606:
1570:
1537:
1487:
1242:
1209:
1176:
1108:
1070:
1047:
1024:
984:
946:
910:
887:
822:
781:
754:
727:
700:
599:as a consequence of having only triangular faces.
572:symmetrically transforms the cuboctahedron into a
2928:
2862:, London: Tarquin Publications, pp. 63–70,
1256:
2890:Sultan Qaboos University Journal for Science
2468:, pp. 11–19, §2. Spherical tensegrities.
2078:but in the dimensional relationships between
559:the rigid-edge cuboctahedron transformation,
2845:
2651:, pp. 16–17, Elasticity Multiplication.
2405:
2049:
549:
2886:"Symmetry of the Pyritohedron and Lattices"
2875:The Borromean Rings: A new logo for the IMU
2787:Computers and Mathematics with Applications
2171:, and the reflex edge chords shorten from 4
2504:). In an actual tensegrity structure this
38:
2901:
2836:
2799:
1869:{\displaystyle 2{\sqrt {2}}\approx 2.828}
1607:{\displaystyle 2{\sqrt {2}}\approx 2.828}
1488:{\displaystyle 2{\sqrt {2}}\approx 2.828}
1273:has a dynamic structural rigidity called
1267:elastic-edge cuboctahedron transformation
947:{\displaystyle 2{\sqrt {3}}\approx 3.464}
823:{\displaystyle 2{\sqrt {3}}\approx 3.464}
2816:
2779:
2687:, Abstract; "This article addresses the
2684:
2672:
2588:
2573:
2520:Topology and Geometry in Polymer Science
1974:{\displaystyle {\sqrt {5}}\approx 2.236}
1571:{\displaystyle {\sqrt {6}}\approx 2.449}
1283:
1243:{\displaystyle {\sqrt {3}}\approx 1.732}
1210:{\displaystyle {\sqrt {5}}\approx 2.236}
1109:{\displaystyle {\sqrt {6}}\approx 2.449}
985:{\displaystyle {\sqrt {3}}\approx 1.732}
553:
89:pyritohedral symmetry of the icosahedron
2726:
2558:
2511:
2429:
570:rigid-edge cuboctahedron transformation
14:
2929:
2756:
2740:
2700:
2660:
2648:
2636:
2477:
2465:
119:Cyclical cuboctahedron transformations
44:Progressions between a cuboctahedron,
2044:tensegrity icosahedron transformation
1398:Elastic-edge kinematic cuboctahedra
587:The cuboctahedron does not actually
2922:. International Mathematical Union.
2817:Itoh, Jin-ichi; Nara, Chie (2021).
24:
2765:. University of California Press.
620:Rigid-edge kinematic cuboctahedra
25:
2958:
2912:
2903:10.24200/squjs.vol21iss2pp139-149
2500:subcycles (connected in a single
2102:of these polyhedra occurs in the
2879:International Mathematical Union
2736:(3rd ed.). New York: Dover.
2033:usually have some elasticity in
1426:
1419:
1412:
1405:
648:
641:
634:
627:
300:
295:
290:
285:
280:
271:
266:
261:
256:
251:
242:
237:
232:
227:
222:
213:
208:
203:
198:
193:
154:
147:
140:
133:
95:Rigid and kinematic cuboctahedra
2761:Geodesic Math and How to Use It
2719:
2552:
2338:
2311:
2266:
2253:
2188:
2131:Notice that the contraction is
2100:static instance of this nesting
1330:Legendre's three-square theorem
33:Kinematics of the cuboctahedron
2147:
2125:
2109:
2092:
13:
1:
2392:
510:the transformation follows a
2801:10.1016/0898-1221(89)90160-0
2579:curve of the two cylinders."
2122:times the octahedron radius.
2115:The cuboctahedron radius is
1838:
1714:
1617:
1457:
1433:
1401:
1394:
1081:
957:
792:
679:
655:
623:
616:
308:
185:
161:
129:
122:
7:
2873:; see the video itself at "
2528:10.1007/978-1-4612-1712-1_9
2068:radial equilateral symmetry
1388:icosahedron resting shape.
1257:Elastic-edge transformation
782:{\displaystyle {\sqrt {6}}}
755:{\displaystyle {\sqrt {6}}}
728:{\displaystyle {\sqrt {6}}}
701:{\displaystyle {\sqrt {6}}}
610:of the octahedron that has
10:
2963:
2838:10.1007/s00022-021-00575-6
2750:Everything I Know Sessions
2663:, p. 12, Equilibrium.
2639:, p. 14, Equilibrium.
2374:{\displaystyle 2\times 4}
2056:Jitterbug transformations
2050:Jitterbug transformations
1397:
619:
550:Rigid-edge transformation
186:
125:
87:, in accordance with the
37:
32:
2780:Verheyen, H. F. (1989).
2406:Gunn & Sullivan 2008
2085:
18:Jitterbug transformation
2506:nondeterministic choice
2139:each time it traversed
516:orientable double cover
126:Kinematic cuboctahedra
2942:Quasiregular polyhedra
2742:Fuller, R. Buckminster
2375:
2026:tensegrity icosahedron
1998:
1975:
1942:
1870:
1829:
1806:
1783:
1733:
1705:
1682:
1659:
1636:
1608:
1572:
1539:
1489:
1379:
1275:infinitesimal mobility
1271:tensegrity icosahedron
1244:
1211:
1178:
1110:
1072:
1049:
1026:
986:
948:
912:
889:
824:
783:
756:
729:
702:
565:
2854:, in Sarhangi, Reza;
2757:Kenner, Hugh (1976).
2376:
1999:
1976:
1943:
1871:
1830:
1807:
1784:
1734:
1706:
1683:
1660:
1637:
1609:
1573:
1540:
1490:
1287:
1245:
1212:
1179:
1111:
1073:
1050:
1027:
987:
949:
913:
890:
825:
784:
757:
730:
703:
557:
533:and are related by a
2947:Tensile architecture
2746:"Vector Equilibrium"
2685:Itoh & Nara 2021
2589:Itoh & Nara 2021
2549:; see Table 2, p. 97
2359:
2248:radially equilateral
2196:Jessen's icosahedron
2076:structural integrity
1988:
1955:
1883:
1847:
1819:
1796:
1746:
1723:
1695:
1672:
1649:
1626:
1585:
1552:
1502:
1466:
1448:Jessen's icosahedron
1289:Jessen's icosahedron
1224:
1191:
1123:
1090:
1062:
1039:
999:
966:
925:
902:
837:
801:
769:
742:
715:
688:
670:Jessen's icosahedron
578:Jessen's icosahedron
540:Jessen's icosahedron
176:Jessen's icosahedron
81:Jessen's icosahedron
2824:Journal of Geometry
2351:, whose edges form
2106:regular 4-polytope.
1443:Regular icosahedron
665:Regular icosahedron
597:structural rigidity
574:regular icosahedron
518:of the octahedron.
171:Regular icosahedron
77:regular icosahedron
69:structural rigidity
2937:Archimedean solids
2482:Buckminster Fuller
2371:
2064:Alicia Boole Stott
2060:Buckminster Fuller
1994:
1971:
1938:
1930:
1866:
1825:
1802:
1779:
1771:
1729:
1701:
1678:
1655:
1632:
1604:
1568:
1535:
1527:
1485:
1453:Regular octahedron
1380:
1240:
1207:
1174:
1155:
1106:
1068:
1045:
1022:
1014:
982:
944:
908:
885:
866:
820:
779:
752:
725:
698:
675:Regular octahedron
566:
181:Regular octahedron
83:, and the regular
2920:"Borromean Rings"
2869:978-0-9665201-9-4
2848:Sullivan, John M.
2733:Regular Polytopes
2537:978-0-387-98580-0
2438:Euclidean 4-space
2349:golden rectangles
2300:
2205:unit short radius
2008:
2007:
2004:
1997:{\displaystyle 2}
1981:
1963:
1948:
1929:
1926:
1911:
1909:
1876:
1858:
1835:
1828:{\displaystyle 0}
1812:
1805:{\displaystyle 1}
1789:
1770:
1767:
1739:
1732:{\displaystyle 2}
1711:
1704:{\displaystyle 4}
1688:
1681:{\displaystyle 4}
1665:
1658:{\displaystyle 4}
1642:
1635:{\displaystyle 4}
1614:
1596:
1578:
1560:
1545:
1526:
1523:
1495:
1477:
1254:
1253:
1250:
1232:
1217:
1199:
1184:
1166:
1154:
1150:
1148:
1116:
1098:
1078:
1071:{\displaystyle 0}
1055:
1048:{\displaystyle 1}
1032:
1013:
1009:
992:
974:
954:
936:
918:
911:{\displaystyle 4}
895:
877:
865:
859:
830:
812:
789:
777:
762:
750:
735:
723:
708:
696:
604:pseudoicosahedron
504:
503:
57:
56:
46:pseudoicosahedron
16:(Redirected from
2954:
2923:
2907:
2905:
2872:
2856:Séquin, Carlo H.
2842:
2840:
2813:
2803:
2794:(1–3): 203–250.
2776:
2764:
2753:
2737:
2713:
2698:
2692:
2682:
2676:
2670:
2664:
2658:
2652:
2646:
2640:
2634:
2628:
2586:
2580:
2571:
2565:
2564:
2556:
2550:
2548:
2515:
2509:
2475:
2469:
2463:
2457:
2454:Koca et al. 2016
2451:
2445:
2427:
2421:
2418:Koca et al. 2016
2415:
2409:
2403:
2386:
2380:
2378:
2377:
2372:
2342:
2336:
2334:
2333:
2327:
2326:
2315:
2309:
2307:
2306:
2299:
2297:
2296:
2293:
2290:
2284:
2282:
2281:
2270:
2264:
2257:
2251:
2245:
2244:
2238:
2237:
2227:
2226:
2220:
2219:
2213:
2212:
2192:
2186:
2181:
2180:
2167:
2166:
2151:
2145:
2129:
2123:
2121:
2120:
2113:
2107:
2096:
2003:
2001:
2000:
1995:
1984:
1980:
1978:
1977:
1972:
1964:
1959:
1951:
1947:
1945:
1944:
1939:
1931:
1928:
1927:
1922:
1913:
1912:
1910:
1905:
1894:
1888:
1879:
1875:
1873:
1872:
1867:
1859:
1854:
1843:
1834:
1832:
1831:
1826:
1815:
1811:
1809:
1808:
1803:
1792:
1788:
1786:
1785:
1780:
1772:
1769:
1768:
1763:
1751:
1742:
1738:
1736:
1735:
1730:
1719:
1710:
1708:
1707:
1702:
1691:
1687:
1685:
1684:
1679:
1668:
1664:
1662:
1661:
1656:
1645:
1641:
1639:
1638:
1633:
1622:
1613:
1611:
1610:
1605:
1597:
1592:
1581:
1577:
1575:
1574:
1569:
1561:
1556:
1548:
1544:
1542:
1541:
1536:
1528:
1525:
1524:
1519:
1507:
1498:
1494:
1492:
1491:
1486:
1478:
1473:
1462:
1430:
1423:
1416:
1409:
1395:
1377:
1375:
1374:
1371:
1368:
1362:
1360:
1359:
1356:
1353:
1347:
1345:
1344:
1341:
1338:
1327:
1326:
1321:
1320:
1315:
1314:
1309:
1308:
1303:
1302:
1297:
1296:
1249:
1247:
1246:
1241:
1233:
1228:
1220:
1216:
1214:
1213:
1208:
1200:
1195:
1187:
1183:
1181:
1180:
1175:
1167:
1162:
1160:
1156:
1149:
1144:
1133:
1132:
1119:
1115:
1113:
1112:
1107:
1099:
1094:
1086:
1077:
1075:
1074:
1069:
1058:
1054:
1052:
1051:
1046:
1035:
1031:
1029:
1028:
1023:
1015:
1005:
1004:
995:
991:
989:
988:
983:
975:
970:
962:
953:
951:
950:
945:
937:
932:
921:
917:
915:
914:
909:
898:
894:
892:
891:
886:
878:
873:
871:
867:
861:
860:
855:
846:
833:
829:
827:
826:
821:
813:
808:
797:
788:
786:
785:
780:
778:
773:
765:
761:
759:
758:
753:
751:
746:
738:
734:
732:
731:
726:
724:
719:
711:
707:
705:
704:
699:
697:
692:
684:
652:
645:
638:
631:
617:
580:, and a regular
500:
498:
497:
494:
491:
485:
483:
482:
479:
476:
470:
468:
467:
464:
461:
453:
451:
450:
447:
444:
438:
436:
435:
432:
429:
423:
421:
420:
417:
414:
406:
404:
403:
400:
397:
391:
389:
388:
385:
382:
376:
374:
373:
370:
367:
359:
357:
356:
353:
350:
344:
342:
341:
338:
335:
329:
327:
326:
323:
320:
311:Mirror dihedrals
305:
304:
303:
299:
298:
294:
293:
289:
288:
284:
283:
276:
275:
274:
270:
269:
265:
264:
260:
259:
255:
254:
247:
246:
245:
241:
240:
236:
235:
231:
230:
226:
225:
218:
217:
216:
212:
211:
207:
206:
202:
201:
197:
196:
158:
151:
144:
137:
123:
101:Cauchy's theorem
42:
30:
29:
21:
2962:
2961:
2957:
2956:
2955:
2953:
2952:
2951:
2927:
2926:
2918:
2915:
2910:
2870:
2846:Gunn, Charles;
2773:
2752:. Philadelphia.
2728:Coxeter, H.S.M.
2722:
2717:
2716:
2699:
2695:
2683:
2679:
2671:
2667:
2659:
2655:
2647:
2643:
2635:
2631:
2620:
2616:
2612:
2604:
2600:
2596:
2587:
2583:
2572:
2568:
2557:
2553:
2538:
2516:
2512:
2476:
2472:
2464:
2460:
2452:
2448:
2442:snub octahedron
2428:
2424:
2416:
2412:
2404:
2400:
2395:
2390:
2389:
2383:integer lattice
2360:
2357:
2356:
2353:Borromean rings
2343:
2339:
2331:
2329:
2324:
2322:
2316:
2312:
2304:
2302:
2294:
2291:
2288:
2287:
2285:
2279:
2277:
2271:
2267:
2258:
2254:
2242:
2240:
2235:
2233:
2224:
2222:
2217:
2215:
2210:
2208:
2193:
2189:
2178:
2176:
2164:
2162:
2152:
2148:
2130:
2126:
2118:
2116:
2114:
2110:
2097:
2093:
2088:
2052:
2013:
1989:
1986:
1985:
1958:
1956:
1953:
1952:
1921:
1914:
1904:
1893:
1889:
1886:
1884:
1881:
1880:
1853:
1848:
1845:
1844:
1820:
1817:
1816:
1797:
1794:
1793:
1762:
1755:
1749:
1747:
1744:
1743:
1724:
1721:
1720:
1696:
1693:
1692:
1673:
1670:
1669:
1650:
1647:
1646:
1627:
1624:
1623:
1591:
1586:
1583:
1582:
1555:
1553:
1550:
1549:
1518:
1511:
1505:
1503:
1500:
1499:
1472:
1467:
1464:
1463:
1372:
1369:
1366:
1365:
1363:
1357:
1354:
1351:
1350:
1348:
1342:
1339:
1336:
1335:
1333:
1332:and the angles
1324:
1322:
1318:
1316:
1312:
1310:
1306:
1304:
1300:
1298:
1294:
1292:
1259:
1227:
1225:
1222:
1221:
1194:
1192:
1189:
1188:
1161:
1143:
1130:
1126:
1124:
1121:
1120:
1093:
1091:
1088:
1087:
1063:
1060:
1059:
1040:
1037:
1036:
1002:
1000:
997:
996:
969:
967:
964:
963:
931:
926:
923:
922:
903:
900:
899:
872:
854:
847:
844:
840:
838:
835:
834:
807:
802:
799:
798:
772:
770:
767:
766:
745:
743:
740:
739:
718:
716:
713:
712:
691:
689:
686:
685:
552:
495:
492:
489:
488:
486:
480:
477:
474:
473:
471:
465:
462:
459:
458:
456:
448:
445:
442:
441:
439:
433:
430:
427:
426:
424:
418:
415:
412:
411:
409:
401:
398:
395:
394:
392:
386:
383:
380:
379:
377:
371:
368:
365:
364:
362:
354:
351:
348:
347:
345:
339:
336:
333:
332:
330:
324:
321:
318:
317:
315:
301:
296:
291:
286:
281:
279:
272:
267:
262:
257:
252:
250:
243:
238:
233:
228:
223:
221:
214:
209:
204:
199:
194:
192:
188:Coxeter mirrors
121:
97:
53:
28:
23:
22:
15:
12:
11:
5:
2960:
2950:
2949:
2944:
2939:
2925:
2924:
2914:
2913:External links
2911:
2909:
2908:
2881:
2868:
2843:
2814:
2777:
2772:978-0520029248
2771:
2754:
2738:
2723:
2721:
2718:
2715:
2714:
2693:
2677:
2665:
2653:
2641:
2629:
2618:
2614:
2610:
2602:
2598:
2594:
2581:
2566:
2551:
2536:
2510:
2470:
2458:
2446:
2422:
2410:
2397:
2396:
2394:
2391:
2388:
2387:
2370:
2367:
2364:
2337:
2310:
2265:
2252:
2187:
2146:
2124:
2108:
2090:
2089:
2087:
2084:
2051:
2048:
2012:
2009:
2006:
2005:
1993:
1982:
1970:
1967:
1962:
1949:
1937:
1934:
1925:
1920:
1917:
1908:
1903:
1900:
1897:
1892:
1877:
1865:
1862:
1857:
1852:
1841:
1837:
1836:
1824:
1813:
1801:
1790:
1778:
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1766:
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1740:
1728:
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1534:
1531:
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1514:
1510:
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1484:
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1471:
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1456:
1455:
1450:
1445:
1440:
1435:
1432:
1431:
1424:
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1403:
1400:
1399:
1258:
1255:
1252:
1251:
1239:
1236:
1231:
1218:
1206:
1203:
1198:
1185:
1173:
1170:
1165:
1159:
1153:
1147:
1142:
1139:
1136:
1129:
1117:
1105:
1102:
1097:
1084:
1080:
1079:
1067:
1056:
1044:
1033:
1021:
1018:
1012:
1008:
993:
981:
978:
973:
960:
956:
955:
943:
940:
935:
930:
919:
907:
896:
884:
881:
876:
870:
864:
858:
853:
850:
843:
831:
819:
816:
811:
806:
795:
791:
790:
776:
763:
749:
736:
722:
709:
695:
682:
678:
677:
672:
667:
662:
657:
654:
653:
646:
639:
632:
625:
622:
621:
551:
548:
502:
501:
454:
407:
360:
313:
307:
306:
277:
248:
219:
190:
184:
183:
178:
173:
168:
163:
160:
159:
152:
145:
138:
131:
128:
127:
120:
117:
96:
93:
55:
54:
43:
35:
34:
26:
9:
6:
4:
3:
2:
2959:
2948:
2945:
2943:
2940:
2938:
2935:
2934:
2932:
2921:
2917:
2916:
2904:
2899:
2895:
2891:
2887:
2882:
2880:
2876:
2871:
2865:
2861:
2857:
2853:
2849:
2844:
2839:
2834:
2830:
2826:
2825:
2820:
2815:
2811:
2807:
2802:
2797:
2793:
2789:
2788:
2783:
2778:
2774:
2768:
2763:
2762:
2755:
2751:
2747:
2743:
2739:
2735:
2734:
2729:
2725:
2724:
2710:
2706:
2702:
2697:
2690:
2686:
2681:
2674:
2673:Verheyen 1989
2669:
2662:
2657:
2650:
2645:
2638:
2633:
2626:
2622:
2606:
2590:
2585:
2578:
2575:
2574:Verheyen 1989
2570:
2562:
2555:
2547:
2543:
2539:
2533:
2529:
2525:
2521:
2514:
2507:
2503:
2499:
2495:
2491:
2487:
2483:
2479:
2474:
2467:
2462:
2455:
2450:
2443:
2439:
2435:
2431:
2426:
2419:
2414:
2407:
2402:
2398:
2384:
2368:
2365:
2362:
2354:
2350:
2347:
2341:
2320:
2314:
2275:
2269:
2262:
2256:
2249:
2231:
2206:
2201:
2197:
2191:
2184:
2174:
2170:
2160:
2156:
2150:
2142:
2138:
2134:
2128:
2112:
2105:
2101:
2095:
2091:
2083:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2047:
2045:
2041:
2036:
2030:
2027:
2022:
2019:
1991:
1983:
1968:
1965:
1960:
1950:
1935:
1932:
1923:
1918:
1915:
1906:
1901:
1898:
1895:
1890:
1878:
1863:
1860:
1855:
1850:
1842:
1839:
1822:
1814:
1799:
1791:
1776:
1773:
1764:
1759:
1756:
1752:
1741:
1726:
1718:
1716:Chord radius
1715:
1698:
1690:
1675:
1667:
1652:
1644:
1629:
1621:
1618:
1601:
1598:
1593:
1588:
1580:
1565:
1562:
1557:
1547:
1532:
1529:
1520:
1515:
1512:
1508:
1497:
1482:
1479:
1474:
1469:
1461:
1458:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1438:Cuboctahedron
1436:
1434:
1429:
1425:
1422:
1418:
1415:
1411:
1408:
1404:
1402:
1396:
1393:
1389:
1386:
1331:
1290:
1286:
1282:
1280:
1276:
1272:
1268:
1264:
1237:
1234:
1229:
1219:
1204:
1201:
1196:
1186:
1171:
1168:
1163:
1157:
1151:
1145:
1140:
1137:
1134:
1127:
1118:
1103:
1100:
1095:
1085:
1082:
1065:
1057:
1042:
1034:
1019:
1016:
1010:
1006:
994:
979:
976:
971:
961:
959:Chord radius
958:
941:
938:
933:
928:
920:
905:
897:
882:
879:
874:
868:
862:
856:
851:
848:
841:
832:
817:
814:
809:
804:
796:
793:
774:
764:
747:
737:
720:
710:
693:
683:
680:
676:
673:
671:
668:
666:
663:
661:
660:Cuboctahedron
658:
656:
651:
647:
644:
640:
637:
633:
630:
626:
624:
618:
615:
613:
609:
605:
600:
598:
594:
590:
585:
583:
579:
575:
571:
562:
556:
547:
543:
541:
536:
532:
531:Russian dolls
527:
525:
521:
517:
513:
509:
508:Topologically
455:
408:
361:
314:
312:
309:
278:
249:
220:
191:
189:
182:
179:
177:
174:
172:
169:
167:
166:Cuboctahedron
164:
162:
157:
153:
150:
146:
143:
139:
136:
132:
130:
124:
116:
114:
108:
106:
102:
92:
90:
86:
82:
78:
74:
70:
66:
65:cuboctahedron
62:
51:
47:
41:
36:
31:
19:
2893:
2889:
2878:
2859:
2828:
2822:
2791:
2785:
2760:
2749:
2732:
2720:Bibliography
2709:elastic-edge
2708:
2704:
2696:
2680:
2668:
2656:
2644:
2632:
2624:
2608:
2592:
2584:
2569:
2560:
2554:
2519:
2513:
2485:
2473:
2461:
2449:
2436:embedded in
2430:Coxeter 1973
2425:
2413:
2401:
2340:
2318:
2313:
2273:
2268:
2255:
2229:
2204:
2200:reflex edges
2199:
2190:
2182:
2172:
2168:
2158:
2154:
2149:
2140:
2127:
2111:
2094:
2053:
2043:
2039:
2034:
2031:
2023:
2017:
2014:
1840:Long radius
1390:
1384:
1381:
1278:
1274:
1266:
1260:
1083:Long radius
611:
608:double cover
601:
592:
588:
586:
569:
567:
560:
544:
528:
505:
109:
105:rigid system
98:
58:
2701:Fuller 1975
2661:Kenner 1976
2649:Kenner 1976
2637:Kenner 1976
2502:Möbius loop
2490:stall point
2478:Fuller 1975
2466:Kenner 1976
2137:Mobius loop
2072:octet truss
1261:There is a
514:: it is an
512:Möbius loop
115:structure.
113:octet truss
2931:Categories
2896:(2): 139.
2705:rigid-edge
2393:References
2346:orthogonal
2042:cables. A
1263:tensegrity
582:octahedron
520:Physically
85:octahedron
73:kinematics
50:octahedron
2730:(1973) .
2366:×
2319:contracts
2080:polytopes
1966:≈
1933:≈
1861:≈
1774:≈
1599:≈
1563:≈
1530:≈
1480:≈
1235:≈
1202:≈
1169:≈
1101:≈
1017:≈
977:≈
939:≈
880:≈
815:≈
2858:(eds.),
2850:(2008),
2744:(1975).
2486:unstable
2434:3-sphere
2104:600-cell
1277:and can
522:it is a
61:skeleton
2810:0994201
2689:24-cell
2546:1655039
2330:√
2323:√
2303:√
2298:
2286:
2278:√
2274:expands
2261:columns
2241:√
2234:√
2230:product
2223:√
2216:√
2209:√
2177:√
2163:√
2117:√
2038:struts
1376:
1364:
1361:
1349:
1346:
1334:
1323:√
1317:√
1311:√
1305:√
1299:√
1293:√
535:helical
499:
487:
484:
472:
469:
457:
452:
440:
437:
425:
422:
410:
405:
393:
390:
378:
375:
363:
358:
346:
343:
331:
328:
316:
2866:
2831:(13).
2808:
2769:
2544:
2534:
2498:chiral
2157:where
2133:chiral
1619:Chord
1269:. The
794:Chord
589:become
524:spinor
79:, the
2494:cycle
2086:Notes
1969:2.236
1936:2.351
1864:2.828
1777:1.236
1602:2.828
1566:2.449
1533:2.472
1483:2.828
1459:Edge
1238:1.732
1205:2.236
1172:2.330
1104:2.449
1020:1.225
980:1.732
942:3.464
883:3.963
818:3.464
681:Edge
595:have
63:of a
2864:ISBN
2767:ISBN
2623:(2)
2607:(1)
2532:ISBN
2194:The
2175:to 2
2141:both
2035:both
1279:only
576:, a
568:The
59:The
48:and
2898:doi
2877:",
2833:doi
2829:112
2796:doi
2524:doi
2232:of
2058:by
2040:and
2018:all
1385:all
1328:of
612:two
561:not
2933::
2894:21
2892:.
2888:.
2827:.
2821:.
2806:MR
2804:.
2792:17
2790:.
2784:.
2748:.
2542:MR
2540:,
2530:,
2250:).
2098:A
1896:10
1367:𝝅
1352:𝝅
1337:𝝅
1135:10
593:do
526:.
490:𝝅
475:𝝅
460:𝝅
443:𝝅
428:𝝅
413:𝝅
396:𝝅
381:𝝅
366:𝝅
349:𝝅
334:𝝅
319:𝝅
91:.
2906:.
2900::
2841:.
2835::
2812:.
2798::
2775:.
2675:.
2627:"
2621:.
2619:3
2617:v
2615:2
2613:v
2611:1
2605:.
2603:3
2601:a
2599:2
2597:a
2595:1
2526::
2385:.
2369:4
2363:2
2332:2
2325:6
2305:2
2295:ϕ
2292:/
2289:4
2280:6
2243:2
2236:2
2225:2
2218:5
2211:6
2183:d
2179:3
2173:d
2169:d
2165:6
2159:d
2155:d
2119:2
1992:2
1961:5
1924:5
1919:+
1916:1
1907:5
1902:2
1899:+
1891:2
1856:2
1851:2
1823:0
1800:1
1765:5
1760:+
1757:1
1753:4
1727:2
1699:4
1676:4
1653:4
1630:4
1594:2
1589:2
1558:6
1521:5
1516:+
1513:1
1509:8
1475:2
1470:2
1378:.
1373:4
1370:/
1358:3
1355:/
1343:2
1340:/
1325:6
1319:5
1313:4
1307:3
1301:2
1295:1
1230:3
1197:5
1164:6
1158:)
1152:4
1146:5
1141:2
1138:+
1128:(
1096:6
1066:0
1043:1
1011:2
1007:6
972:3
934:3
929:2
906:4
875:6
869:)
863:2
857:5
852:+
849:1
842:(
810:3
805:2
775:6
748:6
721:6
694:6
496:2
493:/
481:4
478:/
466:3
463:/
449:2
446:/
434:2
431:/
419:2
416:/
402:2
399:/
387:5
384:/
372:3
369:/
355:2
352:/
340:3
337:/
325:4
322:/
20:)
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