61:
1590:
5157:
1735:
1760:
1890:, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point (and therefore of any constructible length) is a power of 2. In particular, any constructible point (or length) is an
2890:. These can be taken three at a time to yield 139 distinct nontrivial problems of constructing a triangle from three points. Of these problems, three involve a point that can be uniquely constructed from the other two points; 23 can be non-uniquely constructed (in fact for infinitely many solutions) but only if the locations of the points obey certain constraints; in 74 the problem is constructible in the general case; and in 39 the required triangle exists but is not constructible.
3225:
5144:
2154:
2242:, which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple straightedge-and-compass construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining the constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations.
1347:
3216:. It is known that one cannot solve an irreducible polynomial of prime degree greater or equal to 7 using the neusis construction, so it is not possible to construct a regular 23-gon or 29-gon using this tool. Benjamin and Snyder proved that it is possible to construct the regular 11-gon, but did not give a construction. It is still open as to whether a regular 25-gon or 31-gon is constructible using this tool.
1355:
1244:, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see
1785:
38:
1934:
2971:
For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just as powerful.
2970:
A point has a solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools.
2950:
was called solid; the third category included all constructions that did not fall into either of the other two categories. This categorization meshes nicely with the modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described
2949:
The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than the circle), then it
2497:
The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable. With modern methods, however, these straightedge-and-compass constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and
2245:
Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using straightedge and
2538:
can be constructed with ruler and compass alone, namely those constructed from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason.
1637:
For example, starting with just two distinct points, we can create a line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and
2626:
cannot in general be constructed. See Note that results proven here are mostly a consequence of the non-constructivity of conics. If the initial conic is considered as a given, then the proof must be reviewed to check if other distinct conic needs to be generated. As an example, constructions for
2149:{\displaystyle {\begin{aligned}\cos {\left({\frac {2\pi }{17}}\right)}&=\,-{\frac {1}{16}}\,+\,{\frac {1}{16}}{\sqrt {17}}\,+\,{\frac {1}{16}}{\sqrt {34-2{\sqrt {17}}}}\\&\qquad +\,{\frac {1}{8}}{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\end{aligned}}}
3123:
What if, together with the straightedge and compass, we had a tool that could (only) trisect an arbitrary angle? Such constructions are solid constructions, but there exist numbers with solid constructions that cannot be constructed using such a tool. For example, we cannot double the cube with
2166:
The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots in the complex numbers). The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a
2913:
Various attempts have been made to restrict the allowable tools for constructions under various rules, in order to determine what is still constructible and how it may be constructed, as well as determining the minimum criteria necessary to still be able to construct everything that compass and
3172:
gave constructions involving the use of a markable ruler. This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects the two given lines, such that the distance between the points of
2961:
that can be broken down into a tower of fields where each extension has degree two. A complex number that has a solid construction has degree with prime factors of only two and three, and lies in a field extension that is at the top of a tower of fields where each extension has degree 2 or 3.
2563:
Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication,
1919:
under addition modulo 2π (which corresponds to multiplication of the points on the unit circle viewed as complex numbers). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example, the regular
2926:) to construct anything with just a compass if it can be constructed with a ruler and compass, provided that the given data and the data to be found consist of discrete points (not lines or circles). The truth of this theorem depends on the truth of
1408:, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a long and checkered history. In any case, the equivalence is why this feature is not stipulated in the definition of the ideal compass.
2662:
3142:
is more powerful than straightedge-and-compass construction. Folds satisfying the Huzita–Hatori axioms can construct exactly the same set of points as the extended constructions using a compass and conic drawing tool. Therefore,
1419:. That is, it must have a finite number of steps, and not be the limit of ever closer approximations. (If an unlimited number of steps is permitted, some otherwise-impossible constructions become possible by means of
2390:
1939:
2319:
2582:
Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2
2445:
2542:
Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity.
2202:
Given any such interpretation of a set of points as complex numbers, the points constructible using valid straightedge-and-compass constructions alone are precisely the elements of the smallest
2627:
normals of a parabola are known, but they need to use an intersection between circle and the parabola itself. So they are not constructible in the sense that the parabola is not constructible.
1393:
can be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse (i.e. fold after being taken off the page, erasing its 'stored' radius).
3208:
The neusis construction is more powerful than a conic drawing tool, as one can construct complex numbers that do not have solid constructions. In fact, using this tool one can solve some
1597:
All straightedge-and-compass constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are:
2990:, and others as being based on a solid construction, but this has been disputed, as other interpretations are possible. The quadrature of the circle does not have a solid construction.
1641:
Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry is equivalent to an axiomatic
1315:). Many of these problems are easily solvable provided that other geometric transformations are allowed; for example, neusis construction can be used to solve the former two problems.
1400:
Actual compasses do not collapse and modern geometric constructions often use this feature. A 'collapsing compass' would appear to be a less powerful instrument. However, by the
2905:. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined.
2680:) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass?
1865:
3553:
1575:
1415:. "Eyeballing" distances (looking at the construction and guessing at its accuracy) or using markings on a ruler, are not permitted. Each construction must also
2937:
It is impossible to take a square root with just a ruler, so some things that cannot be constructed with a ruler can be constructed with a compass; but (by the
3101:, 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100... (sequence
2246:
compass from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots.
2516:, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass.
3913:
1546:
published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing
1378:
is an infinitely long edge with no markings on it. It can only be used to draw a line segment between two points, or to extend an existing line segment.
1915:
between the angles that are constructible and the points that are constructible on any constructible circle. The angles that are constructible form an
1334:
is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of
3367:
2218:
less than π). The elements of this field are precisely those that may be expressed as a formula in the original points using only the operations of
5181:
5017:
1535:
showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for a regular polygon of
3256:
of certain numbers. The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits.
1700:
4617:
1189:
5095:
3185:) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a
3108:
3071:
2819:
3232:
Given a straight line segment called AB, could this be divided in three new equal segments and in many parts required by the use of
4398:
3934:
2602:) cannot be trisected. The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a
2325:
2259:
3906:
3467:
4942:
1248:. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the
17:
1295:
are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved impossible by
3406:
3280:
2568:
over the rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass.
2396:
3115:
Like the question with Fermat primes, it is an open question as to whether there are an infinite number of
Pierpont primes.
4637:
3739:
1801:
5005:
4408:
3124:
such a tool. On the other hand, every regular n-gon that has a solid construction can be constructed using such a tool.
2954:) has a planar construction. A complex number that includes also the extraction of cube roots has a solid construction.
2826:
There are known to be an infinitude of constructible regular polygons with an even number of sides (because if a regular
1831:
containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form
1581:, and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle.
3147:
can also be used to solve cubic equations (and hence quartic equations), and thus solve two of the classical problems.
5072:
3899:
1550:
of lengths. He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary.
3567:
2565:
1434:, rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be
1227:
255:
2489:
ratio. None of these are in the fields described, hence no straightedge-and-compass construction for these exists.
3360:"Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas"
1287:
impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so.
2987:
1695:
1490:, or regular polygons with other numbers of sides. Nor could they construct the side of a cube whose volume is
1362:
The "straightedge" and "compass" of straightedge-and-compass constructions are idealized versions of real-world
5041:
4974:
4607:
4487:
3462:
2974:
The ancient Greeks knew that doubling the cube and trisecting an arbitrary angle both had solid constructions.
2661:
1684:
1182:
1136:
742:
201:
2686:
in 1796 showed that a regular 17-sided polygon can be constructed, and five years later showed that a regular
1705:
3510:
1528:
to trisect an arbitrary angle; but these methods also cannot be followed with just straightedge and compass.
1482:
of 3, 4, or 5 sides (or one with twice the number of sides of a given polygon). But they could not construct
1478:, a square whose area is twice that of another square, a square having the same area as a given polygon, and
5110:
4867:
4755:
3930:
3286:
3139:
2938:
2643:
proved the theorem that there is no ruler-and-compass construction for the general solution of the ancient
1275:
4819:
4750:
2923:
2863:
2498:
the Greeks knew how to solve them without the constraint of working only with straightedge and compass.)
1823:
Using the equations for lines and circles, one can show that the points at which they intersect lie in a
1813:
1401:
1271:
1245:
31:
3586:
Pascal
Schreck, Pascal Mathis, Vesna Marinkoviċ, and Predrag Janičiċ. "Wernick's list: A final update",
2545:
A method which comes very close to approximating the "quadrature of the circle" can be achieved using a
4446:
4262:
2623:
2192:
1649:
first realized this, and used it to prove the impossibility of some constructions; only much later did
1300:
1157:
767:
4237:
4981:
4952:
4312:
4167:
3783:
E. Benjamin, C. Snyder, "On the construction of the regular hendecagon by marked ruler and compass",
3165:
1834:
1521:
1175:
3863:
2957:
In the language of fields, a complex number that is planar has degree a power of two, and lies in a
5115:
4849:
4392:
3604:
3213:
3133:
1459:
1385:
can have an arbitrarily large radius with no markings on it (unlike certain real-world compasses).
1284:
144:
3480:
5077:
5053:
4927:
4862:
4803:
4740:
4730:
4466:
4385:
4247:
4157:
4037:
1554:
570:
250:
107:
4347:
5100:
5048:
4947:
4775:
4725:
4710:
4705:
4476:
4277:
4212:
4202:
4152:
3270:
2231:
1525:
1367:
1256:
below.) More formally, the only permissible constructions are those granted by the first three
1234:
646:
357:
235:
120:
3294:, a mathematician who has made a sideline of collecting false straightedge-and-compass proofs.
3193:
is constructible. Using a markable ruler, regular polygons with solid constructions, like the
1257:
37:
5148:
5000:
4844:
4785:
4662:
4585:
4533:
4335:
4242:
4092:
3843:
2931:
2656:
2520:
2486:
1578:
1463:
1454:
first attempted straightedge-and-compass constructions, and they discovered how to construct
418:
379:
338:
333:
186:
3426:
2849:
1633:
Creating the one point or two points in the intersection of two circles (if they intersect).
1283:
first conceived straightedge-and-compass constructions, and a number of ancient problems in
5125:
5065:
5029:
4872:
4695:
4642:
4612:
4602:
4511:
4374:
4267:
4182:
4137:
4117:
3962:
3947:
3814:
3706:
3558:
3539:
3359:
2927:
2721:
2683:
2644:
2636:
2618:
can be constructed, but the segment from any point in the plane to the nearest point on an
2160:
1720:
1670:
1646:
1497:
1405:
1327:
1261:
1086:
1009:
857:
762:
284:
179:
93:
3064:, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97... (sequence
1589:
1560:
1270:
It turns out to be the case that every point constructible using straightedge and compass
8:
5105:
5024:
5012:
4993:
4957:
4877:
4795:
4780:
4770:
4720:
4715:
4657:
4526:
4418:
4282:
4272:
4172:
4142:
4082:
4057:
3982:
3972:
3957:
3169:
3156:
2958:
2894:
2867:
2707:
2513:
2507:
2203:
1654:
1512:, but these cannot be constructed by straightedge and compass. In the fifth century BCE,
1487:
1424:
1249:
1091:
1035:
948:
802:
782:
707:
597:
468:
458:
321:
196:
191:
174:
149:
137:
89:
84:
65:
3852:
3818:
2253:(taking one of the two viewpoints above) are constructible as these may be expressed as
5161:
5120:
5060:
4988:
4854:
4829:
4647:
4622:
4590:
4428:
4187:
4132:
4097:
3992:
3802:
3698:
3638:
3591:
3527:
1050:
777:
617:
245:
169:
159:
130:
115:
5156:
4834:
4745:
4595:
4521:
4494:
4257:
4077:
4067:
4002:
3922:
3877:
3874:
3822:
3423:
3402:
3233:
3182:
2930:, which is not first-order in nature. Examples of compass-only constructions include
2898:
2859:
2801:
2793:
2785:
2777:
2558:
2235:
2207:
1775:
1750:
1734:
1491:
1451:
1308:
1280:
1121:
909:
887:
812:
671:
397:
326:
218:
164:
125:
2902:
2871:
1690:
1504:
showed that the volume of the cube could be doubled by finding the intersections of
1475:
1111:
1040:
837:
747:
5036:
4907:
4824:
4580:
4568:
4516:
4252:
3694:
3642:
3630:
3523:
3519:
3505:
3491:
Azad, H., and
Laradji, A., "Some impossible constructions in elementary geometry",
3291:
3190:
2640:
2577:
2535:
2215:
1891:
1642:
1483:
1304:
1101:
842:
552:
430:
365:
223:
208:
73:
2717:
The first few constructible regular polygons have the following numbers of sides:
4677:
4667:
4561:
4287:
3702:
3535:
3061:
2673:
2546:
2470:
2180:
1925:
1602:
1479:
524:
387:
230:
213:
154:
60:
1677:
1096:
1065:
999:
847:
792:
727:
4937:
4932:
4760:
4652:
4632:
4460:
4017:
3987:
3718:
3275:
3265:
3202:
3198:
3186:
3018:
2773:
2711:
2599:
2478:
2227:
2196:
1759:
1638:
one of these new points completes the construction of an equilateral triangle.
1543:
1531:
No progress on the unsolved problems was made for two millennia, until in 1796
1323:
1296:
1152:
1060:
1004:
969:
877:
787:
757:
717:
622:
3181:
to the point. In this expanded scheme, we can trisect an arbitrary angle (see
2171:. In addition there is a dense set of constructible angles of infinite order.
1126:
737:
5175:
4897:
4765:
4735:
4556:
4364:
4307:
3826:
3245:
2781:
2765:
2168:
1916:
1824:
1728:
Straightedge-and-compass constructions corresponding to algebraic operations
1650:
1131:
1116:
1045:
862:
822:
772:
547:
510:
477:
315:
311:
3891:
2519:
Squaring the circle has been proved impossible, as it involves generating a
4839:
4627:
4302:
4062:
4052:
3857:
3682:
3253:
3224:
3057:
3053:
3045:
2875:
2797:
2789:
2769:
2757:
2749:
2699:
2691:
2690:-sided polygon can be constructed with straightedge and compass if the odd
2249:
For example, the real part, imaginary part and modulus of a point or ratio
1921:
1817:
1616:
1431:
1390:
1241:
1070:
1019:
832:
687:
602:
392:
4453:
4341:
4047:
4032:
3618:
3098:
3049:
2883:
2850:
Constructing a triangle from three given characteristic points or lengths
2753:
2474:
2239:
2223:
2211:
1627:
1471:
1335:
1106:
979:
797:
732:
660:
632:
607:
1346:
4439:
4358:
4297:
4292:
4232:
4217:
4162:
4147:
4102:
4042:
4027:
4007:
3977:
3942:
3634:
3531:
3161:
3094:
3090:
3041:
2975:
2893:
Twelve key lengths of a triangle are the three side lengths, the three
2703:
2614:
The line segment from any point in the plane to the nearest point on a
1517:
1501:
1430:
Stated this way, straightedge-and-compass constructions appear to be a
964:
943:
933:
923:
882:
827:
722:
712:
612:
463:
4192:
4177:
4127:
4022:
4012:
3997:
3967:
3882:
3431:
3249:
2745:
1912:
1547:
1505:
1455:
1396:
Lines and circles constructed have infinite precision and zero width.
974:
692:
655:
519:
491:
2842:-gon, etc.). However, there are only 31 known constructible regular
2214:
operations (to avoid ambiguity, we can specify the square root with
4329:
4107:
3952:
3421:
3194:
3037:
3033:
2983:
2887:
2879:
2855:
2761:
2729:
2677:
2666:
2219:
1894:, though not every algebraic number is constructible; for example,
1509:
1420:
1252:
is still impermissible and this is what unmarked really means: see
1203:
1055:
1014:
984:
872:
867:
817:
542:
501:
449:
343:
306:
52:
3621:(1990). "On strict strong constructibility with a compass alone".
4902:
4227:
4222:
4122:
4112:
4087:
3311:
Godfried
Toussaint, "A new look at Euclid’s second proposition,"
3209:
3144:
3025:-gon admits a solid, but not planar, construction if and only if
2941:) given a single circle and its center, they can be constructed.
2809:
2805:
2741:
2737:
2733:
2619:
1623:
1513:
1319:
1292:
989:
702:
496:
440:
240:
42:
3766:
A. Baragar, "Constructions using a Twice-Notched
Straightedge",
1660:
30:"Constructive geometry" redirects here. Not to be confused with
4197:
4072:
3785:
Mathematical
Proceedings of the Cambridge Philosophical Society
3685:(1989), "On Archimedes' construction of the regular heptagon",
3177:("inclination", "tendency" or "verging"), because the new line
3173:
intersection equals the given segment. This the Greeks called
2979:
2917:
2725:
2615:
2603:
2595:
2587:
1609:
1386:
1331:
938:
928:
807:
752:
627:
590:
578:
533:
486:
404:
69:
3803:"The Computation of Certain Numbers Using a Ruler and Compass"
1710:
Drawing a line through a given point parallel to a given line.
1665:
The most-used straightedge-and-compass constructions include:
1630:
in the intersection of a line and a circle (if they intersect)
1354:
4573:
4551:
4207:
2564:
division, and taking square roots. This follows because its
2477:
not divisible by 3) require ratios which are the solution to
2385:{\displaystyle \mathrm {Im} (z)={\frac {z-{\bar {z}}}{2i}}\;}
1532:
1520:
to both trisect the general angle and square the circle, and
1467:
1363:
1288:
1230:
1223:
994:
918:
852:
697:
301:
296:
3721:: "Angle trisection, the heptagon, and the triskaidecagon",
2590:(72° = 360°/5) can be trisected, but the angle of
2314:{\displaystyle \mathrm {Re} (z)={\frac {z+{\bar {z}}}{2}}\;}
3508:(1998), "Reflections on Reflection in a Spherical Mirror",
3103:
3066:
2814:
2206:
containing the original set of points and closed under the
1784:
1276:
straightedge alone if given a single circle and its center.
585:
435:
3283:, most of them show straightedge-and-compass constructions
2647:(billiard problem or reflection from a spherical mirror).
3872:
2986:, which was interpreted by medieval Arabic commentators,
1820:. Finally we can write these vectors as complex numbers.
1612:
that contains one point and has a center at another point
1886:
Since the field of constructible points is closed under
1816:
made of two lines, and represent points of our plane by
2440:{\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}.\;}
1701:
Constructing a line through a point tangent to a circle
3655:
2951:
2399:
2328:
2262:
1937:
1837:
1812:
One can associate an algebra to our geometry using a
1563:
3592:
http://forumgeom.fau.edu/FG2016volume16/FG201610.pdf
3554:"Don solves the last puzzle left by ancient Greeks"
1706:
Constructing a circle through 3 noncollinear points
1330:, and an angle is constructible if and only if its
2439:
2384:
2313:
2148:
1859:
1569:
3664:P. Hummel, "Solid constructions using ellipses",
3343:Famous Problems of Geometry and How to Solve Them
2174:
1341:
5173:
2650:
3239:
3086:-gon has no solid construction is the sequence
3021:(primes of the form 23+1). Therefore, regular
2808:, 128, 136, 160, 170, 192, 204, 240, 255, 256,
1645:, replacing its elements by symbols. Probably
1524:in the second century BCE showed how to use a
3921:
3907:
3800:
3602:Posamentier, Alfred S., and Lehmann, Ingmar.
3219:
2997:-gon has a solid construction if and only if
2830:-gon is constructible, then so is a regular 2
2195:allows us to consider the points as a set of
1661:Common straightedge-and-compass constructions
1486:except in particular cases, or a square with
1312:
1183:
3392:
3390:
3368:Journal de Mathématiques Pures et Appliquées
2918:Constructing with only ruler or only compass
1474:of given lengths. They could also construct
1226:, and other geometric figures using only an
3794:
2908:
2492:
1272:may also be constructed using compass alone
3914:
3900:
3396:
2436:
2381:
2310:
1584:
1190:
1176:
59:
3551:
3387:
3353:
3351:
2944:
2609:
2183:, selecting any one of them to be called
2062:
2020:
2016:
1998:
1994:
1980:
1411:Each construction must be mathematically
3401:. Mineola, N.Y.: Dover. pp. 29–30.
3345:, Dover Publications, 1982 (orig. 1969).
3228:Trisection of a straight edge procedure.
3223:
3205:give constructions for several of them.
2660:
1783:
1758:
1733:
1588:
1353:
1345:
36:
3504:
3357:
2457:(except for special angles such as any
2191:, together with an arbitrary choice of
1906:
1714:
14:
5182:Compass and straightedge constructions
5174:
4943:Latin translations of the 12th century
3348:
3337:
3335:
3333:
3331:
3329:
3327:
3325:
3323:
3321:
3017:is a product of zero or more distinct
2965:
2501:
290:Straightedge and compass constructions
4673:Straightedge and compass construction
3895:
3873:
3681:
3617:
3422:
3281:List of interactive geometry software
3118:
1208:straightedge-and-compass construction
4638:Incircle and excircles of a triangle
3741:Geometric Exercises in Paper Folding
3315:, Vol. 15, No. 3, (1993), pp. 12-24.
2630:
2552:
1903:is algebraic but not constructible.
1253:
3737:
3446:
3318:
3013:are some non-negative integers and
2846:-gons with an odd number of sides.
2571:
2512:The most famous of these problems,
2167:power of two and a set of distinct
27:Method of drawing geometric objects
24:
3853:Construction with the Compass Only
3699:10.1111/j.1600-0498.1989.tb00848.x
3150:
3060:, 26, 27, 28, 35, 36, 37, 38, 39,
2333:
2330:
2267:
2264:
1222:– is the construction of lengths,
25:
5193:
3837:
3768:The American Mathematical Monthly
3754:Conway, John H. and Richard Guy:
3552:Highfield, Roger (1 April 1997),
3397:Kazarinoff, Nicholas D. (2003) .
2922:It is possible (according to the
256:Noncommutative algebraic geometry
5155:
5142:
3358:Wantzel, Pierre-Laurent (1837).
1240:The idealized ruler, known as a
3864:Angle Trisection by Hippocrates
3777:
3760:
3748:
3731:
3712:
3675:
3658:
3649:
3611:
3596:
3580:
3545:
3498:
2988:Bartel Leendert van der Waerden
2710:; the conjecture was proven by
2058:
1860:{\displaystyle x+y={\sqrt {k}}}
1488:the same area as a given circle
45:with a straightedge and compass
4975:A History of Greek Mathematics
4488:The Quadrature of the Parabola
3524:10.1080/00029890.1998.12004920
3485:
3479:Instructions for trisecting a
3473:
3455:
3440:
3415:
3313:The Mathematical Intelligencer
3305:
3140:mathematical theory of origami
2425:
2364:
2343:
2337:
2298:
2277:
2271:
2175:Relation to complex arithmetic
1516:used a curve that he called a
1404:in Proposition 2 of Book 1 of
1342:Straightedge and compass tools
1338:but of no higher-order roots.
1212:ruler-and-compass construction
649:- / other-dimensional
13:
1:
3844:Regular polygon constructions
3511:American Mathematical Monthly
3298:
2706:that this condition was also
2651:Constructing regular polygons
2179:Given a set of points in the
1309:doubling the volume of a cube
1305:trisecting an arbitrary angle
4756:Intersecting secants theorem
3807:Journal of Integer Sequences
3427:"Trigonometry Angles--Pi/17"
3287:Mathematics of paper folding
3252:that can be used to compute
3240:Computation of binary digits
2982:construction of the regular
1452:ancient Greek mathematicians
1322:, a length is constructible
1281:Ancient Greek mathematicians
7:
4751:Intersecting chords theorem
4618:Doctrine of proportionality
3495:88, November 2004, 548–551.
3259:
3214:not solvable using radicals
1928:) is constructible because
1814:Cartesian coordinate system
1696:Mirroring a point in a line
1619:of two (non-parallel) lines
1539:sides to be constructible.
1402:compass equivalence theorem
1246:compass equivalence theorem
32:Constructive solid geometry
10:
5198:
4447:On the Sphere and Cylinder
4400:On the Sizes and Distances
3220:Trisect a straight segment
3197:, are constructible; and
3154:
3131:
3127:
2834:-gon and hence a regular 4
2665:Construction of a regular
2654:
2575:
2556:
2505:
1718:
1615:Creating the point at the
1492:twice the volume of a cube
1484:one third of a given angle
1445:
1313:§ impossible constructions
29:
5149:Ancient Greece portal
5138:
5088:
4966:
4953:Philosophy of mathematics
4923:
4916:
4890:
4868:Ptolemy's table of chords
4812:
4794:
4693:
4686:
4542:
4504:
4321:
3929:
3923:Ancient Greek mathematics
3666:The Pi Mu Epsilon Journal
3608:, Prometheus Books, 2012.
3248:gave a ruler-and-compass
2187:and another to be called
1727:
4820:Aristarchus's inequality
4393:On Conoids and Spheroids
3738:Row, T. Sundara (1966).
3605:The Secrets of Triangles
2939:Poncelet–Steiner theorem
2909:Restricted constructions
2872:internal angle bisectors
2854:Sixteen key points of a
2493:Impossible constructions
1350:Straightedge and compass
145:Non-Archimedean geometry
4928:Ancient Greek astronomy
4741:Inscribed angle theorem
4731:Greek geometric algebra
4386:Measurement of a Circle
3848:The Math Forum @ Drexel
3791:(3), 409 -- 424 (2014).
3774:(2), 151 -- 164 (2002).
3728:(1988), no. 3, 185-194.
2924:Mohr–Mascheroni theorem
1687:from a point to a line.
1653:find a complete set of
1593:The basic constructions
1585:The basic constructions
251:Noncommutative geometry
5162:Mathematics portal
4948:Non-Euclidean geometry
4903:Mouseion of Alexandria
4776:Tangent-secant theorem
4726:Geometric mean theorem
4711:Exterior angle theorem
4706:Angle bisector theorem
4410:On Sizes and Distances
3801:Simon Plouffe (1998).
3672:(8), 429 -- 435 (2003)
3271:Geometric cryptography
3229:
3183:Archimedes' trisection
2945:Extended constructions
2864:midpoints of its sides
2669:
2610:Distance to an ellipse
2455:trisection of an angle
2441:
2386:
2315:
2150:
1861:
1827:of the smallest field
1805:
1802:geometric mean theorem
1779:
1754:
1671:perpendicular bisector
1594:
1571:
1359:
1351:
1220:classical construction
1216:Euclidean construction
219:Discrete/Combinatorial
46:
18:Classical construction
4850:Pappus's area theorem
4786:Theorem of the gnomon
4663:Quadratrix of Hippias
4586:Circles of Apollonius
4534:Problem of Apollonius
4512:Constructible numbers
4336:Archimedes Palimpsest
3590:16, 2016, pp. 69–80.
3227:
2664:
2657:Constructible polygon
2521:transcendental number
2442:
2387:
2316:
2151:
1924:(the seventeen-sided
1862:
1787:
1762:
1737:
1592:
1579:transcendental number
1572:
1476:half of a given angle
1357:
1349:
202:Discrete differential
40:
5066:prehistoric counting
4863:Ptolemy's inequality
4804:Apollonius's theorem
4643:Method of exhaustion
4613:Diophantine equation
4603:Circumscribed circle
4420:On the Moving Sphere
3570:on November 23, 2004
3559:Electronic Telegraph
3493:Mathematical Gazette
3134:Huzita–Hatori axioms
3082:for which a regular
2684:Carl Friedrich Gauss
2397:
2326:
2260:
1935:
1907:Constructible angles
1835:
1721:Constructible number
1715:Constructible points
1570:{\displaystyle \pi }
1561:
1328:constructible number
5152: •
4958:Neusis construction
4878:Spiral of Theodorus
4771:Pythagorean theorem
4716:Euclidean algorithm
4658:Lune of Hippocrates
4527:Squaring the circle
4283:Theon of Alexandria
3958:Aristaeus the Elder
3819:1998JIntS...1...13P
3756:The Book of Numbers
3723:Amer. Math. Monthly
3623:Journal of Geometry
3588:Forum Geometricorum
3463:Squaring the circle
3399:Ruler and the Round
3157:Neusis construction
3029:is in the sequence
2966:Solid constructions
2812:, 272... (sequence
2514:squaring the circle
2508:Squaring the circle
2502:Squaring the circle
2483:squaring the circle
1825:quadratic extension
1655:axioms for geometry
1494:with a given side.
1250:neusis construction
469:Pythagorean theorem
41:Creating a regular
4845:Menelaus's theorem
4835:Irrational numbers
4648:Parallel postulate
4623:Euclidean geometry
4591:Apollonian circles
4133:Isidore of Miletus
3878:"Angle Trisection"
3875:Weisstein, Eric W.
3744:. New York: Dover.
3635:10.1007/BF01222890
3424:Weisstein, Eric W.
3230:
2932:Napoleon's problem
2914:straightedge can.
2870:, the feet of its
2866:, the feet of its
2670:
2566:minimal polynomial
2437:
2382:
2311:
2146:
2144:
1857:
1806:
1780:
1755:
1691:Bisecting an angle
1685:perpendicular line
1605:through two points
1595:
1567:
1421:infinite sequences
1360:
1352:
47:
5169:
5168:
5134:
5133:
4886:
4885:
4873:Ptolemy's theorem
4746:Intercept theorem
4596:Apollonian gasket
4522:Doubling the cube
4495:The Sand Reckoner
3506:Neumann, Peter M.
3408:978-0-486-42515-3
3234:intercept theorem
2928:Archimedes' axiom
2645:Alhazen's problem
2631:Alhazen's problem
2559:Doubling the cube
2553:Doubling the cube
2536:algebraic numbers
2451:Doubling the cube
2431:
2428:
2379:
2367:
2308:
2301:
2236:complex conjugate
2208:complex conjugate
2159:as discovered by
2140:
2138:
2136:
2112:
2110:
2089:
2071:
2049:
2047:
2029:
2014:
2007:
1992:
1966:
1855:
1810:
1809:
1776:intercept theorem
1751:intercept theorem
1669:Constructing the
1406:Euclid's Elements
1291:showed that some
1200:
1199:
1165:
1164:
888:List of geometers
571:Three-dimensional
560:
559:
16:(Redirected from
5189:
5160:
5159:
5147:
5146:
5145:
4921:
4920:
4908:Platonic Academy
4855:Problem II.8 of
4825:Crossbar theorem
4781:Thales's theorem
4721:Euclid's theorem
4691:
4690:
4608:Commensurability
4569:Axiomatic system
4517:Angle trisection
4482:
4472:
4434:
4424:
4414:
4404:
4380:
4370:
4353:
3916:
3909:
3902:
3893:
3892:
3888:
3887:
3831:
3830:
3798:
3792:
3781:
3775:
3764:
3758:
3752:
3746:
3745:
3735:
3729:
3716:
3710:
3709:
3683:Knorr, Wilbur R.
3679:
3673:
3662:
3656:
3653:
3647:
3646:
3615:
3609:
3600:
3594:
3584:
3578:
3577:
3576:
3575:
3566:, archived from
3549:
3543:
3542:
3502:
3496:
3489:
3483:
3477:
3471:
3459:
3453:
3452:
3444:
3438:
3437:
3436:
3419:
3413:
3412:
3394:
3385:
3384:
3382:
3380:
3364:
3355:
3346:
3341:Bold, Benjamin.
3339:
3316:
3309:
3292:Underwood Dudley
3191:quartic equation
3119:Angle trisection
3106:
3069:
2901:, and the three
2817:
2674:regular polygons
2641:Peter M. Neumann
2593:
2585:
2578:Angle trisection
2572:Angle trisection
2533:
2532:
2531:
2530:
2468:
2446:
2444:
2443:
2438:
2432:
2430:
2429:
2421:
2415:
2410:
2391:
2389:
2388:
2383:
2380:
2378:
2370:
2369:
2368:
2360:
2350:
2336:
2320:
2318:
2317:
2312:
2309:
2304:
2303:
2302:
2294:
2284:
2270:
2216:complex argument
2155:
2153:
2152:
2147:
2145:
2141:
2139:
2137:
2132:
2121:
2113:
2111:
2106:
2095:
2090:
2085:
2074:
2072:
2064:
2054:
2050:
2048:
2043:
2032:
2030:
2022:
2015:
2010:
2008:
2000:
1993:
1985:
1972:
1971:
1967:
1962:
1954:
1902:
1901:
1900:
1892:algebraic number
1866:
1864:
1863:
1858:
1856:
1851:
1799:
1798:
1725:
1724:
1576:
1574:
1573:
1568:
1480:regular polygons
1423:converging to a
1326:it represents a
1210:– also known as
1192:
1185:
1178:
906:
905:
425:
424:
358:Zero-dimensional
63:
49:
48:
21:
5197:
5196:
5192:
5191:
5190:
5188:
5187:
5186:
5172:
5171:
5170:
5165:
5154:
5143:
5141:
5130:
5096:Arabian/Islamic
5084:
5073:numeral systems
4962:
4912:
4882:
4830:Heron's formula
4808:
4790:
4682:
4678:Triangle center
4668:Regular polygon
4545:and definitions
4544:
4538:
4500:
4480:
4470:
4432:
4422:
4412:
4402:
4378:
4368:
4351:
4317:
4288:Theon of Smyrna
3933:
3925:
3920:
3846:by Dr. Math at
3840:
3835:
3834:
3799:
3795:
3782:
3778:
3765:
3761:
3753:
3749:
3736:
3732:
3719:Gleason, Andrew
3717:
3713:
3680:
3676:
3663:
3659:
3654:
3650:
3616:
3612:
3601:
3597:
3585:
3581:
3573:
3571:
3550:
3546:
3503:
3499:
3490:
3486:
3478:
3474:
3460:
3456:
3445:
3441:
3420:
3416:
3409:
3395:
3388:
3378:
3376:
3362:
3356:
3349:
3340:
3319:
3310:
3306:
3301:
3262:
3242:
3222:
3159:
3153:
3151:Markable rulers
3136:
3130:
3121:
3102:
3065:
3019:Pierpont primes
2968:
2959:field extension
2947:
2920:
2911:
2903:angle bisectors
2852:
2813:
2659:
2653:
2633:
2612:
2606:construction).
2591:
2583:
2580:
2574:
2561:
2555:
2547:Kepler triangle
2534:. Only certain
2528:
2527:
2525:
2524:
2510:
2504:
2495:
2479:cubic equations
2471:rational number
2466:
2420:
2419:
2414:
2400:
2398:
2395:
2394:
2371:
2359:
2358:
2351:
2349:
2329:
2327:
2324:
2323:
2293:
2292:
2285:
2283:
2263:
2261:
2258:
2257:
2197:complex numbers
2181:Euclidean plane
2177:
2143:
2142:
2131:
2120:
2105:
2094:
2084:
2073:
2063:
2052:
2051:
2042:
2031:
2021:
2009:
1999:
1984:
1973:
1955:
1953:
1949:
1948:
1938:
1936:
1933:
1932:
1926:regular polygon
1909:
1898:
1896:
1895:
1850:
1836:
1833:
1832:
1794:
1792:
1723:
1717:
1663:
1587:
1562:
1559:
1558:
1448:
1344:
1254:Markable rulers
1196:
1167:
1166:
903:
902:
893:
892:
683:
682:
666:
665:
651:
650:
638:
637:
574:
573:
562:
561:
422:
421:
419:Two-dimensional
410:
409:
383:
382:
380:One-dimensional
371:
370:
361:
360:
349:
348:
282:
281:
280:
263:
262:
111:
110:
99:
76:
35:
28:
23:
22:
15:
12:
11:
5:
5195:
5185:
5184:
5167:
5166:
5139:
5136:
5135:
5132:
5131:
5129:
5128:
5123:
5118:
5113:
5108:
5103:
5098:
5092:
5090:
5089:Other cultures
5086:
5085:
5083:
5082:
5081:
5080:
5070:
5069:
5068:
5058:
5057:
5056:
5046:
5045:
5044:
5034:
5033:
5032:
5022:
5021:
5020:
5010:
5009:
5008:
4998:
4997:
4996:
4986:
4985:
4984:
4970:
4968:
4964:
4963:
4961:
4960:
4955:
4950:
4945:
4940:
4938:Greek numerals
4935:
4933:Attic numerals
4930:
4924:
4918:
4914:
4913:
4911:
4910:
4905:
4900:
4894:
4892:
4888:
4887:
4884:
4883:
4881:
4880:
4875:
4870:
4865:
4860:
4852:
4847:
4842:
4837:
4832:
4827:
4822:
4816:
4814:
4810:
4809:
4807:
4806:
4800:
4798:
4792:
4791:
4789:
4788:
4783:
4778:
4773:
4768:
4763:
4761:Law of cosines
4758:
4753:
4748:
4743:
4738:
4733:
4728:
4723:
4718:
4713:
4708:
4702:
4700:
4688:
4684:
4683:
4681:
4680:
4675:
4670:
4665:
4660:
4655:
4653:Platonic solid
4650:
4645:
4640:
4635:
4633:Greek numerals
4630:
4625:
4620:
4615:
4610:
4605:
4600:
4599:
4598:
4593:
4583:
4578:
4577:
4576:
4566:
4565:
4564:
4559:
4548:
4546:
4540:
4539:
4537:
4536:
4531:
4530:
4529:
4524:
4519:
4508:
4506:
4502:
4501:
4499:
4498:
4491:
4484:
4474:
4464:
4461:Planisphaerium
4457:
4450:
4443:
4436:
4426:
4416:
4406:
4396:
4389:
4382:
4372:
4362:
4355:
4345:
4338:
4333:
4325:
4323:
4319:
4318:
4316:
4315:
4310:
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4215:
4210:
4205:
4200:
4195:
4190:
4185:
4180:
4175:
4170:
4165:
4160:
4155:
4150:
4145:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4095:
4090:
4085:
4080:
4075:
4070:
4065:
4060:
4055:
4050:
4045:
4040:
4035:
4030:
4025:
4020:
4015:
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3975:
3970:
3965:
3960:
3955:
3950:
3945:
3939:
3937:
3931:Mathematicians
3927:
3926:
3919:
3918:
3911:
3904:
3896:
3890:
3889:
3870:
3861:
3850:
3839:
3838:External links
3836:
3833:
3832:
3793:
3776:
3759:
3747:
3730:
3711:
3693:(4): 257–271,
3674:
3657:
3648:
3629:(1–2): 12–15.
3610:
3595:
3579:
3544:
3518:(6): 523–528,
3497:
3484:
3472:
3454:
3447:Stewart, Ian.
3439:
3414:
3407:
3386:
3347:
3317:
3303:
3302:
3300:
3297:
3296:
3295:
3289:
3284:
3278:
3276:Geometrography
3273:
3268:
3266:Carlyle circle
3261:
3258:
3241:
3238:
3221:
3218:
3203:Richard K. Guy
3199:John H. Conway
3155:Main article:
3152:
3149:
3132:Main article:
3129:
3126:
3120:
3117:
3113:
3112:
3076:
3075:
2967:
2964:
2946:
2943:
2919:
2916:
2910:
2907:
2851:
2848:
2824:
2823:
2712:Pierre Wantzel
2655:Main article:
2652:
2649:
2639:mathematician
2632:
2629:
2611:
2608:
2576:Main article:
2573:
2570:
2557:Main article:
2554:
2551:
2506:Main article:
2503:
2500:
2494:
2491:
2487:transcendental
2448:
2447:
2435:
2427:
2424:
2418:
2413:
2409:
2406:
2403:
2392:
2377:
2374:
2366:
2363:
2357:
2354:
2348:
2345:
2342:
2339:
2335:
2332:
2321:
2307:
2300:
2297:
2291:
2288:
2282:
2279:
2276:
2273:
2269:
2266:
2228:multiplication
2176:
2173:
2157:
2156:
2135:
2130:
2127:
2124:
2119:
2116:
2109:
2104:
2101:
2098:
2093:
2088:
2083:
2080:
2077:
2070:
2067:
2061:
2057:
2055:
2053:
2046:
2041:
2038:
2035:
2028:
2025:
2019:
2013:
2006:
2003:
1997:
1991:
1988:
1983:
1979:
1976:
1974:
1970:
1965:
1961:
1958:
1952:
1947:
1944:
1941:
1940:
1908:
1905:
1854:
1849:
1846:
1843:
1840:
1808:
1807:
1781:
1756:
1730:
1729:
1719:Main article:
1716:
1713:
1712:
1711:
1708:
1703:
1698:
1693:
1688:
1681:
1674:
1673:from a segment
1662:
1659:
1635:
1634:
1631:
1620:
1613:
1606:
1586:
1583:
1566:
1544:Pierre Wantzel
1447:
1444:
1398:
1397:
1394:
1379:
1343:
1340:
1324:if and only if
1299:in 1837 using
1297:Pierre Wantzel
1285:plane geometry
1233:and a pair of
1198:
1197:
1195:
1194:
1187:
1180:
1172:
1169:
1168:
1163:
1162:
1161:
1160:
1155:
1147:
1146:
1142:
1141:
1140:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1081:
1080:
1076:
1075:
1074:
1073:
1068:
1063:
1058:
1053:
1048:
1043:
1038:
1030:
1029:
1025:
1024:
1023:
1022:
1017:
1012:
1007:
1002:
997:
992:
987:
982:
977:
972:
967:
959:
958:
954:
953:
952:
951:
946:
941:
936:
931:
926:
921:
913:
912:
904:
900:
899:
898:
895:
894:
891:
890:
885:
880:
875:
870:
865:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
684:
680:
679:
678:
675:
674:
668:
667:
664:
663:
658:
652:
645:
644:
643:
640:
639:
636:
635:
630:
625:
623:Platonic Solid
620:
615:
610:
605:
600:
595:
594:
593:
582:
581:
575:
569:
568:
567:
564:
563:
558:
557:
556:
555:
550:
545:
537:
536:
530:
529:
528:
527:
522:
514:
513:
507:
506:
505:
504:
499:
494:
489:
481:
480:
474:
473:
472:
471:
466:
461:
453:
452:
446:
445:
444:
443:
438:
433:
423:
417:
416:
415:
412:
411:
408:
407:
402:
401:
400:
395:
384:
378:
377:
376:
373:
372:
369:
368:
362:
356:
355:
354:
351:
350:
347:
346:
341:
336:
330:
329:
324:
319:
309:
304:
299:
293:
292:
283:
279:
278:
275:
271:
270:
269:
268:
265:
264:
261:
260:
259:
258:
248:
243:
238:
233:
228:
227:
226:
216:
211:
206:
205:
204:
199:
194:
184:
183:
182:
177:
167:
162:
157:
152:
147:
142:
141:
140:
135:
134:
133:
118:
112:
106:
105:
104:
101:
100:
98:
97:
87:
81:
78:
77:
64:
56:
55:
26:
9:
6:
4:
3:
2:
5194:
5183:
5180:
5179:
5177:
5164:
5163:
5158:
5151:
5150:
5137:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5097:
5094:
5093:
5091:
5087:
5079:
5076:
5075:
5074:
5071:
5067:
5064:
5063:
5062:
5059:
5055:
5052:
5051:
5050:
5047:
5043:
5040:
5039:
5038:
5035:
5031:
5028:
5027:
5026:
5023:
5019:
5016:
5015:
5014:
5011:
5007:
5004:
5003:
5002:
4999:
4995:
4992:
4991:
4990:
4987:
4983:
4979:
4978:
4977:
4976:
4972:
4971:
4969:
4965:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4934:
4931:
4929:
4926:
4925:
4922:
4919:
4915:
4909:
4906:
4904:
4901:
4899:
4896:
4895:
4893:
4889:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4859:
4858:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4818:
4817:
4815:
4811:
4805:
4802:
4801:
4799:
4797:
4793:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4766:Pons asinorum
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4736:Hinge theorem
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4703:
4701:
4699:
4698:
4692:
4689:
4685:
4679:
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4649:
4646:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4597:
4594:
4592:
4589:
4588:
4587:
4584:
4582:
4579:
4575:
4572:
4571:
4570:
4567:
4563:
4560:
4558:
4555:
4554:
4553:
4550:
4549:
4547:
4541:
4535:
4532:
4528:
4525:
4523:
4520:
4518:
4515:
4514:
4513:
4510:
4509:
4507:
4503:
4497:
4496:
4492:
4490:
4489:
4485:
4483:
4479:
4475:
4473:
4469:
4465:
4463:
4462:
4458:
4456:
4455:
4451:
4449:
4448:
4444:
4442:
4441:
4437:
4435:
4431:
4427:
4425:
4421:
4417:
4415:
4411:
4407:
4405:
4403:(Aristarchus)
4401:
4397:
4395:
4394:
4390:
4388:
4387:
4383:
4381:
4377:
4373:
4371:
4367:
4363:
4361:
4360:
4356:
4354:
4350:
4346:
4344:
4343:
4339:
4337:
4334:
4332:
4331:
4327:
4326:
4324:
4320:
4314:
4311:
4309:
4308:Zeno of Sidon
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3940:
3938:
3936:
3932:
3928:
3924:
3917:
3912:
3910:
3905:
3903:
3898:
3897:
3894:
3885:
3884:
3879:
3876:
3871:
3869:
3865:
3862:
3860:
3859:
3854:
3851:
3849:
3845:
3842:
3841:
3828:
3824:
3820:
3816:
3812:
3808:
3804:
3797:
3790:
3786:
3780:
3773:
3769:
3763:
3757:
3751:
3743:
3742:
3734:
3727:
3724:
3720:
3715:
3708:
3704:
3700:
3696:
3692:
3688:
3684:
3678:
3671:
3667:
3661:
3652:
3644:
3640:
3636:
3632:
3628:
3624:
3620:
3614:
3607:
3606:
3599:
3593:
3589:
3583:
3569:
3565:
3561:
3560:
3555:
3548:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3512:
3507:
3501:
3494:
3488:
3482:
3476:
3470:
3469:
3464:
3458:
3451:. p. 75.
3450:
3449:Galois Theory
3443:
3434:
3433:
3428:
3425:
3418:
3410:
3404:
3400:
3393:
3391:
3374:
3370:
3369:
3361:
3354:
3352:
3344:
3338:
3336:
3334:
3332:
3330:
3328:
3326:
3324:
3322:
3314:
3308:
3304:
3293:
3290:
3288:
3285:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3263:
3257:
3255:
3254:binary digits
3251:
3247:
3246:Simon Plouffe
3237:
3235:
3226:
3217:
3215:
3211:
3206:
3204:
3200:
3196:
3192:
3188:
3184:
3180:
3176:
3171:
3167:
3163:
3158:
3148:
3146:
3141:
3135:
3125:
3116:
3110:
3105:
3100:
3096:
3092:
3089:
3088:
3087:
3085:
3081:
3073:
3068:
3063:
3059:
3055:
3051:
3047:
3043:
3039:
3035:
3032:
3031:
3030:
3028:
3024:
3020:
3016:
3012:
3008:
3004:
3000:
2996:
2991:
2989:
2985:
2981:
2977:
2972:
2963:
2960:
2955:
2953:
2942:
2940:
2935:
2933:
2929:
2925:
2915:
2906:
2904:
2900:
2896:
2891:
2889:
2885:
2881:
2877:
2873:
2869:
2865:
2861:
2857:
2847:
2845:
2841:
2837:
2833:
2829:
2821:
2816:
2811:
2807:
2803:
2799:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2751:
2747:
2743:
2739:
2735:
2731:
2727:
2723:
2720:
2719:
2718:
2715:
2713:
2709:
2705:
2701:
2700:Fermat primes
2698:are distinct
2697:
2693:
2692:prime factors
2689:
2685:
2681:
2679:
2675:
2668:
2663:
2658:
2648:
2646:
2642:
2638:
2635:In 1997, the
2628:
2625:
2621:
2617:
2607:
2605:
2601:
2597:
2589:
2579:
2569:
2567:
2560:
2550:
2548:
2543:
2540:
2537:
2522:
2517:
2515:
2509:
2499:
2490:
2488:
2484:
2480:
2476:
2472:
2464:
2460:
2456:
2452:
2433:
2422:
2416:
2411:
2407:
2404:
2401:
2393:
2375:
2372:
2361:
2355:
2352:
2346:
2340:
2322:
2305:
2295:
2289:
2286:
2280:
2274:
2256:
2255:
2254:
2252:
2247:
2243:
2241:
2237:
2233:
2229:
2225:
2221:
2217:
2213:
2209:
2205:
2200:
2198:
2194:
2190:
2186:
2182:
2172:
2170:
2169:Fermat primes
2164:
2162:
2133:
2128:
2125:
2122:
2117:
2114:
2107:
2102:
2099:
2096:
2091:
2086:
2081:
2078:
2075:
2068:
2065:
2059:
2056:
2044:
2039:
2036:
2033:
2026:
2023:
2017:
2011:
2004:
2001:
1995:
1989:
1986:
1981:
1977:
1975:
1968:
1963:
1959:
1956:
1950:
1945:
1942:
1931:
1930:
1929:
1927:
1923:
1918:
1917:abelian group
1914:
1904:
1893:
1889:
1884:
1882:
1878:
1874:
1870:
1852:
1847:
1844:
1841:
1838:
1830:
1826:
1821:
1819:
1815:
1803:
1797:
1790:
1786:
1782:
1777:
1773:
1769:
1766: =
1765:
1761:
1757:
1752:
1748:
1744:
1741: =
1740:
1736:
1732:
1731:
1726:
1722:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1686:
1682:
1680:of a segment.
1679:
1675:
1672:
1668:
1667:
1666:
1658:
1656:
1652:
1648:
1644:
1639:
1632:
1629:
1625:
1622:Creating the
1621:
1618:
1614:
1611:
1608:Creating the
1607:
1604:
1601:Creating the
1600:
1599:
1598:
1591:
1582:
1580:
1564:
1556:
1553:Then in 1882
1551:
1549:
1545:
1540:
1538:
1534:
1529:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1443:
1441:
1437:
1433:
1428:
1426:
1422:
1418:
1414:
1409:
1407:
1403:
1395:
1392:
1391:circular arcs
1388:
1384:
1380:
1377:
1373:
1372:
1371:
1369:
1365:
1356:
1348:
1339:
1337:
1333:
1329:
1325:
1321:
1316:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1277:
1273:
1268:
1266:
1265:
1259:
1255:
1251:
1247:
1243:
1238:
1236:
1232:
1229:
1225:
1221:
1217:
1213:
1209:
1205:
1193:
1188:
1186:
1181:
1179:
1174:
1173:
1171:
1170:
1159:
1156:
1154:
1151:
1150:
1149:
1148:
1144:
1143:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1084:
1083:
1082:
1078:
1077:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1037:
1034:
1033:
1032:
1031:
1027:
1026:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
981:
978:
976:
973:
971:
968:
966:
963:
962:
961:
960:
956:
955:
950:
947:
945:
942:
940:
937:
935:
932:
930:
927:
925:
922:
920:
917:
916:
915:
914:
911:
908:
907:
897:
896:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
685:
677:
676:
673:
670:
669:
662:
659:
657:
654:
653:
648:
642:
641:
634:
631:
629:
626:
624:
621:
619:
616:
614:
611:
609:
606:
604:
601:
599:
596:
592:
589:
588:
587:
584:
583:
580:
577:
576:
572:
566:
565:
554:
551:
549:
548:Circumference
546:
544:
541:
540:
539:
538:
535:
532:
531:
526:
523:
521:
518:
517:
516:
515:
512:
511:Quadrilateral
509:
508:
503:
500:
498:
495:
493:
490:
488:
485:
484:
483:
482:
479:
478:Parallelogram
476:
475:
470:
467:
465:
462:
460:
457:
456:
455:
454:
451:
448:
447:
442:
439:
437:
434:
432:
429:
428:
427:
426:
420:
414:
413:
406:
403:
399:
396:
394:
391:
390:
389:
386:
385:
381:
375:
374:
367:
364:
363:
359:
353:
352:
345:
342:
340:
337:
335:
332:
331:
328:
325:
323:
320:
317:
316:Perpendicular
313:
312:Orthogonality
310:
308:
305:
303:
300:
298:
295:
294:
291:
288:
287:
286:
276:
273:
272:
267:
266:
257:
254:
253:
252:
249:
247:
244:
242:
239:
237:
236:Computational
234:
232:
229:
225:
222:
221:
220:
217:
215:
212:
210:
207:
203:
200:
198:
195:
193:
190:
189:
188:
185:
181:
178:
176:
173:
172:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
139:
136:
132:
129:
128:
127:
124:
123:
122:
121:Non-Euclidean
119:
117:
114:
113:
109:
103:
102:
95:
91:
88:
86:
83:
82:
80:
79:
75:
71:
67:
62:
58:
57:
54:
51:
50:
44:
39:
33:
19:
5153:
5140:
4982:Thomas Heath
4973:
4856:
4840:Law of sines
4696:
4672:
4628:Golden ratio
4493:
4486:
4477:
4471:(Theodosius)
4467:
4459:
4452:
4445:
4438:
4429:
4419:
4413:(Hipparchus)
4409:
4399:
4391:
4384:
4375:
4365:
4357:
4352:(Apollonius)
4348:
4340:
4328:
4303:Zeno of Elea
4063:Eratosthenes
4053:Dionysodorus
3881:
3868:cut-the-knot
3867:
3858:cut-the-knot
3856:
3847:
3810:
3806:
3796:
3788:
3784:
3779:
3771:
3767:
3762:
3755:
3750:
3740:
3733:
3725:
3722:
3714:
3690:
3686:
3677:
3669:
3665:
3660:
3651:
3626:
3622:
3619:Avron, Arnon
3613:
3603:
3598:
3587:
3582:
3572:, retrieved
3568:the original
3563:
3557:
3547:
3515:
3509:
3500:
3492:
3487:
3475:
3466:
3457:
3448:
3442:
3430:
3417:
3398:
3377:. Retrieved
3372:
3366:
3342:
3312:
3307:
3243:
3231:
3207:
3178:
3174:
3160:
3137:
3122:
3114:
3083:
3079:
3077:
3026:
3022:
3014:
3010:
3006:
3002:
2998:
2994:
2992:
2973:
2969:
2956:
2948:
2936:
2921:
2912:
2897:, the three
2892:
2876:circumcenter
2853:
2843:
2839:
2835:
2831:
2827:
2825:
2716:
2695:
2687:
2682:
2671:
2634:
2624:eccentricity
2622:of positive
2613:
2581:
2562:
2544:
2541:
2518:
2511:
2496:
2482:
2462:
2458:
2454:
2450:
2449:
2250:
2248:
2244:
2201:
2188:
2184:
2178:
2165:
2158:
1922:heptadecagon
1910:
1888:square roots
1887:
1885:
1880:
1876:
1872:
1868:
1828:
1822:
1811:
1795:
1788:
1771:
1767:
1763:
1746:
1742:
1738:
1676:Finding the
1664:
1640:
1636:
1617:intersection
1596:
1557:showed that
1552:
1541:
1536:
1530:
1496:
1472:square roots
1449:
1439:
1435:
1432:parlour game
1429:
1416:
1412:
1410:
1399:
1382:
1376:straightedge
1375:
1361:
1336:square roots
1318:In terms of
1317:
1301:field theory
1279:
1269:
1263:
1242:straightedge
1239:
1219:
1215:
1211:
1207:
1201:
1020:Parameshvara
833:Parameshvara
603:Dodecahedron
289:
187:Differential
5049:mathematics
4857:Arithmetica
4454:Ostomachion
4423:(Autolycus)
4342:Arithmetica
4118:Hippocrates
4048:Dinostratus
4033:Dicaearchus
3963:Aristarchus
3078:The set of
2884:orthocenter
2704:conjectured
2523:, that is,
2485:requires a
2475:denominator
2240:square root
2224:subtraction
2212:square root
2193:orientation
1911:There is a
1498:Hippocrates
1460:differences
1145:Present day
1092:Lobachevsky
1079:1700s–1900s
1036:Jyeṣṭhadeva
1028:1400s–1700s
980:Brahmagupta
803:Lobachevsky
783:Jyeṣṭhadeva
733:Brahmagupta
661:Hypersphere
633:Tetrahedron
608:Icosahedron
180:Diophantine
5101:Babylonian
5001:arithmetic
4967:History of
4796:Apollonius
4481:(Menelaus)
4440:On Spirals
4359:Catoptrics
4298:Xenocrates
4293:Thymaridas
4278:Theodosius
4263:Theaetetus
4243:Simplicius
4233:Pythagoras
4218:Posidonius
4203:Philonides
4163:Nicomachus
4158:Metrodorus
4148:Menaechmus
4103:Hipparchus
4093:Heliodorus
4043:Diophantus
4028:Democritus
4008:Chrysippus
3978:Archimedes
3973:Apollonius
3943:Anaxagoras
3935:(timeline)
3574:2008-09-24
3481:72˚ angle.
3299:References
3170:Apollonius
3162:Archimedes
2993:A regular
2976:Archimedes
2874:, and its
2461:such that
1683:Drawing a
1628:two points
1548:cube roots
1518:quadratrix
1506:hyperbolas
1502:Menaechmus
1258:postulates
1005:al-Yasamin
949:Apollonius
944:Archimedes
934:Pythagoras
924:Baudhayana
878:al-Yasamin
828:Pythagoras
723:Baudhayana
713:Archimedes
708:Apollonius
613:Octahedron
464:Hypotenuse
339:Similarity
334:Congruence
246:Incidence
197:Symplectic
192:Riemannian
175:Arithmetic
150:Projective
138:Hyperbolic
66:Projecting
4562:Inscribed
4322:Treatises
4313:Zenodorus
4273:Theodorus
4248:Sosigenes
4193:Philolaus
4178:Oenopides
4173:Nicoteles
4168:Nicomedes
4128:Hypsicles
4023:Ctesibius
4013:Cleomedes
3998:Callippus
3983:Autolycus
3968:Aristotle
3948:Anthemius
3883:MathWorld
3827:1530-7638
3687:Centaurus
3432:MathWorld
3375:: 366–372
3250:algorithm
3212:that are
3166:Nicomedes
2895:altitudes
2868:altitudes
2714:in 1837.
2708:necessary
2426:¯
2365:¯
2356:−
2299:¯
2115:−
2100:−
2092:−
2037:−
1982:−
1960:π
1946:
1913:bijection
1624:one point
1565:π
1555:Lindemann
1522:Nicomedes
1510:parabolas
1442:correct.
1417:terminate
1368:compasses
1358:A compass
1303:, namely
1262:Euclid's
1235:compasses
1228:idealized
1122:Minkowski
1041:Descartes
975:Aryabhata
970:Kātyāyana
901:by period
813:Minkowski
788:Kātyāyana
748:Descartes
693:Aryabhata
672:Geometers
656:Tesseract
520:Trapezoid
492:Rectangle
285:Dimension
170:Algebraic
160:Synthetic
131:Spherical
116:Euclidean
5176:Category
5126:Japanese
5111:Egyptian
5054:timeline
5042:timeline
5030:timeline
5025:geometry
5018:timeline
5013:calculus
5006:timeline
4994:timeline
4697:Elements
4543:Concepts
4505:Problems
4478:Spherics
4468:Spherics
4433:(Euclid)
4379:(Euclid)
4376:Elements
4369:(Euclid)
4330:Almagest
4238:Serenus
4213:Porphyry
4153:Menelaus
4108:Hippasus
4083:Eutocius
4058:Domninus
3953:Archytas
3468:MacTutor
3260:See also
3244:In 1998
3210:quintics
3195:heptagon
2984:heptagon
2888:incenter
2880:centroid
2860:vertices
2858:are its
2856:triangle
2702:. Gauss
2678:pentagon
2676:(e.g. a
2667:pentagon
2481:, while
2469:)) is a
2232:division
2220:addition
1867:, where
1800: (
1774: (
1749: (
1678:midpoint
1542:In 1837
1526:conchoid
1464:products
1293:polygons
1274:, or by
1264:Elements
1204:geometry
1112:Poincaré
1056:Minggatu
1015:Yang Hui
985:Virasena
873:Yang Hui
868:Virasena
838:Poincaré
818:Minggatu
598:Cylinder
543:Diameter
502:Rhomboid
459:Altitude
450:Triangle
344:Symmetry
322:Parallel
307:Diagonal
277:Features
274:Concepts
165:Analytic
126:Elliptic
108:Branches
94:Timeline
53:Geometry
5106:Chinese
5061:numbers
4989:algebra
4917:Related
4891:Centers
4687:Results
4557:Central
4228:Ptolemy
4223:Proclus
4188:Perseus
4143:Marinus
4123:Hypatia
4113:Hippias
4088:Geminus
4078:Eudoxus
4068:Eudemus
4038:Diocles
3815:Bibcode
3707:1078083
3643:1537763
3540:1626185
3532:2589403
3379:3 March
3145:origami
3128:Origami
3107:in the
3104:A048136
3070:in the
3067:A051913
2978:gave a
2899:medians
2838:-gon, 8
2818:in the
2815:A003401
2804:, 102,
2620:ellipse
2596:radians
2588:radians
2526:√
1897:√
1879:are in
1818:vectors
1793:√
1651:Hilbert
1643:algebra
1514:Hippias
1446:History
1440:exactly
1387:Circles
1383:compass
1320:algebra
1137:Coxeter
1117:Hilbert
1102:Riemann
1051:Huygens
1010:al-Tusi
1000:Khayyám
990:Alhazen
957:1–1400s
858:al-Tusi
843:Riemann
793:Khayyám
778:Huygens
773:Hilbert
743:Coxeter
703:Alhazen
681:by name
618:Pyramid
497:Rhombus
441:Polygon
393:segment
241:Fractal
224:Digital
209:Complex
90:History
85:Outline
43:hexagon
5121:Indian
4898:Cyrene
4430:Optics
4349:Conics
4268:Theano
4258:Thales
4253:Sporus
4198:Philon
4183:Pappus
4073:Euclid
4003:Carpus
3993:Bryson
3825:
3813:: 13.
3705:
3641:
3538:
3530:
3405:
3175:neusis
3005:where
2980:neusis
2886:, and
2862:, the
2800:, 85,
2796:, 68,
2788:, 51,
2637:Oxford
2616:circle
2604:neusis
2238:, and
1875:, and
1610:circle
1470:, and
1468:ratios
1438:to be
1436:proved
1364:rulers
1332:cosine
1224:angles
1158:Gromov
1153:Atiyah
1132:Veblen
1127:Cartan
1097:Bolyai
1066:Sakabe
1046:Pascal
939:Euclid
929:Manava
863:Veblen
848:Sakabe
823:Pascal
808:Manava
768:Gromov
753:Euclid
738:Cartan
728:Bolyai
718:Atiyah
628:Sphere
591:cuboid
579:Volume
534:Circle
487:Square
405:Length
327:Vertex
231:Convex
214:Finite
155:Affine
70:sphere
5116:Incan
5037:logic
4813:Other
4581:Chord
4574:Axiom
4552:Angle
4208:Plato
4098:Heron
4018:Conon
3639:S2CID
3528:JSTOR
3371:. 1.
3363:(PDF)
3189:or a
3187:cubic
3179:tends
2952:above
2672:Some
2473:with
2204:field
2161:Gauss
1647:Gauss
1577:is a
1533:Gauss
1425:limit
1413:exact
1311:(see
1289:Gauss
1231:ruler
1218:, or
1107:Klein
1087:Gauss
1061:Euler
995:Sijzi
965:Zhang
919:Ahmes
883:Zhang
853:Sijzi
798:Klein
763:Gauss
758:Euler
698:Ahmes
431:Plane
366:Point
302:Curve
297:Angle
74:plane
72:to a
5078:list
4366:Data
4138:Leon
3988:Bion
3823:ISSN
3403:ISBN
3381:2014
3201:and
3168:and
3138:The
3109:OEIS
3072:OEIS
3009:and
2820:OEIS
2453:and
2210:and
1603:line
1508:and
1500:and
1456:sums
1450:The
1389:and
1381:The
1374:The
1366:and
1307:and
1071:Aida
688:Aida
647:Four
586:Cube
553:Area
525:Kite
436:Area
388:Line
4980:by
4694:In
3866:at
3855:at
3789:156
3772:109
3695:doi
3631:doi
3564:676
3520:doi
3516:105
3465:at
3001:=23
2810:257
2806:120
2694:of
2598:(60
2594:/3
2586:/5
2465:/(2
1943:cos
1626:or
1427:.)
1260:of
1202:In
910:BCE
398:ray
5178::
3880:.
3821:.
3809:.
3805:.
3787:,
3770:,
3726:95
3703:MR
3701:,
3691:32
3689:,
3670:11
3668:,
3637:.
3627:38
3625:.
3562:,
3556:,
3536:MR
3534:,
3526:,
3514:,
3429:.
3389:^
3365:.
3350:^
3320:^
3236:.
3164:,
3099:23
3097:,
3095:22
3093:,
3091:11
3062:42
3058:21
3056:,
3054:19
3052:,
3050:18
3048:,
3046:14
3044:,
3042:13
3040:,
3036:,
2934:.
2882:,
2878:,
2802:96
2798:80
2794:64
2792:,
2790:60
2786:48
2784:,
2782:40
2780:,
2778:34
2776:,
2774:32
2772:,
2770:30
2768:,
2766:24
2764:,
2762:20
2760:,
2758:17
2756:,
2754:16
2752:,
2750:15
2748:,
2746:12
2744:,
2742:10
2740:,
2736:,
2732:,
2728:,
2724:,
2549:.
2234:,
2230:,
2226:,
2222:,
2199:.
2163:.
2134:17
2123:34
2108:17
2097:34
2087:17
2076:17
2045:17
2034:34
2027:16
2012:17
2005:16
1990:16
1964:17
1883:.
1871:,
1657:.
1466:,
1462:,
1458:,
1370:.
1267:.
1237:.
1214:,
1206:,
68:a
3915:e
3908:t
3901:v
3886:.
3829:.
3817::
3811:1
3697::
3645:.
3633::
3522::
3461:*
3435:.
3411:.
3383:.
3373:2
3111:)
3084:n
3080:n
3074:)
3038:9
3034:7
3027:n
3023:n
3015:m
3011:b
3007:a
3003:m
2999:n
2995:n
2844:n
2840:n
2836:n
2832:n
2828:n
2822:)
2738:8
2734:6
2730:5
2726:4
2722:3
2696:n
2688:n
2600:°
2592:π
2584:π
2529:π
2467:π
2463:φ
2459:φ
2434:.
2423:z
2417:z
2412:=
2408:|
2405:z
2402:|
2376:i
2373:2
2362:z
2353:z
2347:=
2344:)
2341:z
2338:(
2334:m
2331:I
2306:2
2296:z
2290:+
2287:z
2281:=
2278:)
2275:z
2272:(
2268:e
2265:R
2251:z
2189:1
2185:0
2129:2
2126:+
2118:2
2103:2
2082:3
2079:+
2069:8
2066:1
2060:+
2040:2
2024:1
2018:+
2002:1
1996:+
1987:1
1978:=
1969:)
1957:2
1951:(
1899:2
1881:F
1877:k
1873:y
1869:x
1853:k
1848:=
1845:y
1842:+
1839:x
1829:F
1804:)
1796:a
1791:=
1789:x
1778:)
1772:b
1770:/
1768:a
1764:x
1753:)
1747:b
1745:·
1743:a
1739:x
1537:n
1191:e
1184:t
1177:v
318:)
314:(
96:)
92:(
34:.
20:)
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