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49:
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2344:
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1946:
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squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a
1386:
473:
1786:
1910:
2278:
2663:
arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Larger spherical squares have larger angles.
1314:
773:
709:
Since four squared equals sixteen, a four by four square has an area equal to its perimeter. The only other quadrilateral with such a property is that of a three by six rectangle.
1001:, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if
992:
895:
2110:
2771:
of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih
2205:
1050:
2152:
2014:
of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points (
704:
1086:
3116:"Geometry classes, Problem 331. Square, Point on the Inscribed Circle, Tangency Points. Math teacher Master Degree. College, SAT Prep. Elearning, Online math tutor, LMS"
861:
1998:
951:
825:
2670:, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles.
655:
282:
2331:
1659:
1347:
623:
1934:
1632:
1371:
1601:{\displaystyle {\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}
2868:
is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an
3838:
367:
2320:
3192:
2988:
Zalman
Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 59,
1670:
1797:
3273:
345:
A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals)
3868:
3033:
2613:
2993:
3146:
4403:
129:
119:
101:
4413:
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are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then
111:
137:
716:, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term
124:
106:
2127:
of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to
2216:
3991:
3971:
250:
242:
1229:
93:
4398:
3966:
3923:
3898:
2330:
1117:
3232:
4026:
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1010:
2572:
two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two
3951:
3266:
2869:
31:
735:
3976:
3861:
2835:
963:
866:
558:
2060:
4377:
4317:
3956:
3810:
3803:
3796:
3206:
2803:
2741:
2729:
2625:
2002:
550:
507:(opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely:
2891:
1106:
If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
4261:
4031:
3961:
3903:
3467:
3414:
2415:
1174:
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898:
2652:
In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.
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4408:
4367:
4342:
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4307:
4266:
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3822:
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3139:
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2910:
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2595:
2591:
2515:
1151:
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1019:
538:
180:
2130:
673:
4372:
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3641:
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3478:
3441:
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of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
2776:
2656:
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2472:
2300:
1067:
207:
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2717:
2700:
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has only one inscribed square, with a side coinciding with part of the triangle's longest side.
2352:
1140:
833:
546:
83:
2360:
1951:
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791:
4352:
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2011:
1093:
631:
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258:
142:
73:
48:
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3399:
3388:
3377:
3366:
3357:
3348:
3335:
3313:
3301:
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3283:
2826:
It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
2733:
2724:
2629:
2564:
2304:
1170:
1089:
957:
718:
1637:
1325:
608:
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2523:
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1125:
998:
713:
488:
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188:
184:
69:
62:
3006:
2744:. In fact, for any n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.
3080:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
511:
All four internal angles of a square are equal (each being 360°/4 = 90°, a right angle).
4357:
4337:
4302:
4297:
3928:
3908:
3774:
3187:
John H. Conway, Heidi
Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things,
3143:
3091:
2915:
2865:
2713:
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1919:
1617:
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1159:
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1136:
300:
226:
176:
172:
158:
154:
3195:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
2583:
The fraction of the triangle's area that is filled by the square is no more than 1/2.
4392:
4332:
4183:
4076:
3996:
3938:
3791:
3679:
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3665:
3629:
3622:
3615:
3579:
3572:
3296:
2920:
2831:
2542:
2531:
2479:
2382:
The dihedral symmetries are divided depending on whether they pass through vertices (
1113:
1100:
500:
496:
336:
329:
296:
246:
218:
2378:
1945:
4362:
4232:
4188:
4152:
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2660:
2507:
2427:
2324:
2155:
2124:
1374:
779:
230:
195:
2850:
2736:
with 5 around each vertex, with each square having 72-degree internal angles. The
3053:
2787:
with a common vertex, but the geometric intersection is not considered a vertex.
2712:
Six squares can tile the sphere with 3 squares around each vertex and 120-degree
2687:
Two squares can tile the sphere with 2 squares around each vertex and 180-degree
1088:(about 1.414) times the length of a side of the square. This value, known as the
4271:
4178:
4157:
4147:
3740:
3701:
3651:
3601:
3558:
3528:
3460:
3446:
2940:
2925:
2877:
2453:
1109:
A square has a larger area than any other quadrilateral with the same perimeter.
340:
2707:
4276:
4132:
4122:
4006:
3726:
3710:
3660:
3610:
3567:
3537:
3451:
2887:
2637:
574:
3243:
1173:
inside any regular polygon. The only other polygon with this property is the
4251:
4241:
4218:
4208:
4198:
4127:
4036:
4001:
3782:
3696:
3646:
3596:
3553:
3523:
3492:
2965:
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2795:
2598:
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1132:
602:
582:
521:
504:
492:
468:{\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).}
308:
254:
234:
2478:
Each subgroup symmetry allows one or more degrees of freedom for irregular
1009:
are the area and perimeter enclosed by a quadrilateral, then the following
2390:
for perpendiculars) Cyclic symmetries in the middle column are labeled as
593:
4256:
4246:
4203:
4162:
4091:
4081:
4071:
3890:
3756:
3511:
3507:
3434:
3237:
2935:
2768:
2696:
2691:. Each square covers an entire hemisphere and their vertices lie along a
2554:
905:
2601:, is the problem of constructing a square with the same area as a given
4213:
4193:
4106:
4101:
4096:
4086:
4061:
4016:
3877:
3765:
3735:
3502:
3497:
3488:
3429:
3054:"Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram"
2873:
2816:
A square and a crossed square have the following properties in common:
2791:
2503:
554:
484:
322:
315:
17:
2612:
In 1882, the task was proven to be impossible as a consequence of the
1781:{\displaystyle d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})}
3705:
3655:
3605:
3562:
3532:
3483:
2602:
2541:
subgroup has no degrees of freedom, but can be seen as a square with
1905:{\displaystyle d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),}
221:, which means that it has four sides of equal length and four equal
4066:
3846:
2849:
2802:
is related, as a faceting of the rectangle, both special cases of
2445:
A square is a special case of many lower symmetry quadrilaterals:
2394:
for their central gyration orders. Full symmetry of the square is
2329:
2283:
can also be used to describe the boundary of a square with center
1944:
592:
562:
528:
222:
2458:
A parallelogram with one right angle and two adjacent equal sides
1146:
The square is a highly symmetric object. There are four lines of
3455:
2905:
2319:
The following animations show how to construct a square using a
661:
3850:
2510:
of each other, and have half the symmetry order of the square.
597:
The area of a square is the product of the length of its sides.
2471:
These 6 symmetries express 8 distinct symmetries on a square.
1131:
The square is in two families of polytopes in two dimensions:
527:
The diagonal of a square bisects its internal angle, forming
2115:
specifies the boundary of this square. This equation means "
1349:
is the distance from an arbitrary point in the plane to the
1055:
with equality if and only if the quadrilateral is a square.
2617:
1092:
or
Pythagoras' constant, was the first number proven to be
3076:
Chakerian, G.D. "A Distorted View of
Geometry." Ch. 7 in
332:
with one right vertex angle and two adjacent equal sides
514:
The central angle of a square is equal to 90° (360°/4).
30:
This article is about the polygon. For other uses, see
422:
378:
2219:
2167:
2133:
2063:
1954:
1922:
1800:
1673:
1640:
1620:
1389:
1359:
1328:
1232:
1070:
1022:
966:
920:
869:
836:
794:
738:
676:
634:
611:
370:
270:
237:
with two equal-length adjacent sides. It is the only
3144:
http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
2677:
2273:{\displaystyle \left|x-a\right|+\left|y-b\right|=r.}
4285:
4231:
4171:
4115:
4054:
4045:
3937:
3889:
3233:
Animated course (Construction, Circumference, Area)
725:The area can also be calculated using the diagonal
194:
168:
153:
136:
92:
82:
68:
58:
41:
2272:
2199:
2146:
2104:
1992:
1928:
1904:
1780:
1653:
1626:
1600:
1365:
1341:
1308:
1080:
1044:
986:
945:
889:
855:
819:
767:
698:
649:
617:
467:
276:
3244:Animated applet illustrating the area of a square
3164:"Cyclic Averages of Regular Polygonal Distances"
2064:
2872:of the 4 vertices and 6 edges of the regular 3-
2452:A quadrilateral with four equal sides and four
1309:{\displaystyle 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}
517:The external angle of a square is equal to 90°.
3138:Park, Poo-Sung. "Regular polytope distances",
2605:, by using only a finite number of steps with
3862:
3267:
1154:of order 4 (through 90°, 180° and 270°). Its
348:A convex quadrilateral with successive sides
8:
2809:The interior of a crossed square can have a
1103:with equal diagonals that bisect the angles.
3031:"Properties of equidiagonal quadrilaterals"
2790:A crossed square is sometimes likened to a
4051:
3869:
3855:
3847:
3274:
3260:
3252:
2475:labels these by a letter and group order.
2299:. The square is therefore the shape of a
2295:), and a horizontal or vertical radius of
2449:A rectangle with two adjacent equal sides
2218:
2185:
2172:
2166:
2134:
2132:
2087:
2074:
2062:
1979:
1971:
1963:
1955:
1953:
1921:
1890:
1877:
1858:
1853:
1843:
1838:
1825:
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1805:
1799:
1769:
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1505:
1500:
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1415:
1402:
1397:
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1388:
1358:
1333:
1327:
1297:
1281:
1262:
1246:
1231:
1071:
1069:
1036:
1021:
970:
965:
934:
919:
873:
868:
844:
835:
808:
793:
751:
745:
737:
687:
675:
633:
610:
605:of a square whose four sides have length
495:(one pair of opposite sides parallel), a
453:
440:
421:
409:
396:
377:
369:
269:
2754:
2659:, a square is a polygon whose edges are
2377:
520:The diagonals of a square are equal and
503:or tetragon (four-sided polygon), and a
487:(equal sides, opposite equal angles), a
3839:List of regular polytopes and compounds
2951:
2716:. This is called a spherical cube. The
491:(two pairs of adjacent equal sides), a
257:are all equal in length. A square with
2823:The two diagonals are equal in length.
2418:8. There are 2 dihedral subgroups: Dih
2123:, whichever is larger, equals 1." The
38:
3238:Definition and properties of a square
768:{\displaystyle A={\frac {d^{2}}{2}}.}
722:to mean raising to the second power.
534:All four sides of a square are equal.
7:
2959:
2957:
2955:
2695:. This is called a spherical square
2486:is full symmetry of the square, and
1180:If the inscribed circle of a square
987:{\displaystyle \pi /4\approx 0.7854}
890:{\displaystyle 2/\pi \approx 0.6366}
2820:Opposite sides are equal in length.
2105:{\displaystyle \max(x^{2},y^{2})=1}
1936:is the circumradius of the square.
1099:A square can also be defined as a
25:
3171:International Journal of Geometry
1120:of the plane (the others are the
499:(all opposite sides parallel), a
2890:
2723:
2706:
2681:
2359:
2342:
830:since the area of the circle is
537:Opposite sides of a square are
483:A square is a special case of a
303:it is any one of the following:
127:
122:
117:
109:
104:
99:
47:
2464:A rhombus with all angles equal
2323:. This is possible as 4 = 2, a
339:with four equal sides and four
253:are all equal (90°), and whose
233:). It can also be defined as a
2549:Squares inscribed in triangles
2467:A rhombus with equal diagonals
2349:Square at a given side length,
2334:Square at a given circumcircle
2200:{\displaystyle x^{2}+y^{2}=2.}
2093:
2067:
1980:
1972:
1964:
1956:
1896:
1870:
1775:
1749:
1268:
1236:
1064:The diagonals of a square are
459:
433:
415:
389:
229:angles, π/2 radian angles, or
1:
3162:Meskhishvili, Mamuka (2021).
2614:Lindemann–Weierstrass theorem
1045:{\displaystyle 16A\leq P^{2}}
311:with two adjacent equal sides
2461:A rhombus with a right angle
2147:{\displaystyle {\sqrt {2}}.}
911:, the area of the square is
699:{\displaystyle A=\ell ^{2}.}
569:= 2 case of the families of
3090:Lundsgaard Hansen, Martin.
2775:, order 4. It has the same
2630:algebraic irrational number
2398:and no symmetry is labeled
2210:Alternatively the equation
1353:-th vertex of a square and
1143:for the square is {4}.
1081:{\displaystyle {\sqrt {2}}}
524:each other, meeting at 90°.
4430:
3828:
3255:
2552:
2530:defines the geometry of a
2366:Square at a given diagonal
856:{\displaystyle \pi R^{2},}
785:, the area of a square is
29:
2632:; that is, it is not the
1993:{\displaystyle |x|+|y|=2}
1941:Coordinates and equations
946:{\displaystyle A=4r^{2};}
820:{\displaystyle A=2R^{2};}
565:, {2}. The square is the
318:with a right vertex angle
46:
3092:"Vagn Lundsgaard Hansen"
2607:compass and straightedge
2386:for diagonal) or edges (
2321:compass and straightedge
2010:The coordinates for the
1220:on the inscribed circle,
1011:isoperimetric inequality
994:of that of the square.
650:{\displaystyle P=4\ell }
277:{\displaystyle \square }
4404:Types of quadrilaterals
3240:With interactive applet
2870:orthographic projection
2555:Triangle § Squares
94:Coxeter–Dynkin diagrams
53:A regular quadrilateral
32:Square (disambiguation)
4414:Constructible polygons
3205:Wells, Christopher J.
2858:
2836:uniform star polyhedra
2804:crossed quadrilaterals
2779:as the square, and is
2760:
2648:Non-Euclidean geometry
2576:inscribed squares. An
2514:is the symmetry of an
2506:. These two forms are
2403:
2335:
2274:
2201:
2148:
2106:
2007:
1994:
1930:
1906:
1782:
1655:
1628:
1602:
1367:
1343:
1310:
1082:
1046:
988:
956:hence the area of the
947:
891:
857:
821:
769:
700:
651:
619:
598:
469:
278:
3043:, 14 (2014), 129–144.
2853:
2758:
2626:transcendental number
2522:is the symmetry of a
2502:is the symmetry of a
2494:is the symmetry of a
2381:
2351:right angle by using
2333:
2275:
2202:
2149:
2107:
2003:Cartesian coordinates
1995:
1948:
1931:
1907:
1783:
1656:
1654:{\displaystyle d_{i}}
1629:
1603:
1368:
1344:
1342:{\displaystyle d_{i}}
1311:
1216:, then for any point
1148:reflectional symmetry
1083:
1047:
989:
948:
892:
858:
822:
770:
701:
652:
620:
618:{\displaystyle \ell }
596:
470:
325:with all angles equal
279:
27:Regular quadrilateral
4102:Nonagon/Enneagon (9)
4032:Tangential trapezoid
3211:www.technologyuk.net
2783:. It appears as two
2616:, which proves that
2217:
2165:
2131:
2061:
1952:
1920:
1798:
1671:
1638:
1618:
1387:
1357:
1326:
1230:
1184:has tangency points
1175:equilateral triangle
1122:equilateral triangle
1068:
1020:
964:
918:
899:circumscribed circle
867:
834:
792:
736:
674:
632:
609:
553:square, t{4}, is an
368:
268:
4214:Megagon (1,000,000)
3982:Isosceles trapezoid
3823:pentagonal polytope
3722:Uniform 10-polytope
3282:Fundamental convex
3142:16, 2016, 227–232.
3140:Forum Geometricorum
3041:Forum Geometricorum
3029:Josefsson, Martin,
3011:jwilson.coe.uga.edu
2964:Weisstein, Eric W.
2931:Squaring the square
2911:Pythagorean theorem
2840:tetrahemihexahedron
2668:hyperbolic geometry
2592:Squaring the circle
2587:Squaring the circle
2516:isosceles trapezoid
1863:
1848:
1830:
1815:
1742:
1724:
1706:
1688:
1564:
1546:
1528:
1510:
1461:
1443:
1425:
1407:
1377:of the square, then
1152:rotational symmetry
561:square, h{4}, is a
4184:Icositetragon (24)
3692:Uniform 9-polytope
3642:Uniform 8-polytope
3592:Uniform 7-polytope
3549:Uniform 6-polytope
3519:Uniform 5-polytope
3479:Uniform polychoron
3442:Uniform polyhedron
3290:in dimensions 2–10
3149:2016-10-10 at the
3078:Mathematical Plums
3058:www.mathsisfun.com
3036:2022-09-27 at the
2898:Mathematics portal
2859:
2785:45-45-90 triangles
2777:vertex arrangement
2761:
2657:spherical geometry
2404:
2336:
2270:
2197:
2144:
2102:
2008:
1990:
1926:
1902:
1849:
1834:
1816:
1801:
1778:
1728:
1710:
1692:
1674:
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1624:
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1532:
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1496:
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1429:
1411:
1393:
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1306:
1078:
1042:
984:
943:
887:
853:
817:
765:
696:
647:
615:
599:
589:Perimeter and area
465:
431:
387:
274:
208:Euclidean geometry
4399:Elementary shapes
4386:
4385:
4227:
4226:
4204:Myriagon (10,000)
4189:Triacontagon (30)
4153:Heptadecagon (17)
4143:Pentadecagon (15)
4138:Tetradecagon (14)
4077:Quadrilateral (4)
3947:Antiparallelogram
3844:
3843:
3831:Polytope families
3288:uniform polytopes
3193:978-1-56881-220-5
3007:"Problem Set 1.3"
2970:Wolfram MathWorld
2830:It exists in the
2800:crossed rectangle
2781:vertex-transitive
2748:
2747:
2303:according to the
2158:has the equation
2139:
1929:{\displaystyle R}
1627:{\displaystyle L}
1569:
1466:
1366:{\displaystyle R}
1076:
863:the square fills
760:
430:
386:
291:Characterizations
264:would be denoted
204:
203:
16:(Redirected from
4421:
4199:Chiliagon (1000)
4179:Icositrigon (23)
4158:Octadecagon (18)
4148:Hexadecagon (16)
4052:
3871:
3864:
3857:
3848:
3835:Regular polytope
3396:
3385:
3374:
3333:
3276:
3269:
3262:
3253:
3221:
3220:
3218:
3217:
3207:"Quadrilaterals"
3202:
3196:
3185:
3179:
3178:
3168:
3159:
3153:
3136:
3130:
3129:
3127:
3126:
3112:
3106:
3105:
3103:
3102:
3087:
3081:
3074:
3068:
3067:
3065:
3064:
3050:
3044:
3027:
3021:
3020:
3018:
3017:
3003:
2997:
2986:
2980:
2979:
2977:
2976:
2961:
2900:
2895:
2894:
2734:hyperbolic plane
2730:Squares can tile
2727:
2710:
2685:
2678:
2623:
2490:is no symmetry.
2363:
2346:
2301:topological ball
2279:
2277:
2276:
2271:
2260:
2256:
2238:
2234:
2206:
2204:
2203:
2198:
2190:
2189:
2177:
2176:
2153:
2151:
2150:
2145:
2140:
2135:
2111:
2109:
2108:
2103:
2092:
2091:
2079:
2078:
2053:
2040:
1999:
1997:
1996:
1991:
1983:
1975:
1967:
1959:
1935:
1933:
1932:
1927:
1911:
1909:
1908:
1903:
1895:
1894:
1882:
1881:
1862:
1857:
1847:
1842:
1829:
1824:
1814:
1809:
1787:
1785:
1784:
1779:
1774:
1773:
1761:
1760:
1741:
1736:
1723:
1718:
1705:
1700:
1687:
1682:
1660:
1658:
1657:
1652:
1650:
1649:
1633:
1631:
1630:
1625:
1607:
1605:
1604:
1599:
1594:
1593:
1588:
1584:
1583:
1582:
1570:
1565:
1563:
1558:
1545:
1540:
1527:
1522:
1509:
1504:
1494:
1483:
1482:
1467:
1462:
1460:
1455:
1442:
1437:
1424:
1419:
1406:
1401:
1391:
1372:
1370:
1369:
1364:
1348:
1346:
1345:
1340:
1338:
1337:
1315:
1313:
1312:
1307:
1302:
1301:
1286:
1285:
1267:
1266:
1251:
1250:
1169:A square can be
1116:is one of three
1090:square root of 2
1087:
1085:
1084:
1079:
1077:
1072:
1051:
1049:
1048:
1043:
1041:
1040:
997:Because it is a
993:
991:
990:
985:
974:
958:inscribed circle
952:
950:
949:
944:
939:
938:
904:In terms of the
896:
894:
893:
888:
877:
862:
860:
859:
854:
849:
848:
826:
824:
823:
818:
813:
812:
778:In terms of the
774:
772:
771:
766:
761:
756:
755:
746:
705:
703:
702:
697:
692:
691:
656:
654:
653:
648:
624:
622:
621:
616:
474:
472:
471:
466:
458:
457:
445:
444:
432:
423:
414:
413:
401:
400:
388:
379:
283:
281:
280:
275:
132:
131:
130:
126:
125:
121:
120:
114:
113:
112:
108:
107:
103:
102:
51:
39:
21:
4429:
4428:
4424:
4423:
4422:
4420:
4419:
4418:
4389:
4388:
4387:
4382:
4281:
4235:
4223:
4167:
4133:Tridecagon (13)
4123:Hendecagon (11)
4111:
4047:
4041:
4012:Right trapezoid
3933:
3885:
3875:
3845:
3814:
3807:
3800:
3683:
3676:
3669:
3633:
3626:
3619:
3583:
3576:
3410:Regular polygon
3403:
3394:
3387:
3383:
3376:
3372:
3363:
3354:
3347:
3343:
3331:
3325:
3321:
3309:
3291:
3280:
3249:
3229:
3224:
3215:
3213:
3204:
3203:
3199:
3186:
3182:
3166:
3161:
3160:
3156:
3151:Wayback Machine
3137:
3133:
3124:
3122:
3114:
3113:
3109:
3100:
3098:
3096:www2.mat.dtu.dk
3089:
3088:
3084:
3075:
3071:
3062:
3060:
3052:
3051:
3047:
3038:Wayback Machine
3028:
3024:
3015:
3013:
3005:
3004:
3000:
2987:
2983:
2974:
2972:
2963:
2962:
2953:
2949:
2896:
2889:
2886:
2864:
2848:
2811:polygon density
2774:
2753:
2738:Schläfli symbol
2728:
2720:is {4,3}.
2718:Schläfli symbol
2714:internal angles
2711:
2703:is {4,2}.
2701:Schläfli symbol
2689:internal angles
2686:
2650:
2628:rather than an
2621:
2589:
2578:obtuse triangle
2557:
2551:
2441:
2437:
2433:
2425:
2421:
2413:
2376:
2371:
2370:
2369:
2368:
2367:
2364:
2356:
2355:
2353:Thales' theorem
2350:
2347:
2317:
2309:distance metric
2308:
2246:
2242:
2224:
2220:
2215:
2214:
2181:
2168:
2163:
2162:
2129:
2128:
2083:
2070:
2059:
2058:
2054:. The equation
2051:
2042:
2038:
2029:
2027:
2020:
1950:
1949:
1943:
1918:
1917:
1886:
1873:
1796:
1795:
1765:
1752:
1669:
1668:
1641:
1636:
1635:
1616:
1615:
1574:
1495:
1492:
1488:
1487:
1474:
1392:
1385:
1384:
1355:
1354:
1329:
1324:
1323:
1293:
1277:
1258:
1242:
1228:
1227:
1165:
1141:Schläfli symbol
1126:regular hexagon
1118:regular tilings
1066:
1065:
1061:
1032:
1018:
1017:
999:regular polygon
962:
961:
930:
916:
915:
865:
864:
840:
832:
831:
804:
790:
789:
747:
734:
733:
714:classical times
683:
672:
671:
630:
629:
607:
606:
591:
547:Schläfli symbol
529:adjacent angles
481:
449:
436:
405:
392:
366:
365:
293:
266:
265:
239:regular polygon
148:
128:
123:
118:
116:
115:
110:
105:
100:
98:
84:Schläfli symbol
63:Regular polygon
54:
35:
28:
23:
22:
15:
12:
11:
5:
4427:
4425:
4417:
4416:
4411:
4406:
4401:
4391:
4390:
4384:
4383:
4381:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4338:Pseudotriangle
4335:
4330:
4325:
4320:
4315:
4310:
4305:
4300:
4295:
4289:
4287:
4283:
4282:
4280:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4238:
4236:
4229:
4228:
4225:
4224:
4222:
4221:
4216:
4211:
4206:
4201:
4196:
4191:
4186:
4181:
4175:
4173:
4169:
4168:
4166:
4165:
4160:
4155:
4150:
4145:
4140:
4135:
4130:
4128:Dodecagon (12)
4125:
4119:
4117:
4113:
4112:
4110:
4109:
4104:
4099:
4094:
4089:
4084:
4079:
4074:
4069:
4064:
4058:
4056:
4049:
4043:
4042:
4040:
4039:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3949:
3943:
3941:
3939:Quadrilaterals
3935:
3934:
3932:
3931:
3926:
3921:
3916:
3911:
3906:
3901:
3895:
3893:
3887:
3886:
3876:
3874:
3873:
3866:
3859:
3851:
3842:
3841:
3826:
3825:
3816:
3812:
3805:
3798:
3794:
3785:
3768:
3759:
3748:
3747:
3745:
3743:
3738:
3729:
3724:
3718:
3717:
3715:
3713:
3708:
3699:
3694:
3688:
3687:
3685:
3681:
3674:
3667:
3663:
3658:
3649:
3644:
3638:
3637:
3635:
3631:
3624:
3617:
3613:
3608:
3599:
3594:
3588:
3587:
3585:
3581:
3574:
3570:
3565:
3556:
3551:
3545:
3544:
3542:
3540:
3535:
3526:
3521:
3515:
3514:
3505:
3500:
3495:
3486:
3481:
3475:
3474:
3465:
3463:
3458:
3449:
3444:
3438:
3437:
3432:
3427:
3422:
3417:
3412:
3406:
3405:
3401:
3397:
3392:
3381:
3370:
3361:
3352:
3345:
3339:
3329:
3323:
3317:
3311:
3305:
3299:
3293:
3292:
3281:
3279:
3278:
3271:
3264:
3256:
3251:
3247:
3246:
3241:
3235:
3228:
3227:External links
3225:
3223:
3222:
3197:
3180:
3154:
3131:
3120:gogeometry.com
3107:
3082:
3069:
3045:
3022:
2998:
2981:
2950:
2948:
2945:
2944:
2943:
2938:
2933:
2928:
2923:
2918:
2916:Square lattice
2913:
2908:
2902:
2901:
2885:
2882:
2866:complete graph
2862:
2847:
2844:
2828:
2827:
2824:
2821:
2772:
2765:crossed square
2759:Crossed-square
2752:
2751:Crossed square
2749:
2746:
2745:
2721:
2704:
2649:
2646:
2644:coefficients.
2594:, proposed by
2588:
2585:
2570:right triangle
2561:acute triangle
2553:Main article:
2550:
2547:
2543:directed edges
2480:quadrilaterals
2469:
2468:
2465:
2462:
2459:
2456:
2450:
2439:
2435:
2431:
2423:
2419:
2411:
2375:
2372:
2365:
2358:
2357:
2348:
2341:
2340:
2339:
2338:
2337:
2316:
2313:
2306:
2281:
2280:
2269:
2266:
2263:
2259:
2255:
2252:
2249:
2245:
2241:
2237:
2233:
2230:
2227:
2223:
2208:
2207:
2196:
2193:
2188:
2184:
2180:
2175:
2171:
2143:
2138:
2113:
2112:
2101:
2098:
2095:
2090:
2086:
2082:
2077:
2073:
2069:
2066:
2047:
2034:
2025:
2018:
1989:
1986:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1942:
1939:
1938:
1937:
1925:
1914:
1913:
1912:
1901:
1898:
1893:
1889:
1885:
1880:
1876:
1872:
1869:
1866:
1861:
1856:
1852:
1846:
1841:
1837:
1833:
1828:
1823:
1819:
1813:
1808:
1804:
1790:
1789:
1788:
1777:
1772:
1768:
1764:
1759:
1755:
1751:
1748:
1745:
1740:
1735:
1731:
1727:
1722:
1717:
1713:
1709:
1704:
1699:
1695:
1691:
1686:
1681:
1677:
1663:
1662:
1648:
1644:
1623:
1611:
1610:
1609:
1608:
1597:
1592:
1587:
1581:
1577:
1573:
1568:
1562:
1557:
1553:
1549:
1544:
1539:
1535:
1531:
1526:
1521:
1517:
1513:
1508:
1503:
1499:
1491:
1486:
1481:
1477:
1473:
1470:
1465:
1459:
1454:
1450:
1446:
1441:
1436:
1432:
1428:
1423:
1418:
1414:
1410:
1405:
1400:
1396:
1379:
1378:
1362:
1336:
1332:
1319:
1318:
1317:
1316:
1305:
1300:
1296:
1292:
1289:
1284:
1280:
1276:
1273:
1270:
1265:
1261:
1257:
1254:
1249:
1245:
1241:
1238:
1235:
1222:
1221:
1178:
1167:
1163:
1160:dihedral group
1156:symmetry group
1144:
1137:cross-polytope
1129:
1110:
1107:
1104:
1097:
1075:
1060:
1057:
1053:
1052:
1039:
1035:
1031:
1028:
1025:
983:
980:
977:
973:
969:
954:
953:
942:
937:
933:
929:
926:
923:
886:
883:
880:
876:
872:
852:
847:
843:
839:
828:
827:
816:
811:
807:
803:
800:
797:
776:
775:
764:
759:
754:
750:
744:
741:
707:
706:
695:
690:
686:
682:
679:
658:
657:
646:
643:
640:
637:
614:
590:
587:
543:
542:
535:
532:
525:
518:
515:
512:
480:
477:
476:
475:
464:
461:
456:
452:
448:
443:
439:
435:
429:
426:
420:
417:
412:
408:
404:
399:
395:
391:
385:
382:
376:
373:
364:whose area is
346:
343:
333:
326:
319:
312:
301:if and only if
292:
289:
273:
251:external angle
243:internal angle
202:
201:
198:
192:
191:
170:
166:
165:
162:
155:Internal angle
151:
150:
146:
140:
138:Symmetry group
134:
133:
96:
90:
89:
86:
80:
79:
76:
66:
65:
60:
56:
55:
52:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4426:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4397:
4396:
4394:
4379:
4378:Weakly simple
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4318:Infinite skew
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4290:
4288:
4284:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4239:
4237:
4234:
4233:Star polygons
4230:
4220:
4219:Apeirogon (∞)
4217:
4215:
4212:
4210:
4207:
4205:
4202:
4200:
4197:
4195:
4192:
4190:
4187:
4185:
4182:
4180:
4177:
4176:
4174:
4170:
4164:
4163:Icosagon (20)
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4120:
4118:
4114:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4059:
4057:
4053:
4050:
4044:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3997:Parallelogram
3995:
3993:
3992:Orthodiagonal
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3972:Ex-tangential
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3944:
3942:
3940:
3936:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3900:
3897:
3896:
3894:
3892:
3888:
3883:
3879:
3872:
3867:
3865:
3860:
3858:
3853:
3852:
3849:
3840:
3836:
3832:
3827:
3824:
3820:
3817:
3815:
3808:
3801:
3795:
3793:
3789:
3786:
3784:
3780:
3776:
3772:
3769:
3767:
3763:
3760:
3758:
3754:
3750:
3749:
3746:
3744:
3742:
3739:
3737:
3733:
3730:
3728:
3725:
3723:
3720:
3719:
3716:
3714:
3712:
3709:
3707:
3703:
3700:
3698:
3695:
3693:
3690:
3689:
3686:
3684:
3677:
3670:
3664:
3662:
3659:
3657:
3653:
3650:
3648:
3645:
3643:
3640:
3639:
3636:
3634:
3627:
3620:
3614:
3612:
3609:
3607:
3603:
3600:
3598:
3595:
3593:
3590:
3589:
3586:
3584:
3577:
3571:
3569:
3566:
3564:
3560:
3557:
3555:
3552:
3550:
3547:
3546:
3543:
3541:
3539:
3536:
3534:
3530:
3527:
3525:
3522:
3520:
3517:
3516:
3513:
3509:
3506:
3504:
3501:
3499:
3498:Demitesseract
3496:
3494:
3490:
3487:
3485:
3482:
3480:
3477:
3476:
3473:
3469:
3466:
3464:
3462:
3459:
3457:
3453:
3450:
3448:
3445:
3443:
3440:
3439:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3407:
3404:
3398:
3395:
3391:
3384:
3380:
3373:
3369:
3364:
3360:
3355:
3351:
3346:
3344:
3342:
3338:
3328:
3324:
3322:
3320:
3316:
3312:
3310:
3308:
3304:
3300:
3298:
3295:
3294:
3289:
3285:
3277:
3272:
3270:
3265:
3263:
3258:
3257:
3254:
3250:
3245:
3242:
3239:
3236:
3234:
3231:
3230:
3226:
3212:
3208:
3201:
3198:
3194:
3190:
3184:
3181:
3176:
3172:
3165:
3158:
3155:
3152:
3148:
3145:
3141:
3135:
3132:
3121:
3117:
3111:
3108:
3097:
3093:
3086:
3083:
3079:
3073:
3070:
3059:
3055:
3049:
3046:
3042:
3039:
3035:
3032:
3026:
3023:
3012:
3008:
3002:
2999:
2995:
2994:1-59311-695-0
2991:
2985:
2982:
2971:
2967:
2960:
2958:
2956:
2952:
2946:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2921:Square number
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2903:
2899:
2893:
2888:
2883:
2881:
2879:
2875:
2871:
2867:
2856:
2852:
2845:
2843:
2841:
2837:
2833:
2832:vertex figure
2825:
2822:
2819:
2818:
2817:
2814:
2812:
2807:
2805:
2801:
2797:
2793:
2788:
2786:
2782:
2778:
2770:
2766:
2757:
2750:
2743:
2739:
2735:
2731:
2726:
2722:
2719:
2715:
2709:
2705:
2702:
2698:
2694:
2690:
2684:
2680:
2679:
2676:
2675:
2671:
2669:
2664:
2662:
2658:
2653:
2647:
2645:
2643:
2639:
2635:
2631:
2627:
2619:
2615:
2610:
2608:
2604:
2600:
2597:
2593:
2586:
2584:
2581:
2579:
2575:
2571:
2566:
2562:
2556:
2548:
2546:
2544:
2540:
2535:
2533:
2532:parallelogram
2529:
2525:
2521:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2476:
2474:
2466:
2463:
2460:
2457:
2455:
2451:
2448:
2447:
2446:
2443:
2429:
2417:
2409:
2401:
2397:
2393:
2389:
2385:
2380:
2373:
2362:
2354:
2345:
2332:
2328:
2326:
2322:
2314:
2312:
2310:
2302:
2298:
2294:
2290:
2286:
2267:
2264:
2261:
2257:
2253:
2250:
2247:
2243:
2239:
2235:
2231:
2228:
2225:
2221:
2213:
2212:
2211:
2194:
2191:
2186:
2182:
2178:
2173:
2169:
2161:
2160:
2159:
2157:
2141:
2136:
2126:
2122:
2118:
2099:
2096:
2088:
2084:
2080:
2075:
2071:
2057:
2056:
2055:
2050:
2046:
2037:
2033:
2024:
2017:
2013:
2005:
2004:
1987:
1984:
1976:
1968:
1960:
1947:
1940:
1923:
1915:
1899:
1891:
1887:
1883:
1878:
1874:
1867:
1864:
1859:
1854:
1850:
1844:
1839:
1835:
1831:
1826:
1821:
1817:
1811:
1806:
1802:
1794:
1793:
1791:
1770:
1766:
1762:
1757:
1753:
1746:
1743:
1738:
1733:
1729:
1725:
1720:
1715:
1711:
1707:
1702:
1697:
1693:
1689:
1684:
1679:
1675:
1667:
1666:
1665:
1664:
1646:
1642:
1621:
1613:
1612:
1595:
1590:
1585:
1579:
1575:
1571:
1566:
1560:
1555:
1551:
1547:
1542:
1537:
1533:
1529:
1524:
1519:
1515:
1511:
1506:
1501:
1497:
1489:
1484:
1479:
1475:
1471:
1468:
1463:
1457:
1452:
1448:
1444:
1439:
1434:
1430:
1426:
1421:
1416:
1412:
1408:
1403:
1398:
1394:
1383:
1382:
1381:
1380:
1376:
1360:
1352:
1334:
1330:
1321:
1320:
1303:
1298:
1294:
1290:
1287:
1282:
1278:
1274:
1271:
1263:
1259:
1255:
1252:
1247:
1243:
1239:
1233:
1226:
1225:
1224:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1176:
1172:
1168:
1161:
1157:
1153:
1149:
1145:
1142:
1138:
1134:
1130:
1127:
1123:
1119:
1115:
1114:square tiling
1111:
1108:
1105:
1102:
1101:parallelogram
1098:
1095:
1091:
1073:
1063:
1062:
1058:
1056:
1037:
1033:
1029:
1026:
1023:
1016:
1015:
1014:
1012:
1008:
1004:
1000:
995:
981:
978:
975:
971:
967:
959:
940:
935:
931:
927:
924:
921:
914:
913:
912:
910:
907:
902:
900:
884:
881:
878:
874:
870:
850:
845:
841:
837:
814:
809:
805:
801:
798:
795:
788:
787:
786:
784:
781:
762:
757:
752:
748:
742:
739:
732:
731:
730:
729:according to
728:
723:
721:
720:
715:
710:
693:
688:
684:
680:
677:
670:
669:
668:
666:
663:
644:
641:
638:
635:
628:
627:
626:
612:
604:
595:
588:
586:
584:
580:
576:
572:
568:
564:
560:
556:
552:
548:
545:A square has
540:
536:
533:
530:
526:
523:
519:
516:
513:
510:
509:
508:
506:
502:
501:quadrilateral
498:
497:parallelogram
494:
490:
486:
478:
462:
454:
450:
446:
441:
437:
427:
424:
418:
410:
406:
402:
397:
393:
383:
380:
374:
371:
363:
359:
355:
351:
347:
344:
342:
338:
337:quadrilateral
334:
331:
330:parallelogram
327:
324:
320:
317:
313:
310:
306:
305:
304:
302:
298:
297:quadrilateral
290:
288:
286:
271:
263:
260:
256:
252:
248:
247:central angle
244:
240:
236:
232:
228:
224:
220:
219:quadrilateral
217:
213:
209:
199:
197:
193:
190:
186:
182:
178:
174:
171:
167:
163:
160:
156:
152:
144:
141:
139:
135:
97:
95:
91:
87:
85:
81:
77:
75:
71:
67:
64:
61:
57:
50:
45:
40:
37:
33:
19:
4172:>20 sides
4107:Decagon (10)
4092:Heptagon (7)
4082:Pentagon (5)
4072:Triangle (3)
4021:
3967:Equidiagonal
3818:
3787:
3778:
3770:
3761:
3752:
3732:10-orthoplex
3468:Dodecahedron
3419:
3389:
3378:
3367:
3358:
3349:
3340:
3336:
3326:
3318:
3314:
3306:
3302:
3248:
3214:. Retrieved
3210:
3200:
3183:
3174:
3170:
3157:
3134:
3123:. Retrieved
3119:
3110:
3099:. Retrieved
3095:
3085:
3077:
3072:
3061:. Retrieved
3057:
3048:
3040:
3025:
3014:. Retrieved
3010:
3001:
2984:
2973:. Retrieved
2969:
2860:
2829:
2815:
2808:
2789:
2764:
2762:
2693:great circle
2673:
2672:
2665:
2661:great circle
2654:
2651:
2611:
2590:
2582:
2573:
2558:
2538:
2536:
2527:
2519:
2511:
2499:
2491:
2487:
2483:
2477:
2470:
2454:right angles
2444:
2430:subgroups: Z
2407:
2405:
2399:
2395:
2391:
2387:
2383:
2325:power of two
2318:
2315:Construction
2296:
2292:
2288:
2282:
2209:
2156:circumcircle
2125:circumradius
2120:
2116:
2114:
2048:
2044:
2035:
2031:
2022:
2015:
2009:
2001:
1375:circumradius
1350:
1217:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1054:
1006:
1002:
996:
955:
908:
903:
829:
782:
780:circumradius
777:
726:
724:
717:
711:
708:
664:
659:
600:
578:
570:
566:
544:
482:
361:
357:
353:
349:
341:right angles
299:is a square
294:
284:
261:
231:right angles
211:
205:
196:Dual polygon
149:), order 2×4
36:
4368:Star-shaped
4343:Rectilinear
4313:Equilateral
4308:Equiangular
4272:Hendecagram
4116:11–20 sides
4097:Octagon (8)
4087:Hexagon (6)
4062:Monogon (1)
3904:Equilateral
3741:10-demicube
3702:9-orthoplex
3652:8-orthoplex
3602:7-orthoplex
3559:6-orthoplex
3529:5-orthoplex
3484:Pentachoron
3472:Icosahedron
3447:Tetrahedron
2941:Unit square
2926:Square root
2878:tetrahedron
2473:John Conway
2285:coordinates
2000:plotted on
1150:and it has
1059:Other facts
583:orthoplexes
181:equilateral
4409:4 (number)
4393:Categories
4373:Tangential
4277:Dodecagram
4055:1–10 sides
4046:By number
4027:Tangential
4007:Right kite
3727:10-simplex
3711:9-demicube
3661:8-demicube
3611:7-demicube
3568:6-demicube
3538:5-demicube
3452:Octahedron
3216:2017-12-12
3125:2017-12-12
3101:2017-12-12
3063:2020-09-02
3016:2017-12-12
2975:2020-09-02
2947:References
2638:polynomial
2563:has three
2414:symmetry,
1094:irrational
575:hypercubes
559:alternated
557:, {8}. An
479:Properties
169:Properties
4353:Reinhardt
4262:Enneagram
4252:Heptagram
4242:Pentagram
4209:65537-gon
4067:Digon (2)
4037:Trapezoid
4002:Rectangle
3952:Bicentric
3914:Isosceles
3891:Triangles
3775:orthoplex
3697:9-simplex
3647:8-simplex
3597:7-simplex
3554:6-simplex
3524:5-simplex
3493:Tesseract
2855:3-simplex
2796:butterfly
2674:Examples:
2599:geometers
2565:inscribed
2537:Only the
2496:rectangle
2251:−
2229:−
2154:Then the
1288:−
1253:−
1171:inscribed
1133:hypercube
1030:≤
979:≈
968:π
882:≈
879:π
838:π
685:ℓ
645:ℓ
613:ℓ
603:perimeter
551:truncated
505:rectangle
493:trapezoid
309:rectangle
272:◻
255:diagonals
235:rectangle
4328:Isotoxal
4323:Isogonal
4267:Decagram
4257:Octagram
4247:Hexagram
4048:of sides
3977:Harmonic
3878:Polygons
3829:Topics:
3792:demicube
3757:polytope
3751:Uniform
3512:600-cell
3508:120-cell
3461:Demicube
3435:Pentagon
3415:Triangle
3177:: 58–65.
3147:Archived
3034:Archived
2966:"Square"
2936:Squircle
2884:See also
2769:faceting
2740:is
2697:dihedron
2642:rational
2574:distinct
2426:, and 3
2374:Symmetry
2043:−1 <
2030:−1 <
2012:vertices
1135:and the
1124:and the
906:inradius
660:and the
539:parallel
259:vertices
189:isotoxal
185:isogonal
143:Dihedral
74:vertices
4348:Regular
4293:Concave
4286:Classes
4194:257-gon
4017:Rhombus
3957:Crossed
3766:simplex
3736:10-cube
3503:24-cell
3489:16-cell
3430:Hexagon
3284:regular
2874:simplex
2792:bow tie
2636:of any
2624:) is a
2596:ancient
2504:rhombus
2438:, and Z
2410:has Dih
2028:) with
1373:is the
1162: D
1158:is the
1013:holds:
897:of its
555:octagon
549:{4}. A
531:of 45°.
485:rhombus
323:rhombus
316:rhombus
216:regular
159:degrees
18:Squares
4358:Simple
4303:Cyclic
4298:Convex
4022:Square
3962:Cyclic
3924:Obtuse
3919:Kepler
3706:9-cube
3656:8-cube
3606:7-cube
3563:6-cube
3533:5-cube
3420:Square
3297:Family
3191:
2992:
2846:Graphs
2838:, the
2798:. the
2699:. The
2603:circle
2559:Every
2518:, and
2498:, and
2428:cyclic
2408:square
2052:< 1
2039:< 1
1916:where
1208:, and
1139:. The
982:0.7854
885:0.6366
719:square
522:bisect
249:, and
241:whose
227:degree
223:angles
212:square
177:cyclic
173:Convex
42:Square
4333:Magic
3929:Right
3909:Ideal
3899:Acute
3425:p-gon
3167:(PDF)
2861:The K
2834:of a
2767:is a
2742:{4,5}
2640:with
2508:duals
2422:, Dih
2416:order
563:digon
214:is a
70:Edges
4363:Skew
3987:Kite
3882:List
3783:cube
3456:Cube
3286:and
3189:ISBN
2990:ISBN
2906:Cube
2857:(3D)
2732:the
2634:root
2524:kite
2406:The
2041:and
1792:and
1634:and
1182:ABCD
1005:and
662:area
601:The
577:and
489:kite
285:ABCD
262:ABCD
225:(90-
210:, a
200:Self
72:and
59:Type
3332:(p)
2880:).
2794:or
2666:In
2655:In
2434:, Z
2119:or
2065:max
1614:If
1322:If
1212:on
1204:on
1196:on
1188:on
960:is
712:In
667:is
625:is
206:In
164:90°
88:{4}
4395::
3837:•
3833:•
3813:21
3809:•
3806:k1
3802:•
3799:k2
3777:•
3734:•
3704:•
3682:21
3678:•
3675:41
3671:•
3668:42
3654:•
3632:21
3628:•
3625:31
3621:•
3618:32
3604:•
3582:21
3578:•
3575:22
3561:•
3531:•
3510:•
3491:•
3470:•
3454:•
3386:/
3375:/
3365:/
3356:/
3334:/
3209:.
3175:10
3173:.
3169:.
3118:.
3094:.
3056:.
3009:.
2968:.
2954:^
2842:.
2806:.
2763:A
2618:pi
2609:.
2545:.
2539:g4
2534:.
2528:g2
2526:.
2520:p2
2512:d2
2500:p4
2492:d4
2488:a1
2484:r8
2482:.
2442:.
2400:a1
2396:r8
2327:.
2311:.
2291:,
2195:2.
2021:,
1214:DA
1206:CD
1200:,
1198:BC
1192:,
1190:AB
1128:).
1112:A
1024:16
901:.
585:.
360:,
356:,
352:,
335:A
328:A
321:A
314:A
307:A
295:A
287:.
245:,
187:,
183:,
179:,
175:,
145:(D
3884:)
3880:(
3870:e
3863:t
3856:v
3821:-
3819:n
3811:k
3804:2
3797:1
3790:-
3788:n
3781:-
3779:n
3773:-
3771:n
3764:-
3762:n
3755:-
3753:n
3680:4
3673:2
3666:1
3630:3
3623:2
3616:1
3580:2
3573:1
3402:n
3400:H
3393:2
3390:G
3382:4
3379:F
3371:8
3368:E
3362:7
3359:E
3353:6
3350:E
3341:n
3337:D
3330:2
3327:I
3319:n
3315:B
3307:n
3303:A
3275:e
3268:t
3261:v
3219:.
3128:.
3104:.
3066:.
3019:.
2996:.
2978:.
2876:(
2863:4
2773:2
2622:π
2620:(
2440:1
2436:2
2432:4
2424:1
2420:2
2412:4
2402:.
2392:g
2388:p
2384:d
2307:1
2305:L
2297:r
2293:b
2289:a
2287:(
2268:.
2265:r
2262:=
2258:|
2254:b
2248:y
2244:|
2240:+
2236:|
2232:a
2226:x
2222:|
2192:=
2187:2
2183:y
2179:+
2174:2
2170:x
2142:.
2137:2
2121:y
2117:x
2100:1
2097:=
2094:)
2089:2
2085:y
2081:,
2076:2
2072:x
2068:(
2049:i
2045:y
2036:i
2032:x
2026:i
2023:y
2019:i
2016:x
2006:.
1988:2
1985:=
1981:|
1977:y
1973:|
1969:+
1965:|
1961:x
1957:|
1924:R
1900:,
1897:)
1892:4
1888:L
1884:+
1879:4
1875:R
1871:(
1868:2
1865:=
1860:2
1855:4
1851:d
1845:2
1840:2
1836:d
1832:+
1827:2
1822:3
1818:d
1812:2
1807:1
1803:d
1776:)
1771:2
1767:L
1763:+
1758:2
1754:R
1750:(
1747:2
1744:=
1739:2
1734:4
1730:d
1726:+
1721:2
1716:2
1712:d
1708:=
1703:2
1698:3
1694:d
1690:+
1685:2
1680:1
1676:d
1647:i
1643:d
1622:L
1596:.
1591:2
1586:)
1580:2
1576:R
1572:+
1567:4
1561:2
1556:4
1552:d
1548:+
1543:2
1538:3
1534:d
1530:+
1525:2
1520:2
1516:d
1512:+
1507:2
1502:1
1498:d
1490:(
1485:=
1480:4
1476:R
1472:3
1469:+
1464:4
1458:4
1453:4
1449:d
1445:+
1440:4
1435:3
1431:d
1427:+
1422:4
1417:2
1413:d
1409:+
1404:4
1399:1
1395:d
1361:R
1351:i
1335:i
1331:d
1304:.
1299:2
1295:B
1291:P
1283:2
1279:D
1275:P
1272:=
1269:)
1264:2
1260:E
1256:P
1248:2
1244:H
1240:P
1237:(
1234:2
1218:P
1210:H
1202:G
1194:F
1186:E
1177:.
1166:.
1164:4
1096:.
1074:2
1038:2
1034:P
1027:A
1007:P
1003:A
976:4
972:/
941:;
936:2
932:r
928:4
925:=
922:A
909:r
875:/
871:2
851:,
846:2
842:R
815:;
810:2
806:R
802:2
799:=
796:A
783:R
763:.
758:2
753:2
749:d
743:=
740:A
727:d
694:.
689:2
681:=
678:A
665:A
642:4
639:=
636:P
581:-
579:n
573:-
571:n
567:n
541:.
463:.
460:)
455:2
451:d
447:+
442:2
438:b
434:(
428:2
425:1
419:=
416:)
411:2
407:c
403:+
398:2
394:a
390:(
384:2
381:1
375:=
372:A
362:d
358:c
354:b
350:a
161:)
157:(
147:4
78:4
34:.
20:)
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