938:
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86:
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75:
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43:
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34:
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499:)}; connecting every third vertex of the pentagon will yield an identical result to that of connecting every second vertex. However, the vertices will be reached in the opposite direction, which makes a difference when retrograde polygons are incorporated in higher-dimensional polytopes. For example, an
467:
points regularly spaced in a circular placement. For instance, in a regular pentagon, a five-pointed star can be obtained by drawing a line from the 1st to the 3rd vertex, from the 3rd to the 5th vertex, from the 5th to the 2nd vertex, from the 2nd to the 4th vertex, and from the 4th to the 1st
535:
are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example, {6/2} will appear as a triangle, but can be labeled with two sets of vertices: 1-3 and 4-6. This should be seen not as two overlapping triangles, but as a double-winding single unicursal hexagon.
1358:
Wherever a line segment may be drawn between two sides, the region in which the line segment lies is treated as inside the figure. This is shown in the right hand illustration and commonly occurs when making a physical
585:/2, the lines will be parallel, with both resulting in no further intersection in Euclidean space. However, it may be possible to construct some such polygons in spherical space, similarly to the
1351:. The exterior is given a density of 0, and any region of density > 0 is treated as internal. This is shown in the central illustration and commonly occurs in the mathematical treatment of
1355:. (However, for non-orientable polyhedra, density can only be considered modulo 2 and hence, in those cases, for consistency, the first treatment is sometimes used instead.)
451:-sided simple polygon to another vertex, non-adjacent to the first one, and continuing the process until the original vertex is reached again. Alternatively, for integers
1097:
includes nonperiodic tilings, like that with three regular pentagons and one regular star pentagon fitting around certain vertices, 5.5.5.5/2, and related to modern
1338:
Where a line segment occurs, one side is treated as outside and the other as inside. This is shown in the left hand illustration and commonly occurs in computer
1740:
937:
1555:
1826:
1827:
Branko
Grunbaum and Geoffrey C. Shephard, Tilings by Regular Polygons, Mathematics Magazine #50 (1977), pp. 227–247, and #51 (1978), pp. 205–206
1942:
1861:
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979:
1984:
133:
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1957:
The
Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History
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508:
1599:
561:
polygon. Constructions based on stellation also allow regular polygonal compounds to be obtained in cases where the density
1400:
230:
1083:
These polygons are often seen in tiling patterns. The parametric angle 𝛼 (in degrees or radians) can be chosen to match
1309:
The interior of a star polygon may be treated in different ways. Three such treatments are illustrated for a pentagram.
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have been studied in depth; while star polygons in general appear not to have been formally defined,
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60:, |5/2|, has ten edges and two sets of five vertices. The first is used in definitions of
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two, because each side arises once from an original side and once from an original vertex. Since
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When the area of the polygon is calculated, each of these approaches yields a different result.
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Regular convex and star polygons with 3 to 12 vertices, labeled with their Schläfli symbols
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2443:
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2408:
2127:
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1451:) is also known as a pentalpha or pentangle, and historically has been considered by many
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of vertices are not coprime. When constructing star polygons from stellation, however, if
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176:
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140:
125:
56:, {5/2}, has five vertices (its corner tips) and five intersecting edges, while a concave
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The number of times that the polygonal curve winds around a given region determines its
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1435:
Star polygons feature prominently in art and culture. Such polygons may or may not be
1079:
Uniform tiling § Uniform tilings using star polygons as concave alternating faces
507:; the analogous construction from a retrograde "crossed pentagram" {5/3} results in a
332:
2508:
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2192:
2112:
2054:
1570:
1486:
541:
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192:
90:
233:, but synonyms using other prefixes exist. For example, a nine-pointed polygon or
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Alternatively, a regular star polygon can also be obtained as a sequence of
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can arise through truncation operations on regular simple or star polygons.
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2006:
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53:
723:}, the outer internal and inner external angles, also denoted by 𝛼 and
209:
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2222:
2217:
2212:
2202:
2177:
2132:
1993:
1529:
1407: in this section. Unsourced material may be challenged and removed.
605:-gon, the resulting figure is no longer regular, but can be seen as an
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345:
57:
219:
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of neighboring polygons in a tessellation pattern. In his 1619 work
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601:
When the intersecting line segments are removed from a regular star
1030:
33:
2182:
1962:
1475:
593:; such polygons do not yet appear to have been studied in detail.
590:
25:
1921:
Proc of NATO-ASI Conference on
Polytopes ... etc. (Toronto, 1993)
1630:
Grünbaum & Shephard (1987). Tilings and
Patterns. Section 2.5
459:, it can be considered as being constructed by connecting every
1966:
631:, represents such a star that matches the outline of a regular
1376:
1313:
and
Geoffrey Shephard consider two of them, as regular star
136:
that do not generate new vertices, and the other one to the
700:
degrees, necessarily, and the inner (new) vertices have an
427:
Regular star polygons were first studied systematically by
1756:
Coxeter, The
Densities of the Regular Polytopes I, p. 43:
1945:(Chapter 26, p. 404: Regular star-polytopes Dimension 2)
1814:
the density of a polygon is never altered by truncation.
1105:
Examples of isogonal tilings with isotoxal simple stars
68:, while the second is sometimes used in planar tilings.
1668:
Coxeter, Introduction to
Geometry, second edition, 2.8
443:
Regular star polygons can be created by connecting one
730:, do not have to match those of any regular polygram {
577:/2, the lines will instead diverge infinitely, and if
289:
1470:) also have occult significance, particularly in the
383:≥ 2. The density of a polygon can also be called its
344:
is a self-intersecting, equilateral, and equiangular
124:
identified two primary usages of this terminology by
1856:. London: Thames and Hudson. pp. 183–185, 193.
1626:
1624:
1514:
503:
formed from a prograde pentagram {5/2} results in a
2401:
2347:
2287:
2231:
2170:
2161:
2053:
2005:
621:-gon, alternating vertices at two different radii.
515:, which can be seen as a "crossed triangle" {3/2}
272:reflects the resemblance of these shapes to the
1978:
1556:List of regular polytopes and compounds#Stars
8:
1905:, New York: W. H. Freeman & Co. (1987),
1738:Are Your Polyhedra the Same as My Polyhedra?
1201:
1154:
286:Regular polygon § Regular star polygons
2167:
1985:
1971:
1963:
1103:
768:Examples of isotoxal star simple polygons
1822:
1820:
1423:Learn how and when to remove this message
992:
922:
771:
351:A regular star polygon is denoted by its
1933:, Heidi Burgiel, Chaim Goodman-Strauss,
1682:Coxeter, Harold Scott Macdonald (1973).
1639:Abelson, Harold, diSessa, Andera, 1980,
766:
331:
1620:
1688:. Courier Dover Publications. p.
1655:, Henry George Liddell, Robert Scott,
487:} will result in the same polygon as {
391:of all the vertices, divided by 360°.
1485:) is a frequent geometrical motif in
1317:-gons and as isotoxal concave simple
156:, but also compound figures like the
7:
1927:, Kluwer Academic (1994), pp. 43–70.
1405:adding citations to reliable sources
1543:with circle and dots (star figure)
1466:The {7/2} and {7/3} star polygons (
1503:An eleven pointed star called the
439:Construction via vertex connection
379:(they share no factors) and where
229:). The prefix is normally a Greek
223:(in this case generating the word
14:
1838:Tiling with Regular Star Polygons
1534:
1518:
1381:
1373:Star polygons in art and culture
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523:Degenerate regular star polygons
315:
304:
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73:
41:
32:
1919:; Polyhedra with Hollow Faces,
1392:needs additional citations for
509:pentagrammic crossed-antiprism
479:/2, then the construction of {
1:
1790:} consists of two coincident
1439:, but they are always highly
1109:
993:
923:
597:Isotoxal star simple polygons
367:(the number of vertices) and
203:Star polygon names combine a
1782:is even, the truncation of {
1509:tomb of Shah Nematollah Vali
1334:These three treatments are:
688:|, the outer internal angle
268:), meaning a line. The name
80:Small stellated dodecahedron
71:
30:
27:Two types of star pentagons
1762:is odd, the truncation of {
651:|, or more generally with {
549:Construction via stellation
128:, one corresponding to the
2536:
1853:Islamic Geometric Patterns
1850:Broug, Eric (2008-05-27).
1516:
1370:
1093:, among periodic tilings,
1076:
283:
152:include polygons like the
18:
16:Regular non-convex polygon
1953:Metamorphoses of polygons
1600:Kepler–Poinsot polyhedron
1528:constructed in a regular
1447:The {5/2} star pentagon (
1135:
1037:
1019:
878:
872:
511:. Another example is the
51:
1935:The Symmetries of Things
1481:The {8/3} star polygon (
19:Not to be confused with
1657:A Greek-English Lexicon
1604:uniform star polyhedron
1581:Regular star 4-polytope
1496:; the first is on the
505:pentagrammic antiprism
337:
171:, is a polygon having
1923:, ed. T. Bisztriczky
1576:Pentagramma mirificum
1077:Further information:
713:degrees, necessarily.
335:
284:Further information:
130:regular star polygons
112:Regular star polygons
2218:Nonagon/Enneagon (9)
2148:Tangential trapezoid
1903:Tilings and Patterns
1901:and G. C. Shephard;
1498:emblem of Azerbaijan
1443:. Examples include:
1401:improve this article
756:necessarily (here, {
658:}, which denotes an
628:Tilings and patterns
557:of a convex regular
342:regular star polygon
280:Regular star polygon
258:suffix derives from
163:One definition of a
116:certain notable ones
2330:Megagon (1,000,000)
2098:Isosceles trapezoid
1106:
1073:Examples in tilings
769:
513:tetrahemihexahedron
239:is also known as a
28:
21:Star-shaped polygon
2300:Icositetragon (24)
1879:, CUP, Hbk. 1997,
1743:2016-08-03 at the
1715:Weisstein, Eric W.
1643:, MIT Press, p. 24
1367:In art and culture
1118:"Triangular" stars
1104:
767:
740:𝛼 < 180(1 − 2/
429:Thomas Bradwardine
338:
274:diffraction spikes
134:intersecting edges
26:
2502:
2501:
2343:
2342:
2320:Myriagon (10,000)
2305:Triacontagon (30)
2269:Heptadecagon (17)
2259:Pentadecagon (15)
2254:Tetradecagon (14)
2193:Quadrilateral (4)
2063:Antiparallelogram
1943:978-1-56881-220-5
1863:978-0-500-28721-7
1770:} is naturally {2
1747:, Branko Grünbaum
1699:978-0-486-61480-9
1685:Regular polytopes
1561:Five-pointed star
1547:
1546:
1433:
1432:
1425:
1302:
1301:
1298:not edge-to-edge
1145:"Octagonal" stars
1136:"Hexagonal" stars
1070:
1069:
660:isotoxal concave
420:, independent of
387:: the sum of the
330:
329:
106:is a type of non-
96:
95:
2527:
2315:Chiliagon (1000)
2295:Icositrigon (23)
2274:Octadecagon (18)
2264:Hexadecagon (16)
2168:
1987:
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1973:
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1507:was used on the
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1156:Image of tiling
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1090:Harmonices Mundi
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699:
672:-gon with outer
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463:th point out of
377:relatively prime
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297:
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88:
77:
47:|5/2|
45:
36:
29:
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2525:
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2505:
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2339:
2283:
2249:Tridecagon (13)
2239:Hendecagon (11)
2227:
2163:
2157:
2128:Right trapezoid
2049:
2001:
1991:
1955:, published in
1949:Branko Grünbaum
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1641:Turtle Geometry
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1596:Star polyhedron
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1541:Seal of Solomon
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1203:Vertex config.
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1115:-pointed stars
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1110:Isotoxal simple
1099:Penrose tilings
1095:Johannes Kepler
1085:internal angles
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847:|6/2|
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838:|8/3|
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433:Johannes Kepler
415:
353:Schläfli symbol
320:
309:
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276:of real stars.
201:
169:turtle graphics
144:simple polygons
126:Johannes Kepler
122:Branko Grünbaum
89:
78:
66:uniform tilings
52:A regular star
46:
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17:
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5:
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2244:Dodecagon (12)
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2055:Quadrilaterals
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2011:
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1946:
1931:John H. Conway
1928:
1914:
1896:
1875:Cromwell, P.;
1870:
1869:
1862:
1842:
1840:, Joseph Myers
1830:
1816:
1749:
1730:
1718:"Star Polygon"
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1513:
1512:
1501:
1479:
1464:
1459:cults to have
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1430:
1389:
1387:
1380:
1371:Main article:
1368:
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1199:
1192:
1185:
1178:
1171:
1164:
1157:
1153:
1152:
1143:
1134:
1127:"Square" stars
1125:
1116:
1074:
1071:
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796:
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764:
763:} is concave).
760:
751:
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720:
714:
708:
702:external angle
690:𝛼 = 180(1 − 2
674:internal angle
655:
598:
595:
550:
547:
546:
545:
524:
521:
440:
437:
411:
408:dihedral group
396:symmetry group
385:turning number
328:
327:
324:
313:
302:
281:
278:
205:numeral prefix
200:
197:
185:turning number
183:is called the
108:convex polygon
94:
93:
82:
70:
69:
62:star polyhedra
49:
48:
39:
15:
13:
10:
9:
6:
4:
3:
2:
2532:
2521:
2520:Star polygons
2518:
2516:
2513:
2512:
2510:
2495:
2494:Weakly simple
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2434:Infinite skew
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2406:
2404:
2400:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2355:
2353:
2350:
2349:Star polygons
2346:
2336:
2335:Apeirogon (∞)
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2292:
2290:
2286:
2280:
2279:Icosagon (20)
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2236:
2234:
2230:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2175:
2173:
2169:
2166:
2160:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2113:Parallelogram
2111:
2109:
2108:Orthodiagonal
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2088:Ex-tangential
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2060:
2058:
2056:
2052:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2012:
2010:
2008:
2004:
1999:
1995:
1988:
1983:
1981:
1976:
1974:
1969:
1968:
1965:
1958:
1954:
1950:
1947:
1944:
1940:
1936:
1932:
1929:
1926:
1922:
1918:
1915:
1912:
1911:0-7167-1193-1
1908:
1904:
1900:
1897:
1894:
1893:0-521-66405-5
1890:
1887:. Pbk. 1999,
1886:
1885:0-521-66432-2
1882:
1878:
1874:
1873:
1865:
1859:
1855:
1854:
1846:
1843:
1839:
1834:
1831:
1828:
1823:
1821:
1817:
1811:
1807:
1799:
1795:
1789:
1785:
1781:
1777:
1773:
1769:
1765:
1761:
1753:
1750:
1746:
1742:
1739:
1734:
1731:
1725:
1724:
1719:
1716:
1709:
1706:
1701:
1695:
1691:
1687:
1686:
1678:
1675:
1671:
1670:Star polygons
1665:
1662:
1658:
1654:
1649:
1646:
1642:
1636:
1633:
1627:
1625:
1621:
1614:
1610:
1607:
1605:
1601:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1574:
1572:
1571:Moravian star
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1553:
1549:
1542:
1537:
1533:
1531:
1527:
1521:
1517:
1510:
1506:
1502:
1499:
1495:
1491:
1488:
1484:
1480:
1477:
1473:
1469:
1465:
1463:significance.
1462:
1458:
1454:
1450:
1446:
1445:
1444:
1442:
1438:
1427:
1424:
1416:
1406:
1402:
1396:
1395:
1390:This section
1388:
1384:
1379:
1378:
1374:
1366:
1364:
1357:
1354:
1350:
1349:
1344:
1341:
1337:
1336:
1335:
1332:
1330:
1325:
1323:
1320:
1316:
1312:
1304:
1297:
1285:
1265:
1245:
1225:
1205:
1202:
1197:
1193:
1190:
1186:
1183:
1179:
1176:
1172:
1169:
1165:
1162:
1158:
1155:
1150:
1144:
1141:
1132:
1126:
1123:
1117:
1114:
1108:
1102:
1100:
1096:
1092:
1091:
1086:
1080:
1072:
1063:
1059:
1057:
1050:
1046:
1041:
1032:
1028:
1023:
1014:
1010:
1007:
1003:
988:
984:
981:
977:
974:
970:
967:
963:
960:
956:
953:
949:
946:
942:
939:
935:
930:
918:
915:
912:
909:
906:
903:
900:
897:
892:
889:
888:
884:
881:
875:
869:
866:
865:
855:
846:
837:
828:
819:
810:
801:
792:
786:
780:
776:
772:
759:
750:
743:
737:
733:
726:
719:
715:
707:
703:
697:
693:
687:
683:
679:
678:
677:
675:
671:
668:
664:
663:
654:
650:
646:
642:
638:
634:
630:
629:
624:
620:
617:
614:
611:
608:
604:
596:
594:
592:
588:
584:
580:
576:
572:
568:
564:
560:
556:
548:
543:
539:
538:
537:
534:
530:
522:
520:
518:
514:
510:
506:
502:
498:
494:
490:
486:
482:
478:
474:
469:
466:
462:
458:
454:
450:
447:of a regular
446:
438:
436:
434:
430:
425:
423:
419:
414:
409:
405:
401:
397:
392:
390:
386:
382:
378:
374:
370:
366:
362:
358:
354:
349:
347:
343:
334:
325:
323:
318:
314:
312:
307:
303:
301:
296:
292:
291:
287:
279:
277:
275:
271:
267:
263:
262:
257:
253:
249:
246:
242:
238:
237:
232:
228:
227:
222:
221:
216:
212:
211:
206:
198:
196:
194:
193:spirolaterals
190:
186:
182:
178:
174:
170:
166:
161:
159:
155:
151:
147:
145:
142:
139:
135:
131:
127:
123:
119:
117:
113:
109:
105:
101:
92:
87:
83:
81:
76:
72:
67:
63:
59:
55:
50:
44:
40:
35:
31:
22:
2515:Star symbols
2348:
2288:>20 sides
2223:Decagon (10)
2208:Heptagon (7)
2198:Pentagon (5)
2188:Triangle (3)
2083:Equidiagonal
1956:
1952:
1934:
1924:
1920:
1917:Grünbaum, B.
1902:
1899:Grünbaum, B.
1876:
1852:
1845:
1833:
1809:
1805:
1797:
1793:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1752:
1733:
1721:
1708:
1683:
1677:
1669:
1664:
1659:, on Perseus
1656:
1648:
1640:
1635:
1591:Star (glyph)
1494:architecture
1434:
1419:
1410:
1399:Please help
1394:verification
1391:
1362:
1346:
1333:
1326:
1321:
1318:
1314:
1308:
1148:
1139:
1130:
1121:
1112:
1088:
1082:
1005:
1001:
928:
890:
784:
778:
774:
757:
748:
746:degrees and
741:
738:}; however,
735:
731:
724:
717:
705:
695:
691:
685:
681:
669:
666:
661:
652:
648:
644:
640:
636:
626:
618:
615:
602:
600:
582:
578:
574:
570:
566:
562:
558:
552:
532:
528:
526:
496:
492:
488:
484:
480:
476:
472:
470:
464:
460:
456:
452:
448:
442:
431:, and later
426:
421:
417:
416:, of order 2
412:
403:
399:
393:
380:
368:
364:
360:
356:
350:
341:
339:
270:star polygon
269:
265:
259:
255:
247:
243:, using the
240:
234:
224:
218:
208:
202:
180:
172:
165:star polygon
164:
162:
148:
120:
111:
104:star polygon
103:
97:
91:Tessellation
2484:Star-shaped
2459:Rectilinear
2429:Equilateral
2424:Equiangular
2388:Hendecagram
2232:11–20 sides
2213:Octagon (8)
2203:Hexagon (6)
2178:Monogon (1)
2020:Equilateral
1672:, pp. 36–38
1586:Rub el Hizb
1505:hendecagram
1490:Islamic art
1441:symmetrical
1056:Star figure
565:and amount
555:stellations
389:turn angles
213:, with the
191:), like in
2509:Categories
2489:Tangential
2393:Dodecagram
2171:1–10 sides
2162:By number
2143:Tangential
2123:Right kite
1778:}. But if
1615:References
1566:Magic star
1468:heptagrams
1413:March 2024
1342:rendering.
754:< 180°,
207:, such as
167:, used in
2469:Reinhardt
2378:Enneagram
2368:Heptagram
2358:Pentagram
2325:65537-gon
2183:Digon (2)
2153:Trapezoid
2118:Rectangle
2068:Bicentric
2030:Isosceles
2007:Triangles
1877:Polyhedra
1723:MathWorld
1524:An {8/3}
1457:religious
1449:pentagram
1353:polyhedra
1305:Interiors
662:or convex
501:antiprism
406:} is the
363:}, where
236:enneagram
226:pentagram
154:pentagram
150:Polygrams
64:and star
2444:Isotoxal
2439:Isogonal
2383:Decagram
2373:Octagram
2363:Hexagram
2164:of sides
2093:Harmonic
1994:Polygons
1937:, 2008,
1895:. p. 175
1741:Archived
1609:Starfish
1550:See also
1526:octagram
1483:octagram
1472:Kabbalah
998:polygram
931:-pointed
924:Isotoxal
633:polygram
607:isotoxal
468:vertex.
241:nonagram
231:cardinal
158:hexagram
138:isotoxal
100:geometry
54:pentagon
2464:Regular
2409:Concave
2402:Classes
2310:257-gon
2133:Rhombus
2073:Crossed
1800:/2)}'s;
1530:octagon
1474:and in
1453:magical
1437:regular
1348:density
1324:-gons.
1066:{10/3}
1026:{12/5}
996:regular
994:Related
665:simple
610:concave
587:monogon
517:cuploid
373:density
346:polygon
245:ordinal
217:suffix
189:density
141:concave
58:decagon
2474:Simple
2419:Cyclic
2414:Convex
2138:Square
2078:Cyclic
2040:Obtuse
2035:Kepler
1959:(1994)
1941:
1925:et al.
1909:
1891:
1883:
1860:
1808:/2) =
1696:
1653:γραμμή
1602:, and
1487:Mughal
1461:occult
1359:model.
1044:{8/3}
1035:{5/2}
1017:{9/4}
926:simple
856:
829:
811:
802:
781:|
773:|
643:} as |
613:simple
445:vertex
375:) are
266:grammḗ
261:γραμμή
254:. The
210:penta-
38:{5/2}
2449:Magic
2045:Right
2025:Ideal
2015:Acute
1476:Wicca
1286:3.6.6
1151:= 8)
1142:= 6)
1133:= 4)
1124:= 3)
933:star
919:144°
916:120°
910:135°
907:108°
901:150°
716:For {
711:= 180
680:For |
625:, in
591:digon
573:>
371:(the
322:{7/3}
311:{7/2}
300:{5/2}
256:-gram
252:Latin
250:from
220:-gram
215:Greek
199:Names
177:turns
132:with
2479:Skew
2103:Kite
1998:List
1939:ISBN
1907:ISBN
1889:ISBN
1881:ISBN
1858:ISBN
1694:ISBN
1492:and
1455:and
1255:.6.6
1235:.8.4
1215:.3.3
1053:2{3}
913:90°
904:90°
898:60°
885:72°
882:60°
879:45°
876:36°
873:30°
870:20°
676:𝛼.
589:and
559:core
531:and
455:and
398:of {
394:The
326:...
248:nona
175:≥ 2
102:, a
1758:If
1403:by
1295:.6
1291:π/3
1280:π/3
1271:π/3
1266:3.6
1260:π/3
1251:π/3
1246:6.6
1240:π/4
1231:π/4
1226:8.4
1206:3.3
894:ext
867:𝛼
860:72°
851:60°
842:45°
833:45°
824:36°
815:30°
806:30°
797:20°
752:ext
728:ext
709:ext
527:If
471:If
187:or
98:In
2511::
1951:,
1819:^
1804:2(
1796:/(
1720:.
1692:.
1690:93
1623:^
1598:,
1275:.6
1220:𝛼
1211:𝛼
1101:.
1008:}
862:}
858:{5
853:}
849:{6
844:}
840:{8
835:}
831:{4
826:}
822:{5
817:}
813:{6
808:}
804:{3
799:}
795:{9
790:}
788:𝛼
761:𝛼
721:𝛼
656:𝛼
581:=
519:.
495:−
491:/(
475:≥
435:.
424:.
348:.
340:A
195:.
160:.
146:.
110:.
2000:)
1996:(
1986:e
1979:t
1972:v
1913:.
1866:.
1812:,
1810:q
1806:q
1798:q
1794:p
1792:{
1788:q
1786:/
1784:p
1780:q
1776:q
1774:/
1772:p
1768:q
1766:/
1764:p
1760:q
1726:.
1702:.
1511:.
1500:.
1478:.
1426:)
1420:(
1415:)
1411:(
1397:.
1322:n
1319:2
1315:n
1149:n
1147:(
1140:n
1138:(
1131:n
1129:(
1122:n
1120:(
1113:n
1006:d
1004:/
1002:n
1000:{
929:n
891:β
785:n
783:{
779:d
777:/
775:n
758:n
749:β
744:)
742:n
736:d
734:/
732:n
725:β
718:n
706:β
698:)
696:n
694:/
692:d
686:d
684:/
682:n
670:n
667:2
653:n
649:d
647:/
645:n
641:d
639:/
637:n
635:{
619:n
616:2
603:n
583:p
579:q
575:p
571:q
567:p
563:q
533:q
529:p
497:q
493:p
489:p
485:q
483:/
481:p
477:p
473:q
465:p
461:q
457:q
453:p
449:p
422:q
418:p
413:p
410:D
404:q
402:/
400:p
381:q
369:q
365:p
361:q
359:/
357:p
355:{
264:(
181:q
179:(
173:q
23:.
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