555:
548:
534:
665:
658:
649:
419:
699:
692:
683:
445:
541:
29:
292:
672:
527:
706:
432:
375:
395:
410:
has two vertices and one edge. There are isotoxal decagram forms, which alternates vertices at two radii. Each form has a freedom of one angle. The first is a variation of a double-wound of a pentagon {5}, and last is a variation of a double-wound of a pentagram {5/2}. The middle is a variation of a
919:
Coxeter, The
Densities of the Regular polytopes I, p.43 If d is odd, the truncation of the polygon {p/q} is naturally {2n/d}. But if not, it consists of two coincident {n/(d/2)}'s; two, because each side arises from an original side and once from an original vertex. Thus the density of a polygon is
611:
607:
599:
603:
271:
121:
111:
83:
116:
103:
88:
93:
621:
Deeper truncations of the regular pentagon and pentagram can produce intermediate star polygon forms with ten equally spaced vertices and two edge lengths that remain
98:
606:
through similar reasons. It has six four-dimensional analogues, with two of these being compounds of two self-dual star polytopes, like the pentagram itself; the
827:
The
Lighter Side of Mathematics: Proceedings of the EugĂšne Strens Memorial Conference on Recreational Mathematics and its History, (1994),
722:
584:
768:
278:
948:
370:
For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below.
753:
588:
785:
Sarhangi, Reza (2012), "Polyhedral
Modularity in a Special Class of Decagram Based Interlocking Star Polygons",
846:
129:
615:
1071:
1051:
1046:
1003:
978:
851:
Philosophical
Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences
296:
75:
1106:
1031:
1483:
1056:
941:
476:, but there are also three ten-vertex polygrams which can be interpreted as regular compounds:
264:
1457:
1397:
1036:
786:
743:
1478:
1341:
1111:
1041:
983:
898:
862:
236:
42:
8:
1447:
1422:
1392:
1387:
1061:
592:
465:
172:
866:
1452:
993:
902:
886:
832:
472:. Only one of these polygrams, {10/3} (connecting every third point), forms a regular
320:
291:
63:
1432:
1026:
934:
906:
878:
858:
749:
622:
134:
53:
961:
870:
625:(any two vertices can be transformed into each other by a symmetry of the figure).
407:
1427:
1407:
1402:
1372:
1091:
1066:
998:
894:
664:
657:
648:
418:
316:
180:
176:
49:
698:
691:
682:
444:
1437:
1417:
1382:
1377:
1008:
988:
341:
331:
168:
150:
146:
598:{10/4} can be seen as the two-dimensional equivalent of the three-dimensional
1472:
1412:
1263:
1156:
1076:
1018:
882:
554:
547:
533:
540:
28:
1442:
1312:
1268:
1232:
1222:
1217:
874:
671:
473:
312:
201:
187:
164:
1351:
1258:
1237:
1227:
805:
Regular polytopes, p 93-95, regular star polygons, regular star compounds
386:
246:
849:; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra".
468:, represented by symbol {10/n}, containing the same vertices as regular
1356:
1212:
1202:
1086:
251:
346:
336:
1331:
1321:
1298:
1288:
1278:
1207:
1116:
1081:
890:
526:
488:
226:
216:
705:
431:
1336:
1326:
1283:
1242:
1171:
1161:
1151:
970:
495:
374:
304:
231:
221:
926:
1293:
1273:
1186:
1181:
1176:
1166:
1141:
1096:
957:
469:
1101:
600:
compound of small stellated dodecahedron and great dodecahedron
604:
compound of great icosahedron and great stellated dodecahedron
394:
315:. There is one regular decagram, containing the vertices of a
1146:
481:
845:
788:
Bridges 2012: Mathematics, Music, Art, Architecture, Culture
385:
Decagrams have been used as one of the decorative motifs in
814:
Coxeter, Introduction to
Geometry, second edition, 2.8
723:
List of regular polytopes and compounds § Stars
583:{10/2} can be seen as the 2D equivalent of the 3D
1470:
629:Isogonal truncations of pentagon and pentagram
942:
272:
949:
935:
319:, but connected by every third point. Its
279:
265:
679:
645:
612:compound of two grand stellated 120-cells
520:
784:
585:compound of dodecahedron and icosahedron
480:{10/5} is a compound of five degenerate
299:, here in a Quran from the 14th century.
290:
1471:
771:, Henry George Liddell, Robert Scott,
741:
616:Polytope compound#Compounds with duals
401:
930:
595:in their respective dual positions.
956:
365:
13:
459:
14:
1495:
589:compound of 120-cell and 600-cell
464:A regular decagram is a 10-sided
704:
697:
690:
681:
670:
663:
656:
647:
553:
546:
539:
532:
525:
443:
430:
417:
393:
373:
119:
114:
109:
101:
96:
91:
86:
81:
27:
847:Coxeter, Harold Scott MacDonald
703:
680:
669:
646:
608:compound of two great 120-cells
591:; that is, the compound of two
545:
538:
531:
524:
380:
913:
838:
821:
808:
799:
778:
762:
735:
1:
728:
614:. A full list can be seen at
521:
748:, Springer, pp. 28â29,
560:
501:
494:{10/2} is a compound of two
487:{10/4} is a compound of two
415:
7:
716:
10:
1500:
411:regular decagram, {10/3}.
297:Islamic geometric patterns
1365:
1311:
1251:
1195:
1134:
1125:
1017:
969:
829:Metamorphoses of polygons
676:t{5/4} = {10/4} = 2{5/2}
636:
516:
186:
160:
145:
128:
74:
62:
48:
38:
26:
21:
920:unaltered by truncation.
295:Decagrams are common in
773:A Greek-English Lexicon
710:t{5/2} = {10/2} = 2{5}
76:CoxeterâDynkin diagrams
16:10-pointed star polygon
875:10.1098/rsta.1954.0003
300:
742:Barnes, John (2012),
294:
1182:Nonagon/Enneagon (9)
1112:Tangential trapezoid
593:pentagonal polytopes
354:suffix derives from
43:Regular star polygon
1294:Megagon (1,000,000)
1062:Isosceles trapezoid
867:1954RSPTA.246..401C
630:
402:Isotoxal variations
1264:Icositetragon (24)
794:, pp. 165â174
628:
362:) meaning a line.
301:
33:A regular decagram
1466:
1465:
1307:
1306:
1284:Myriagon (10,000)
1269:Triacontagon (30)
1233:Heptadecagon (17)
1223:Pentadecagon (15)
1218:Tetradecagon (14)
1157:Quadrilateral (4)
1027:Antiparallelogram
859:The Royal Society
714:
713:
623:vertex-transitive
581:
580:
457:
456:
289:
288:
196:
195:
1491:
1279:Chiliagon (1000)
1259:Icositrigon (23)
1238:Octadecagon (18)
1228:Hexadecagon (16)
1132:
1131:
951:
944:
937:
928:
927:
921:
917:
911:
910:
842:
836:
825:
819:
812:
806:
803:
797:
795:
793:
782:
776:
766:
760:
758:
745:Gems of Geometry
739:
708:
701:
694:
687:t{5/3} = {10/3}
685:
674:
667:
660:
651:
642:Double covering
631:
627:
574:{10/4} = 2{5/2}
557:
550:
543:
536:
529:
502:
447:
434:
421:
414:
413:
408:isotoxal polygon
397:
377:
366:Regular decagram
281:
274:
267:
198:
197:
124:
123:
122:
118:
117:
113:
112:
106:
105:
104:
100:
99:
95:
94:
90:
89:
85:
84:
31:
22:Regular decagram
19:
18:
1499:
1498:
1494:
1493:
1492:
1490:
1489:
1488:
1469:
1468:
1467:
1462:
1361:
1315:
1303:
1247:
1213:Tridecagon (13)
1203:Hendecagon (11)
1191:
1127:
1121:
1092:Right trapezoid
1013:
965:
955:
925:
924:
918:
914:
843:
839:
833:Branko GrĂŒnbaum
826:
822:
813:
809:
804:
800:
791:
783:
779:
767:
763:
756:
740:
736:
731:
719:
709:
686:
675:
652:
641:
462:
460:Related figures
452:
448:
439:
435:
426:
422:
404:
383:
368:
321:SchlÀfli symbol
317:regular decagon
285:
256:
208:
140:
120:
115:
110:
108:
107:
102:
97:
92:
87:
82:
80:
69:
64:SchlÀfli symbol
34:
17:
12:
11:
5:
1497:
1487:
1486:
1481:
1464:
1463:
1461:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1420:
1418:Pseudotriangle
1415:
1410:
1405:
1400:
1395:
1390:
1385:
1380:
1375:
1369:
1367:
1363:
1362:
1360:
1359:
1354:
1349:
1344:
1339:
1334:
1329:
1324:
1318:
1316:
1309:
1308:
1305:
1304:
1302:
1301:
1296:
1291:
1286:
1281:
1276:
1271:
1266:
1261:
1255:
1253:
1249:
1248:
1246:
1245:
1240:
1235:
1230:
1225:
1220:
1215:
1210:
1208:Dodecagon (12)
1205:
1199:
1197:
1193:
1192:
1190:
1189:
1184:
1179:
1174:
1169:
1164:
1159:
1154:
1149:
1144:
1138:
1136:
1129:
1123:
1122:
1120:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1023:
1021:
1019:Quadrilaterals
1015:
1014:
1012:
1011:
1006:
1001:
996:
991:
986:
981:
975:
973:
967:
966:
954:
953:
946:
939:
931:
923:
922:
912:
837:
820:
807:
798:
777:
761:
754:
733:
732:
730:
727:
726:
725:
718:
715:
712:
711:
702:
695:
688:
678:
677:
668:
661:
654:
644:
643:
638:
635:
579:
578:
577:{10/5} = 5{2}
575:
572:
569:
568:{10/2} = 2{5}
566:
565:{10/1} = {10}
563:
559:
558:
551:
544:
537:
530:
523:
519:
518:
515:
512:
509:
506:
500:
499:
492:
485:
461:
458:
455:
454:
450:
441:
437:
428:
424:
403:
400:
399:
398:
382:
379:
367:
364:
332:numeral prefix
311:is a 10-point
287:
286:
284:
283:
276:
269:
261:
258:
257:
255:
254:
249:
244:
239:
234:
229:
224:
219:
213:
210:
209:
205:
204:
194:
193:
190:
184:
183:
162:
158:
157:
154:
147:Internal angle
143:
142:
138:
132:
130:Symmetry group
126:
125:
78:
72:
71:
66:
60:
59:
56:
46:
45:
40:
36:
35:
32:
24:
23:
15:
9:
6:
4:
3:
2:
1496:
1485:
1484:Star polygons
1482:
1480:
1477:
1476:
1474:
1459:
1458:Weakly simple
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1398:Infinite skew
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1370:
1368:
1364:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1319:
1317:
1314:
1313:Star polygons
1310:
1300:
1299:Apeirogon (â)
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1257:
1256:
1254:
1250:
1244:
1243:Icosagon (20)
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1200:
1198:
1194:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1139:
1137:
1133:
1130:
1124:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1077:Parallelogram
1075:
1073:
1072:Orthodiagonal
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1052:Ex-tangential
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1024:
1022:
1020:
1016:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
976:
974:
972:
968:
963:
959:
952:
947:
945:
940:
938:
933:
932:
929:
916:
908:
904:
900:
896:
892:
888:
884:
880:
876:
872:
868:
864:
860:
856:
852:
848:
841:
834:
830:
824:
817:
816:Star polygons
811:
802:
790:
789:
781:
774:
770:
765:
757:
755:9783642309649
751:
747:
746:
738:
734:
724:
721:
720:
707:
700:
696:
693:
689:
684:
673:
666:
662:
659:
655:
650:
639:
634:Quasiregular
633:
632:
626:
624:
619:
617:
613:
609:
605:
601:
596:
594:
590:
586:
576:
573:
570:
567:
564:
561:
556:
552:
549:
542:
535:
528:
514:Star polygon
513:
510:
507:
504:
503:
497:
493:
490:
486:
483:
479:
478:
477:
475:
471:
467:
446:
442:
433:
429:
420:
416:
412:
409:
396:
392:
391:
390:
388:
378:
376:
371:
363:
361:
357:
353:
349:
348:
343:
339:
338:
333:
329:
324:
322:
318:
314:
310:
306:
298:
293:
282:
277:
275:
270:
268:
263:
262:
260:
259:
253:
250:
248:
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
214:
212:
211:
207:
206:
203:
202:Star polygons
200:
199:
191:
189:
185:
182:
178:
174:
170:
166:
163:
159:
155:
152:
148:
144:
136:
133:
131:
127:
79:
77:
73:
67:
65:
61:
57:
55:
51:
47:
44:
41:
37:
30:
25:
20:
1346:
1252:>20 sides
1187:Decagon (10)
1172:Heptagon (7)
1162:Pentagon (5)
1152:Triangle (3)
1047:Equidiagonal
915:
854:
850:
840:
828:
823:
815:
810:
801:
787:
780:
775:, on Perseus
772:
764:
744:
737:
653:t{5} = {10}
640:Quasiregular
620:
597:
582:
474:star polygon
463:
405:
384:
381:Applications
372:
369:
359:
355:
351:
345:
335:
327:
325:
313:star polygon
308:
302:
241:
188:Dual polygon
1479:10 (number)
1448:Star-shaped
1423:Rectilinear
1393:Equilateral
1388:Equiangular
1352:Hendecagram
1196:11â20 sides
1177:Octagon (8)
1167:Hexagon (6)
1142:Monogon (1)
984:Equilateral
387:girih tiles
340:, with the
330:combines a
323:is {10/3}.
247:hendecagram
173:equilateral
1473:Categories
1453:Tangential
1357:Dodecagram
1135:1â10 sides
1126:By number
1107:Tangential
1087:Right kite
729:References
517:Compounds
489:pentagrams
252:dodecagram
161:Properties
1433:Reinhardt
1342:Enneagram
1332:Heptagram
1322:Pentagram
1289:65537-gon
1147:Digon (2)
1117:Trapezoid
1082:Rectangle
1032:Bicentric
994:Isosceles
971:Triangles
907:202575183
883:0080-4614
637:Isogonal
511:Compound
496:pentagons
326:The name
237:enneagram
227:heptagram
217:pentagram
1408:Isotoxal
1403:Isogonal
1347:Decagram
1337:Octagram
1327:Hexagram
1128:of sides
1057:Harmonic
958:Polygons
717:See also
610:and the
466:polygram
328:decagram
309:decagram
305:geometry
242:decagram
232:octagram
222:hexagram
181:isotoxal
177:isogonal
135:Dihedral
54:vertices
1428:Regular
1373:Concave
1366:Classes
1274:257-gon
1097:Rhombus
1037:Crossed
899:0062446
863:Bibcode
861:: 411.
857:(916).
818:p.36-38
587:and 4D
571:{10/3}
562:Symbol
508:Convex
470:decagon
360:grammÄs
356:ÎłÏαΌΌáżÏ
344:suffix
151:degrees
1438:Simple
1383:Cyclic
1378:Convex
1102:Square
1042:Cyclic
1004:Obtuse
999:Kepler
905:
897:
889:
881:
769:ÎłÏÎ±ÎŒÎŒÎź
752:
522:Image
491:2{5/2}
482:digons
449:{(5/4)
436:{(5/3)
423:{(5/2)
350:. The
169:cyclic
70:t{5/3}
68:{10/3}
1413:Magic
1009:Right
989:Ideal
979:Acute
903:S2CID
891:91532
887:JSTOR
792:(PDF)
505:Form
498:2{5}.
352:-gram
347:-gram
342:Greek
337:deca-
50:Edges
1443:Skew
1067:Kite
962:List
879:ISSN
750:ISBN
484:5{2}
307:, a
192:self
165:star
52:and
39:Type
871:doi
855:246
602:or
406:An
303:In
156:72°
1475::
901:.
895:MR
893:.
885:.
877:.
869:.
853:.
831:,
618:.
453:}
440:}
427:}
389:.
334:,
179:,
175:,
171:,
167:,
139:10
137:(D
58:10
964:)
960:(
950:e
943:t
936:v
909:.
873::
865::
844:*
835:.
796:.
759:.
451:α
438:α
425:α
358:(
280:e
273:t
266:v
153:)
149:(
141:)
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