268:
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1570:
34:
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29:
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410:
1584:
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629:
980:. Since three faces meet at each vertex it has 20 × 9 / 3 = 60 vertices (these are the outermost layer of visible vertices and form the tips of the "spines") and 20 × 9 / 2 = 90 edges (each edge of the star polyhedron includes and connects two of the 180 visible edges).
578:
429:
The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The
435:
421:
As a simple, visible surface polyhedron, the outward form of the final stellation is composed of 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an
308:
stated a set of stellation rules for the regular icosahedron and gave a systematic enumeration of the fifty-nine stellations which conform to those rules. The complete stellation is referenced as the eighth in the book. In
863:
373:
Stellation diagram of the icosahedron with numbered cells. The complete icosahedron is formed from all the cells in the stellation, but only the outermost regions, labelled "13" in the diagram, are visible.
1120:
381:
of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges.
926:
573:{\displaystyle {\sqrt {{\frac {3}{2}}\left(3+{\sqrt {5}}\right)}}\,:\,{\sqrt {{\frac {1}{2}}\left(25+11{\sqrt {5}}\right)}}\,:\,{\sqrt {{\frac {1}{2}}\left(97+43{\sqrt {5}}\right)}}\,.}
771:
734:
704:
791:
681:
199:. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by
802:
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315:
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125:
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872:
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1119:"... and some odd solids including the echidnahedron (my name; its actually the final stellation of the icosahedron)."
261:
204:
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241:
237:
1662:
1541:
1526:
278:
extended the stellation theory beyond regular forms, and identified ten stellations of the icosahedron, including the
1669:
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1648:
1641:
1546:
1536:
1531:
1521:
1361:
319:, the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index number W
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304:
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having 20 faces corresponding to the 20 faces of the underlying icosahedron. Each face is an irregular 9/4
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published a list of twenty stellation forms (twenty-two including reflective copies), also including the
1676:
1611:
646:
257:
166:
50:
1396:
401:, because it includes all cells in the stellation diagram up to and including the outermost "h" layer.
33:
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652:
28:
1707:
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1309:
The Final
Stellation of the Icosahedron: An Advanced Mathematical Model to Cut Out and Glue Together
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Certain forms of the icosahedron and a method for deriving and designating higher polyhedra
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992:
969:
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can be constructed by 12 sets of faces, each folded into a group of five pyramids.
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188:
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946:
Twenty 9/4 polygon faces (one face is drawn yellow with 9 vertices labeled.)
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1346:. Proc. Internat. Math. Congress, Toronto. Vol. 1. pp. 701–708.
176:
628:
260:
in 1812 that there are only four regular star polyhedrons, known as the
335:
1698:
The stellation process on the icosahedron creates a number of related
1472:
1468:
1111:
339:
327:
1257:
Coxeter, H.S.M.; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999) .
858:{\displaystyle S={\frac {1}{20}}(13211+{\sqrt {174306161}})a^{2}\,,}
663:
When regarded as a three-dimensional solid object with edge lengths
1121:
geometry.research; "polyhedra database"; August 30, 1995, 12:00 am.
1457:
http://www.georgehart.com/virtual-polyhedra/vrml/echidnahedron.wrl
983:
When regarded as a star icosahedron, the complete stellation is a
408:
368:
164:
1452:
1397:
Towards stellating the icosahedron and faceting the dodecahedron
968:
The complete stellation can also be seen as a self-intersecting
343:
1387:
With instructions for constructing a model of the echidnahedron
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171:
3D model of the final stellation of the icosahedron
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21:
1329:(1810). "Memoire sur les polygones et polyèdres".
920:
857:
785:
765:
728:
698:
675:
572:
397:. The Du Val symbol of the complete stellation is
350:and which curls up in a ball to protect itself.
1207:Vielecke und Vielflache: Theorie und Geschichte
921:{\displaystyle V=(210+90{\sqrt {5}})a^{3}\,.}
389:, according to a set of rules put forward by
8:
797:) the complete icosahedron has surface area
1214:] (in German). Leipzig: B.G. Treubner.
1311:. Tarquin Publications, Norfolk, England.
1212:Polygons and Polyhedra: Theory and History
385:enumerates the stellations of the regular
1090:
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1307:Jenkins, Gerald; Bear, Magdalen (1985).
1135:
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766:{\displaystyle \varphi ^{2}a{\sqrt {2}}}
583:Convex hulls of each sphere of vertices
430:radii of these spheres are in the ratio
275:
266:
1066:
1026:
1003:
283:
16:Outermost stellation of the icosahedron
18:
1167:
1165:
1163:
326:In 1995, Andrew Hume named it in his
7:
1408:"Fifty nine icosahedron stellations"
207:. It can be viewed as an irregular,
248:in 1809 rediscovered two more, the
185:final stellation of the icosahedron
22:Final stellation of the icosahedron
14:
1448:59 Stellations of the Icosahedron
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1106:may be credited to Andrew Hume,
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627:
616:
32:
27:
1238:(3rd ed.). Dover. 3.6 6.2
244:as regular polyhedra. However,
205:Kepler–Poinsot polyhedron
1484:stellations of the icosahedron
1473:Polyhedron database, model 141
1443:Stellations of the icosahedron
1292:. Cambridge University Press.
1240:Stellating the Platonic solids
901:
882:
838:
822:
1:
729:{\displaystyle \varphi ^{2}a}
342:that is covered with coarse
242:great stellated dodecahedron
238:small stellated dodecahedron
1542:Compound of five tetrahedra
1527:Medial triambic icosahedron
1331:J. De l'École Polytechnique
1286:Cromwell, Peter R. (1997).
338:or spiny anteater, a small
330:polyhedral database as the
1749:
1697:
1547:Compound of ten tetrahedra
1537:Compound of five octahedra
1532:Great triambic icosahedron
1522:Small triambic icosahedron
1480:
1362:Cambridge University Press
1045:, Taf. XI, Fig. 14, 1900).
614:
362:
1646:
1581:
1510:
1500:
1495:
1263:(3rd ed.). Tarquin.
1260:The Fifty-Nine Icosahedra
699:{\displaystyle \varphi a}
383:The Fifty Nine Icosahedra
365:The Fifty-Nine Icosahedra
305:The Fifty Nine Icosahedra
262:Kepler–Poinsot polyhedron
236:process, recognizing the
26:
786:{\displaystyle \varphi }
93:As a simple polyhedron:
40:orthographic projections
1342:Wheeler, A. H. (1924).
203:after the discovery of
1557:Excavated dodecahedron
991:(face-transitive) and
922:
859:
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700:
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574:
418:
405:As a simple polyhedron
374:
272:
172:
152:As a star polyhedron:
63:As a star polyhedron:
1728:Polyhedral stellation
1079:Coxeter et al. (1999)
1011:Coxeter et al. (1999)
995:(vertex-transitive).
987:, because it is both
923:
860:
788:
768:
731:
701:
678:
657:Complete icosahedron
647:truncated icosahedron
575:
412:
372:
270:
258:Augustin-Louis Cauchy
256:. This was proved by
170:
51:Stellated icosahedron
1708:icosahedral symmetry
1517:(Convex) icosahedron
1352:Wenninger, Magnus J.
931:As a star polyhedron
873:
803:
777:
740:
710:
687:
667:
436:
424:Euler characteristic
298:, H. T. Flather and
1242:, pp. 96–104.
1115:polyhedron database
584:
395:complete stellation
302:in their 1938 book
288:complete stellation
280:complete stellation
1497:Uniform duals
1462:2021-12-31 at the
1424:Weisstein, Eric W.
1405:Weisstein, Eric W.
1174:Weisstein, Eric W.
1153:2008-10-07 at the
918:
855:
783:
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673:
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419:
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273:
254:great dodecahedron
197:stellation diagram
173:
1714:
1713:
1552:Great icosahedron
1502:Regular compounds
1371:978-0-521-09859-5
1357:Polyhedron models
1318:978-0-906212-48-6
1270:978-1-899618-32-3
1235:Regular Polytopes
1221:978-1-4181-6590-1
1013:, p. 30–31;
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676:{\displaystyle a}
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316:Polyhedron Models
250:great icosahedron
187:is the outermost
163:
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154:vertex-transitive
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1562:Final stellation
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1417:
1393:) by Ralph Jones
1375:
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1157:at polyhedra.org
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1091:Wenninger (1971)
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1015:Wenninger (1971)
1008:
985:noble polyhedron
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415:polyhedral model
393:, including the
292:H. S. M. Coxeter
271:Brückner's model
229:Harmonices Mundi
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1464:Wayback Machine
1427:"Echidnahedron"
1422:
1421:
1403:
1402:
1399:by Guy Inchbald
1383:
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1230:Coxeter, H.S.M.
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1177:"Echidnahedron"
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1155:Wayback Machine
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1142:
1136:Cromwell (1997)
1134:
1127:
1123:
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1055:Brückner (1900)
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1043:Brückner (1900)
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1031:Cromwell (1997)
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970:star polyhedron
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391:J. C. P. Miller
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361:
359:As a stellation
356:
354:Interpretations
322:
276:Brückner (1900)
224:Johannes Kepler
221:
213:star polyhedron
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158:face-transitive
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1138:, p. 259.
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1067:Wheeler (1924)
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1033:, p. 259.
1027:Poinsot (1810)
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1202:Brückner, Max
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1148:Echidnahedron
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1104:echidnahedron
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1093:, p. 65.
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1017:, p. 65.
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380:
371:
366:
358:
353:
351:
349:
345:
341:
337:
333:
332:echidnahedron
329:
324:
318:
317:
312:
307:
306:
301:
297:
293:
289:
285:
281:
277:
269:
265:
263:
259:
255:
251:
247:
246:Louis Poinsot
243:
239:
235:
231:
230:
225:
218:
216:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
168:
159:
155:
151:
147:
141:
137:
132:
129:
127:
123:
118:
111:
104:
97:
88:
81:
74:
67:
62:
60:
56:
52:
49:
45:
41:
35:
30:
25:
20:
1715:
1561:
1507:Regular star
1481:
1430:
1411:
1356:
1343:
1334:
1330:
1308:
1288:
1259:
1239:
1234:
1211:
1206:
1180:
1143:
1103:
1098:
1086:
1074:
1062:
1050:
1038:
1022:
1006:
982:
974:star polygon
967:
869:
866:
799:
795:golden ratio
662:
623:Dodecahedron
611:92 vertices
608:60 vertices
605:12 vertices
602:20 vertices
432:
428:
420:
398:
394:
382:
376:
331:
325:
314:
303:
300:J. F. Petrie
287:
279:
274:
232:applied the
227:
222:
201:Max Brückner
184:
180:
174:
139:
135:
116:
109:
102:
95:
86:
79:
72:
65:
867:and volume
634:Icosahedron
387:icosahedron
193:icosahedron
131:icosahedral
89:= −10
59:Euler char.
53:, 8th of 59
1722:Categories
1194:References
959:2-isogonal
644:Nonuniform
597:All three
379:stellation
334:after the
234:stellation
219:Background
189:stellation
149:Properties
1733:Polyhedra
1704:compounds
1700:polyhedra
1432:MathWorld
1413:MathWorld
1289:Polyhedra
1182:MathWorld
1108:developer
1102:The name
989:isohedral
978:enneagram
961:9/4 faces
834:174306161
781:φ
745:φ
715:φ
691:φ
311:Wenninger
296:P. du Val
1482:Notable
1460:Archived
1354:(1971).
1337:: 16–48.
1232:(1973).
1204:(1900).
1151:Archived
993:isogonal
313:'s book
181:complete
177:geometry
1511:Others
1492:Regular
1455:model:
1279:0676126
1110:of the
793:is the
773:(where
591:Middle
336:echidna
226:in his
191:of the
1469:Netlib
1368:
1315:
1296:
1277:
1267:
1246:
1218:
1112:netlib
594:Outer
588:Inner
426:of 2.
348:spines
340:mammal
328:Netlib
211:, and
209:simple
179:, the
1706:with
1210:[
999:Notes
976:, or
826:13211
105:= 270
98:= 180
1702:and
1453:VRML
1391:.doc
1366:ISBN
1313:ISBN
1294:ISBN
1265:ISBN
1244:ISBN
1216:ISBN
736:and
377:The
346:and
344:hair
252:and
240:and
119:= 2)
112:= 92
82:= 60
75:= 90
68:= 20
47:Type
886:210
183:or
175:In
1724::
1710:.
1471::
1429:.
1410:.
1364:.
1360:.
1333:.
1275:MR
1273:.
1179:.
1162:^
1128:^
1029:;
892:90
818:20
706:,
683:,
550:43
544:97
505:11
499:25
413:A
323:.
321:42
294:,
290:.
282:.
264:.
215:.
156:,
107:,
100:,
77:,
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1374:.
1335:9
1321:.
1302:.
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1069:.
1057:.
916:.
910:3
906:a
902:)
897:5
889:+
883:(
880:=
877:V
853:,
847:2
843:a
839:)
829:+
823:(
815:1
810:=
807:S
759:2
754:a
749:2
724:a
719:2
694:a
671:a
568:.
561:)
555:5
547:+
540:(
534:2
531:1
523::
516:)
510:5
502:+
495:(
489:2
486:1
478::
471:)
465:5
460:+
457:3
453:(
447:2
444:3
399:H
144:)
140:h
136:I
133:(
117:χ
114:(
110:V
103:E
96:F
91:)
87:χ
84:(
80:V
73:E
66:F
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