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This is the only non-degenerate uniform polyhedron with more than six faces meeting at a vertex. Each vertex has 4 squares which pass through the vertex central axis (and thus through the centre of the figure), alternating with two triangles and two pentagrams. Another unusual feature is that the
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518:
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406:(quasirhombicosidodecahedron) by a branched cover: there is a function from the great dirhombicosidodecahedron to the quasirhombicosidodecahedron that is 2-to-1 everywhere, except for the vertices.
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If the definition of a uniform polyhedron is relaxed to allow any even number of faces adjacent to an edge, then this definition gives rise to one further polyhedron: the
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755:{\displaystyle M={\begin{pmatrix}1/2&-\phi /2&1/(2\phi )\\\phi /2&1/(2\phi )&-1/2\\1/(2\phi )&1/2&\phi /2\end{pmatrix}}}
283:, except that instead of the non-snub faces being surrounded by snub triangles as in most snub polyhedra, they are surrounded by snub squares.
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Philosophical
Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences
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which has the same vertices and edges but with a different arrangement of triangular faces.
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513:{\displaystyle p={\begin{pmatrix}\phi ^{-{\frac {1}{2}}}\\0\\\phi ^{-1}\end{pmatrix}}}
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For a great dirhombicosidodecahedron whose edge length is 1, the circumradius is
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are the vertices of a great dirhombicosidodecahedron. The edge length equals
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1401:; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra",
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The vertices and edges are also shared with the uniform compounds of
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This is also the only uniform polyhedron that cannot be made by the
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with an even number of minus signs. The transformations
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1138:constitute the group of rotational symmetries of a
1016:constitute the group of rotational symmetries of a
854:, counterclockwise. Let the linear transformations
1611:http://www.mathconsult.ch/showroom/unipoly/75.html
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286:It has been nicknamed "Miller's monster" (after
181:3D model of a great dirhombicosidodecahedron.
1616:http://www.software3d.com/MillersMonster.php
279:. This symbol suggests that it is a sort of
239:from a spherical triangle. It has a special
1299:{\displaystyle R={\frac {1}{2}}{\sqrt {2}}}
321:. 180 of the 240 edges are shared with the
298:enumerated the uniform polyhedra in 1954).
900:be the transformations which send a point
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1253:{\displaystyle {\frac {1}{2}}{\sqrt {2}}}
1496:, Geometriae Dedicata 47, 57-110, 1993.
409:
174:
1493:Uniform Solution for Uniform Polyhedra.
1623:
404:nonconvex great rhombicosidodecahedron
16:Uniform star polyhedron with 124 faces
1592:
395:Compound of twenty tetrahemihexahedra
1546:"75: great dirhombicosidodecahedron"
1525:
893:{\displaystyle T_{0},\ldots ,T_{11}}
301:
982:{\displaystyle (\pm x,\pm y,\pm z)}
232:faces all occur in coplanar pairs.
13:
1543:
402:This polyhedron is related to the
195:great snub disicosidisdodecahedron
14:
1642:
1585:
373:Great disnub dirhombidodecahedron
349:Great snub dodecicosidodecahedron
323:great snub dodecicosidodecahedron
308:great disnub dirhombidodecahedron
1597:"Great dirhombicosidodecahedron"
1564:"Great Dirhombicosidodecahedron"
1385:Interior view, modulo-2 filling
1379:
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1361:
1339:{\displaystyle r={\frac {1}{2}}}
785:is the rotation around the axis
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27:
1399:Coxeter, Harold Scott MacDonald
1090:{\displaystyle (i=0,\ldots ,11}
175:
22:Great dirhombicosidodecahedron
1556:
1537:
1522:Mathematica J. 3, 48-57, 1993.
1468:. Cambridge University Press.
1131:{\displaystyle j=0,\ldots ,4)}
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360:Great dirhombicosidodecahedron
191:great dirhombicosidodecahedron
1:
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384:Compound of twenty octahedra
327:
199:nonconvex uniform polyhedron
142:Great dirhombicosidodecacron
18:
7:
1226:, and the midradius equals
1199:{\displaystyle {\sqrt {2}}}
1175:{\displaystyle T_{i}M^{j}p}
816:{\displaystyle (1,0,\phi )}
275:relating it to a spherical
10:
1647:
1350:
1206:, the circumradius equals
1050:{\displaystyle T_{i}M^{j}}
26:
21:
931:{\displaystyle (x,y,z)}
847:{\displaystyle 2\pi /5}
208:. It has 124 faces (40
40:Uniform star polyhedron
1528:"3D uniform polyhedra"
1423:10.1098/rsta.1954.0003
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1020:. The transformations
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1142:. Then the 60 points
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1009:{\displaystyle T_{i}}
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539:
537:{\displaystyle \phi }
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410:Cartesian coordinates
319:20 tetrahemihexahedra
296:M. S. Longuet-Higgins
180:
158:4.5/3.4.3.4.5/2.4.3/2
90:| 3/2 5/3 3 5/2
63:= 60 (χ = −56)
1367:Traditional filling
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237:Wythoff construction
71:40{3}+60{4}+24{5/2}
1526:Klitzing, Richard.
1415:1954RSPTA.246..401C
1140:regular icosahedron
1018:regular tetrahedron
1594:Weisstein, Eric W.
1519:Uniform Polyhedra.
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201:, indexed last as
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1631:Uniform polyhedra
1466:Polyhedron Models
1462:Wenninger, Magnus
1389:
1388:
1376:Modulo-2 filling
1334:
1310:Its midradius is
1294:
1287:
1248:
1241:
1219:{\displaystyle 1}
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940:even permutations
778:{\displaystyle M}
561:{\displaystyle M}
548:. Let the matrix
474:
427:{\displaystyle p}
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302:Related polyhedra
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1576:
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1566:. Archived from
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1502:Kaleido software
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1409:(916): 401–450,
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292:H. S. M. Coxeter
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110:Index references
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1544:Maeder, Roman.
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823:by an angle of
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288:J. C. P. Miller
281:snub polyhedron
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138:Dual polyhedron
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77:Coxeter diagram
59:
17:
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1586:External links
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241:Wythoff symbol
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165:Bowers acronym
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96:Symmetry group
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86:Wythoff symbol
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68:Faces by sides
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1570:on 2018-10-18
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277:quadrilateral
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148:Vertex figure
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1572:. Retrieved
1568:the original
1558:
1549:
1539:
1517:
1515:Mäder, R. E.
1492:
1465:
1406:
1402:
1354:
1309:
1262:
765:
568:be given by
546:golden ratio
523:
434:be given by
413:
401:
359:
315:20 octahedra
312:
305:
285:
234:
230:
194:
190:
184:
60:
55:
51:
1550:MathConsult
1510:dual images
1490:Har'El, Z.
338:Convex hull
290:, who with
1574:2022-07-24
1498:Zvi Har’El
1392:References
218:pentagrams
1602:MathWorld
1455:202575183
1431:0080-4614
1117:…
1079:…
971:±
962:±
953:±
875:…
834:π
808:ϕ
734:ϕ
713:ϕ
681:−
673:ϕ
646:ϕ
636:ϕ
609:ϕ
606:−
532:ϕ
495:−
491:ϕ
464:−
460:ϕ
224:, and 60
216:, and 24
210:triangles
104:, , *532
1625:Category
1464:(1974).
226:vertices
187:geometry
47:Elements
1484:1738087
1447:0062446
1411:Bibcode
1351:Gallery
938:to the
544:is the
268:⁄
258:⁄
249:⁄
220:), 240
214:squares
197:) is a
169:Gidrid
54:= 124,
1506:Images
1482:
1472:
1453:
1445:
1437:
1429:
524:where
189:, the
1451:S2CID
1439:91532
1435:JSTOR
222:edges
212:, 60
58:= 240
1480:OCLC
1470:ISBN
1427:ISSN
294:and
193:(or
36:Type
1419:doi
1407:246
942:of
317:or
185:In
132:119
1627::
1599:.
1548:.
1508:,
1504:,
1500:,
1478:.
1449:,
1443:MR
1441:,
1433:,
1425:,
1417:,
1405:,
1260:.
1097:,
1085:11
886:11
325:.
263:3
244:|
228:.
205:75
127:,
125:92
120:,
118:75
1605:.
1577:.
1552:.
1530:.
1486:.
1421::
1413::
1346:.
1332:2
1329:1
1324:=
1321:r
1306:.
1292:2
1285:2
1282:1
1277:=
1274:R
1246:2
1239:2
1236:1
1214:1
1192:2
1170:p
1165:j
1161:M
1155:i
1151:T
1126:)
1123:4
1120:,
1114:,
1111:0
1108:=
1105:j
1082:,
1076:,
1073:0
1070:=
1067:i
1064:(
1043:j
1039:M
1033:i
1029:T
1002:i
998:T
977:)
974:z
968:,
965:y
959:,
956:x
950:(
926:)
923:z
920:,
917:y
914:,
911:x
908:(
882:T
878:,
872:,
867:0
863:T
842:5
838:/
831:2
811:)
805:,
802:0
799:,
796:1
793:(
773:M
762:.
748:)
742:2
738:/
729:2
725:/
721:1
716:)
710:2
707:(
703:/
699:1
692:2
688:/
684:1
676:)
670:2
667:(
663:/
659:1
654:2
650:/
639:)
633:2
630:(
626:/
622:1
617:2
613:/
601:2
597:/
593:1
587:(
582:=
579:M
556:M
520:,
506:)
498:1
483:0
472:2
469:1
453:(
448:=
445:p
422:p
273:,
270:2
266:5
260:3
256:5
251:2
247:3
203:U
129:W
122:C
115:U
102:h
100:I
61:V
56:E
52:F
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