507:
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29:
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is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a
591:
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as a rectified great stellated dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the
762:
A truncation process applied to the great stellated dodecahedron produces a series of uniform polyhedra. Truncating edges down to points produces the
497:
407:
The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the
483:
A net of a great stellated dodecahedron (surface geometry); twenty isosceles triangular pyramids, arranged like the faces of an icosahedron.
374:
of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the
415:
which has pentagonal polytope faces and simplex vertex figures until it can no longer be stellated; that is, it is its final stellation.
443:
1338:
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907:
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triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a
233:
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1222:
Philosophical
Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
1197:
459:
with a density of 7. (One spherical pentagram face is shown above, outlined in blue, filled in yellow)
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8:
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If the pentagrammic faces are broken into triangles, it is topologically related to the
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653:{\displaystyle {\text{Surface Area}}=15{\Bigl (}{\sqrt {5+2{\sqrt {5}}}}{\Bigr )}E^{2}}
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119:
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324:
99:
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928:
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287:
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148:
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799:
580:{\displaystyle {\text{Circumradius}}={\tfrac {E}{4}}(3+{\sqrt {5}}){\sqrt {3}}}
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129:
28:
1297:
1166:
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1229:
437:
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811:
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1307:
1289:
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397:
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338:
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720:{\displaystyle {\text{Volume}}={\tfrac {5}{4}}(3+{\sqrt {5}})E^{3}}
312:
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784:
inscribed within and sharing the edges of the icosahedron.
522:
For a great stellated dodecahedron with edge length E,
677:
540:
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530:
351:
faces, with three pentagrams meeting at each vertex.
396:, with the same face connectivity, but much taller
719:
652:
579:
385:Shaving the triangular pyramids off results in an
635:
608:
1651:
511:Complete net of a great stellated dodecahedron.
1339:
1353:
492:It can be constructed as the third of three
786:
1346:
1332:
442:Transparent great stellated dodecahedron (
307:3D model of a great stellated dodecahedron
1188:
378:, is related in a similar fashion to the
733:
300:
1303:"Three stellations of the dodecahedron"
1216:
496:of the dodecahedron, and referenced as
1652:
1446:nonconvex great rhombicosidodecahedron
908:Truncated great stellated dodecahedron
1327:
1298:
729:
738:Animated truncation sequence from {
13:
347:It is composed of 12 intersecting
14:
1681:
1564:great stellapentakis dodecahedron
1549:medial pentagonal hexecontahedron
1534:small stellapentakis dodecahedron
1451:great truncated icosidodecahedron
1269:
1579:great pentagonal hexecontahedron
1554:medial disdyakis triacontahedron
1539:medial deltoidal hexecontahedron
1172:
1165:
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281:
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27:
1574:great disdyakis triacontahedron
1569:great deltoidal hexecontahedron
792:Stellations of the dodecahedron
455:This polyhedron can be made as
301:
1529:medial rhombic triacontahedron
704:
688:
567:
551:
1:
1559:great rhombic triacontahedron
1220:(1954). "Uniform Polyhedra".
1182:
22:Great stellated dodecahedron
1496:great dodecahemidodecahedron
1486:small dodecahemidodecahedron
1426:truncated dodecadodecahedron
1416:truncated great dodecahedron
1386:great stellated dodecahedron
1376:small stellated dodecahedron
1281:Great stellated dodecahedron
935:
893:
827:Great stellated dodecahedron
817:Small stellated dodecahedron
778:great stellated dodecahedron
422:
317:great stellated dodecahedron
18:
7:
1501:great icosihemidodecahedron
1491:small icosihemidodecahedron
1441:truncated great icosahedron
1319:Uniform polyhedra and duals
517:
10:
1686:
1624:great dodecahemidodecacron
1614:small dodecahemidodecacron
1511:small dodecahemicosahedron
1506:great dodecahemicosahedron
1198:Cambridge University Press
1629:great icosihemidodecacron
1619:small icosihemidodecacron
1587:
1519:
1459:
1399:
1361:
803:
789:
418:
321:Kepler–Poinsot polyhedron
40:Kepler–Poinsot polyhedron
35:
26:
21:
16:Kepler–Poinsot polyhedron
1670:Kepler–Poinsot polyhedra
1639:small dodecahemicosacron
1634:great dodecahemicosacron
1421:rhombidodecadodecahedron
1355:Star-polyhedra navigator
1436:great icosidodecahedron
1431:snub dodecadodecahedron
764:great icosidodecahedron
337:,3}. It is one of four
1590:uniform polyhedra with
1544:small rhombidodecacron
1246:10.1098/rsta.1954.0003
759:
721:
654:
581:
308:
1660:Polyhedral stellation
805:Kepler–Poinsot solids
737:
722:
655:
582:
370:, as well as being a
306:
1592:infinite stellations
1400:Uniform truncations
665:
592:
528:
364:vertex configuration
53:regular dodecahedron
1520:Duals of nonconvex
1471:tetrahemihexahedron
1238:1954RSPTA.246..401C
413:pentagonal polytope
394:triakis icosahedron
366:, with the regular
358:, although not its
1588:Duals of nonconvex
1481:octahemioctahedron
1476:cubohemioctahedron
1460:Nonconvex uniform
1411:dodecadodecahedron
1402:of Kepler-Poinsot
1381:great dodecahedron
1369:regular polyhedra)
1300:Weisstein, Eric W.
1285:Uniform polyhedron
1277:Weisstein, Eric W.
822:Great dodecahedron
782:great dodecahedron
760:
717:
686:
650:
577:
549:
469:Stellation facets
426:Transparent model
402:great dodecahedron
356:vertex arrangement
309:
120:Face configuration
1665:Regular polyhedra
1647:
1646:
1599:tetrahemihexacron
1522:uniform polyhedra
1391:great icosahedron
1194:Polyhedron Models
1190:Wenninger, Magnus
1180:
1179:
915:icosidodecahedron
892:
891:
768:great icosahedron
730:Related polyhedra
702:
685:
671:
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376:great icosahedron
342:regular polyhedra
299:
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288:Great icosahedron
1677:
1609:octahemioctacron
1604:hexahemioctacron
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325:Schläfli symbol
293:dual polyhedron
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149:Coxeter diagram
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100:Schläfli symbol
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1270:External links
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938:Coxeter-Dynkin
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800:Platonic solid
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354:It shares its
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195:Symmetry group
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130:Wythoff symbol
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81:Faces by sides
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1463:hemipolyhedra
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1230:Royal Society
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1207:0-521-09859-9
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904:dodecahedron
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411:-dimensional
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360:vertex figure
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276:Vertex figure
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76:= 20 (χ = 2)
75:
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812:Dodecahedron
790:
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761:
748:, 3} to {3,
661:
597:Surface Area
588:
533:Circumradius
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408:
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368:dodecahedron
353:
349:pentagrammic
346:
316:
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73:
68:
64:
1367:(nonconvex
1232:: 401–450.
931:icosahedron
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494:stellations
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380:icosahedron
209:, , (*532)
134:3 | 2
1654:Categories
1183:References
372:stellation
243:Properties
215:References
47:Stellation
1404:polyhedra
1365:polyhedra
1308:MathWorld
1290:MathWorld
1262:202575183
920:Truncated
902:stellated
775:truncated
444:Animation
398:isosceles
339:nonconvex
250:nonconvex
1192:(1974).
1141:Picture
518:Formulas
313:geometry
60:Elements
1234:Bibcode
1228:(916).
940:diagram
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332:⁄
323:, with
267:⁄
247:Regular
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670:Volume
430:Tiling
419:Images
315:, the
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1258:S2CID
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