Tridiminished icosahedron (EntityTopic, 10)
From Hi.gher. Space
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{{STS Shape | {{STS Shape | ||
| name=Tridiminished icosahedron | | name=Tridiminished icosahedron | ||
| + | | image=<[#embed [hash BYG0WBQWWCPDRE0BK8YAPNR162]]> | ||
| dim=3 | | dim=3 | ||
| - | | elements= | + | | elements=8, 15, 9 |
| + | | sym=[[Pyroaxial symmetry|C<sub>3v</sub>]] | ||
| genus=0 | | genus=0 | ||
| extra={{STS Polytope | | extra={{STS Polytope | ||
| flayout={{FLD|a3|i|adesr|di|asl|i|a3|cr}} | | flayout={{FLD|a3|i|adesr|di|asl|i|a3|cr}} | ||
| + | | bowers=Teddi | ||
}}}} | }}}} | ||
| + | The '''tridiminished icosahedron''', abbreviated '''TDI''', is the 63rd [[Johnson solid]], J63. It suddenly became important when the [[bixylodiminished hydrochoron]] was discovered. It is the only CRF 3D member of the family of [[ursatope]]s, and appears as cells in the majority of the higher-dimensional ones. | ||
| - | The ''' | + | == Relationship with J83 == |
| + | J83 can be constructed as a sort of informal expansion of J63. See the [[tridiminished rhodoperihedron]] page for the visualisation. | ||
| + | |||
| + | == Equations == | ||
| + | *The [[hypervolume]]s of a tridiminished icosahedron with side length ''l'' are given by: | ||
| + | <blockquote>total edge length = 15''l''<br> | ||
| + | surface area = {{Over|1|4}} · (3√(25+10√5) + 5√3) {{DotHV}}<br> | ||
| + | volume = {{Over|(15+7√5)|24}} {{DotHV|3}}</blockquote> | ||
| + | |||
| + | <[#polytope [id 47]]> | ||
{{Trishapes}} | {{Trishapes}} | ||
Latest revision as of 13:34, 12 March 2016
The tridiminished icosahedron, abbreviated TDI, is the 63rd Johnson solid, J63. It suddenly became important when the bixylodiminished hydrochoron was discovered. It is the only CRF 3D member of the family of ursatopes, and appears as cells in the majority of the higher-dimensional ones.
Relationship with J83
J83 can be constructed as a sort of informal expansion of J63. See the tridiminished rhodoperihedron page for the visualisation.
Equations
- The hypervolumes of a tridiminished icosahedron with side length l are given by:
total edge length = 15l
surface area = 1∕4 · (3√(25+10√5) + 5√3) · l2
volume = (15+7√5)∕24 · l3
Incidence matrix
Dual: J63 dual
| # | TXID | Va | Vb | Vc | Ea | Eb | Ec | Ed | 3a | 5a | 3b | 3c | Type | Name |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | Va | = point | ; top vertices | |||||||||||
| 1 | Vb | = point | ; mid vertices | |||||||||||
| 2 | Vc | = point | ; bot vertices | |||||||||||
| 3 | Ea | 2 | 0 | 0 | = digon | ; top to pen | ||||||||
| 4 | Eb | 1 | 1 | 0 | = digon | ; pen to pen | ||||||||
| 5 | Ec | 0 | 1 | 1 | = digon | ; pen to lat | ||||||||
| 6 | Ed | 0 | 0 | 2 | = digon | ; lat to bot | ||||||||
| 7 | 3a | 3 | 0 | 0 | 3 | 0 | 0 | 0 | = triangle | ; top | ||||
| 8 | 5a | 2 | 2 | 1 | 1 | 2 | 2 | 0 | = pentagon | ; pen | ||||
| 9 | 3b | 0 | 1 | 2 | 0 | 0 | 2 | 1 | = triangle | ; lat | ||||
| 10 | 3c | 0 | 0 | 3 | 0 | 0 | 0 | 3 | = triangle | ; bot | ||||
| 11 | C1a | 3 | 3 | 3 | 3 | 3 | 6 | 3 | 1 | 3 | 3 | 1 | = tridiminished icosahedron | ; |
Usage as facets
- 48× 1-facets of a bixylodiminished hydrochoron
- 96× 1-facets of a rectified snub demitesseract
- 96× 1-facets of a truncated snub demitesseract
- 32× 1-facets of a D4.11
- 12× 1-facets of a D4.7
| Notable Trishapes | |
| Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
| Direct truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
| Mesotruncates: | stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |
| Snubs: | snub staurohedron • snub rhodohedron |
| Curved: | sphere • torus • cylinder • cone • frustum • crind |
