Icosahedral truncate (EntityTopic, 11)
From Hi.gher. Space
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| - | {{Shape | + | <[#ontology [kind topic] [cats 3D Uniform Polytope]]> |
| - | | image= | + | {{STS Shape |
| + | | image=<[#embed [hash 8PVAGX6A245BRXX2ZYGYJYN5H5] [width 180]]> | ||
| dim=3 | | dim=3 | ||
| elements=32, 90, 60 | | elements=32, 90, 60 | ||
| + | | sym=[[Rhodohedral symmetry|I<sub>h</sub>, H<sub>3</sub>, [5,3], (*532)]] | ||
| genus=0 | | genus=0 | ||
| - | |||
| - | |||
| ssc={G5<sup>3</sup>}X6 | | ssc={G5<sup>3</sup>}X6 | ||
| + | | ssc2=Ki6 | ||
| + | | extra={{STS Polytope | ||
| + | | flayout={{FLD|a5|i|a3|er}} | ||
| + | | dual=[[Pentakis dodecahedron]] | ||
| + | | bowers=Ti | ||
| + | }}{{STS Uniform polytope | ||
| + | | schlaefli=t{[[Pentagon|3,]]5} | ||
| + | | conway=t[[Icosahedron|s]][[Tetrahedron|Y3]] | ||
| vfigure=Isosceles [[triangle]] | | vfigure=Isosceles [[triangle]] | ||
| vlayout=[[Pentagon|5]]⋅[[Hexagon|6]]<sup>2</sup> | | vlayout=[[Pentagon|5]]⋅[[Hexagon|6]]<sup>2</sup> | ||
| - | + | }}}} | |
| - | + | == Equations == | |
| - | + | ||
| - | }} | + | |
| - | + | ||
| - | + | ||
| - | + | ||
*Variables: | *Variables: | ||
<blockquote>''l'' ⇒ length of edges of the truncated icosahedron</blockquote> | <blockquote>''l'' ⇒ length of edges of the truncated icosahedron</blockquote> | ||
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surface area = 3(3<sup>2<sup>-1</sup></sup>10+5<sup>2<sup>-1</sup></sup>(5+5<sup>2<sup>-1</sup></sup>2)<sup>2<sup>-1</sup></sup>)''l''<sup>2</sup><br> | surface area = 3(3<sup>2<sup>-1</sup></sup>10+5<sup>2<sup>-1</sup></sup>(5+5<sup>2<sup>-1</sup></sup>2)<sup>2<sup>-1</sup></sup>)''l''<sup>2</sup><br> | ||
volume = (125+5<sup>2<sup>-1</sup></sup>43)''l''<sup>3</sup>4<sup>-1</sup></blockquote> | volume = (125+5<sup>2<sup>-1</sup></sup>43)''l''<sup>3</sup>4<sup>-1</sup></blockquote> | ||
| - | |||
| - | |||
| - | [[ | + | <[#polytope [id 24]]> |
| + | |||
| + | {{Trishapes}} | ||
Latest revision as of 11:29, 2 March 2014
Equations
- Variables:
l ⇒ length of edges of the truncated icosahedron
- The hypervolumes of a truncated icosahedron are given by:
total edge length = 90l
surface area = 3(32-110+52-1(5+52-12)2-1)l2
volume = (125+52-143)l34-1
Incidence matrix
Dual: pentakis dodecahedron
| # | TXID | Va | Ea | Eb | 5a | 6a | Type | Name |
|---|---|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | |||||
| 1 | Ea | 2 | = digon | ; | ||||
| 2 | Eb | 2 | = digon | ; | ||||
| 3 | 5a | 5 | 0 | 5 | = pentagon | ; | ||
| 4 | 6a | 6 | 3 | 3 | = hexagon | ; | ||
| 5 | C1a | 60 | 30 | 60 | 12 | 20 | = truncated icosahedron | ; |
Usage as facets
- 24× 1-facets of a truncated snub demitesseract
- 1× 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 |
