Icosahedron (EntityTopic, 12)
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The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron. | The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron. | ||
| + | ==Coordinates== | ||
| + | The coordinates of an icosahedron with edge length 2 are: | ||
| + | <blockquote>(0, ±1, ±φ)<br>(±1, ±φ, 0)<br>(±φ, 0, ±1)</blockquote> | ||
== Equations == | == Equations == | ||
*The [[hypervolume]]s of an icosahedron with side length ''l'' are given by: | *The [[hypervolume]]s of an icosahedron with side length ''l'' are given by: | ||
Latest revision as of 14:50, 26 March 2017
The icosahedron is one of the five Platonic solids. It contains 20 triangles joined five to a vertex. It can also be considered as a snubbed tetrahedron.
Coordinates
The coordinates of an icosahedron with edge length 2 are:
(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)
Equations
- The hypervolumes of an icosahedron with side length l are given by:
total edge length = 30l
surface area = 5√3 · l2
volume = 5(3+√5)∕12 · l3
Relation to other polyhedra
Non-convex polyhedra sharing vertex coordinates with the icosahedron can be morphed into J91 (bilbiro) and J92 (thawro) as seen in the following animations:
The above animations are not embedded into the page due to a wiki bug. Please leave them as links and do not embed them.
Incidence matrix
Dual: dodecahedron
| # | TXID | Va | Ea | 3a | Type | Name |
|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | |||
| 1 | Ea | 2 | = digon | ; | ||
| 2 | 3a | 3 | 3 | = triangle | ; | |
| 3 | C1a | 12 | 30 | 20 | = icosahedron | ; |
Usage as facets
- 6× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron
| 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 |
