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.
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 · l^{2}
volume = ^{5(3+√5)}∕_{12} · l^{3}
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 |