# Icosahedron (EntityTopic, 12)

(Difference between revisions)
 Revision as of 02:18, 26 March 2017 (view source)← Older edit Latest revision as of 14:50, 26 March 2017 (view source) Line 25: Line 25: 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: +
(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)
== 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 Type Name Va Ea 3a 0 Va = point ; 1 Ea 2 = digon ; 2 3a 3 3 = triangle ; 3 C1a 12 30 20 = icosahedron ;

## Usage as facets

 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