# Icosahedron (EntityTopic, 12)

(Difference between revisions)
 Revision as of 08:23, 18 March 2014 (view source)Hayate (Talk | contribs) (add morphing animations for bilbiro and thawro)← Older edit Revision as of 02:18, 26 March 2017 (view source)Newer edit → Line 4: Line 4: | image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]> | image=<[#embed [hash 9F9FGF6HTQC8EZ51GC34JEKCKN] [width 150]]> | dim=3 | dim=3 - | elements=20, 30, 12 + | elements=20 [[triangle]]s, 30 [[digon]]s, 12 [[point]]s | sym=[[Rhodohedral symmetry|Ih, H3, [5,3], (*532)]] | sym=[[Rhodohedral symmetry|Ih, H3, [5,3], (*532)]] | genus=0 | genus=0 Line 18: Line 18: | wythoff=5 | 2 3 | wythoff=5 | 2 3 | schlaefli={[[Triangle|3,]]5} | schlaefli={[[Triangle|3,]]5} + | dynkin=o5o3x, s3s3s | conway=s[[Tetrahedron|Y3]] | conway=s[[Tetrahedron|Y3]] | vlayout=[[Triangle|3]]5 | vlayout=[[Triangle|3]]5 | vfigure=Regular [[pentagon]], edge 1 | vfigure=Regular [[pentagon]], edge 1 }}}} }}}} + 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. + == 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:

## Revision as of 02:18, 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.

## 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