Icosahedron

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In geometry, an icosahedron (/ˌkɵsəˈhdrən/ or /ˌkɒsəˈhdrən/) is a polyhedron with 20 faces. The name comes from Greek είκοσι (eíkosi), meaning "twenty", and έδρα (hédra), meaning "seat". The plural can be either "icosahedra" (-/drə/) or "icosahedrons".

There are many kinds of icosahedron, with some being more symmetrical than others. The most well known is the regular convex or Platonic icosahedron.

Regular icosahedra[edit]

The most symmetrical are the two kinds of regular icosahedron. Each has 20 equilateral triangle faces with five meeting at each of its twelve vertices.

Convex regular icosahedron[edit]

Main article: Regular icosahedron

The convex regular icosahedron is one of the five regular Platonic solids and is represented by its Schläfli symbol {3, 5}.

The dual polyhedron is the regular dodecahedron {5, 3} having three regular pentagonal faces around each vertex.

Great icosahedron[edit]

Main article: Great icosahedron

The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra. Its Schläfli symbol is {3, 5/2}.

The dual polyhedron is the great stellated dodecahedron {5/2, 3}, having three regular star pentagonal faces around each vertex.

Stellated icosahedra[edit]

Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the rsulting figure retains the overall symmetry of the parent figure.

In their book The fifty nine icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron.

Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them.

Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such.

Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
(Convex) icosahedron Small triambic icosahedron Medial triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.

Pyritohedral symmetry[edit]

Pyritohedral and tetrahedral symmetries
Pseudoicosahedron-2.pngPseudoicosahedron-1.png
Pseudoicosahedron-4.pngPseudoicosahedron-3.png
Four views of an icosahedron with tetrahedral symmetry, with eight equilateral triangles (red and yellow), and 12 blue isosceles triangles. Yellow and red triangles are the same color in pyritohedral symmetry.
Coxeter diagrams CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png (pyritohedral) Uniform polyhedron-43-h01.svg
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png (tetrahedral) Uniform polyhedron-33-s012.svg
Schläfli symbol s{3,4}
sr{3,3} or s\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}
Faces 20 triangles:
8 equilateral
12 isosceles
Edges 30 (6 short + 24 long)
Vertices 12
Symmetry group Th, [4,3+], (3*2), order 24
Rotation group Td, [3,3]+, (332), order 12
Dual polyhedron Pyritohedron
Properties convex
Pseudoicosahedron flat.png
Net
Construction from the vertices of a truncated octahedron, showing internal rectangles with edge length ratios of 2:1.

A regular icosahedron can be constructed with pyritohedral symmetry, and is called a snub octahedron or snub tetratetrahedron or snub tetrahedron. this can be seen as an alternated truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.

Pyritohedral symmetry has the symbol (3*2), [4,3+], with order 24. Tetrahedral symmetry has the symbol (332), [3,3]+, with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles.

Cartesian coordinates[edit]

The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.

This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (φ, 1, 0), where φ is the golden ratio.[1]

Jessen's icosahedron[edit]

Main article: Jessen's icosahedron

In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently such that the figure is non-convex. It has right dihedral angles.

It is scissors congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.

Other symmetries[edit]

Rhombic icosahedron[edit]

Main article: Rhombic icosahedron

The rhombic icosahedron is a zonohedron made up of 20 congruent rhombs. It can be derived from the rhombic triacontahedron by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not face-transitive.

Tetrahedral colouring[edit]

20 triangles can also be arranged with tetrahedral symmetry (332), [3,3]+, seen as the 8 triangles marked (colored) in alternating pairs of four, with order 12. These symmetries offer Coxeter diagrams: CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png and CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png respectfully, each representing the lower symmetry to the regular icosahedron CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png, (*532), [5,3] icosahedral symmetry of order 120.

Pyramid and prism symmetries[edit]

Common icosahedra with pyramid and prism symmetries include:

  • 19-sided pyramid (plus 1 base = 20).
  • 18-sided prism (plus 2 ends = 20).
  • 9-sided antiprism (2 sets of 9 sides + 2 ends = 20).
  • 10-sided bipyramid (2 sets of 10 sides = 20).
  • 10-sided trapezohedron (2 sets of 10 sides = 20).

Johnson solids[edit]

Several Johnson solids are icosahedra:[2]

J22 J35 J36 J59 J60 J92
Gyroelongated triangular cupola.png
Gyroelongated triangular cupola
Elongated triangular orthobicupola.png
Elongated triangular orthobicupola
Elongated triangular gyrobicupola.png
Elongated triangular gyrobicupola
Parabiaugmented dodecahedron.png
Parabiaugmented dodecahedron
Metabiaugmented dodecahedron.png
Metabiaugmented dodecahedron
Triangular hebesphenorotunda.png
Triangular hebesphenorotunda
Johnson solid 22 net.png Johnson solid 35 net.png Johnson solid 36 net.png Johnson solid 59 net.png Johnson solid 60 net.png Johnson solid 92 net.png
16 triangles
3 squares
 
1 hexagon
8 triangles
12 squares
8 triangles
12 squares
10 triangles
 
10 pentagons
10 triangles
 
10 pentagons
13 triangles
3 squares
3 pentagons
1 hexagon

References[edit]

  1. ^ John Baez (September 11, 2011). "Fool's Gold". 
  2. ^ Icosahedron on Mathworld.