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In geometry, an icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/[1]) is a polyhedron with 20 faces. The name comes from Greek εἴκοσι (eíkosi), meaning 'twenty', and ἕδρα (hédra), meaning 'seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons". There are many kinds of icosahedra, with some being more symmetrical than others. The best known is the Platonic, convex regular icosahedron.

Contents

1 Regular icosahedra

1.1 Convex regular icosahedron 1.2 Great icosahedron

2 Stellated icosahedra 3 Pyritohedral
Pyritohedral
symmetry

3.1 Cartesian coordinates 3.2 Jessen's icosahedron

4 Other icosahedra

4.1 Rhombic icosahedron 4.2 Pyramid
Pyramid
and prism symmetries 4.3 Johnson solids

5 See also 6 References

Regular icosahedra[edit]

Two kinds of regular icosahedra

Convex regular icosahedron

Great icosahedron

There are two objects, one convex and one concave, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a great icosahedron. Convex regular icosahedron[edit] Main article: Regular icosahedron The convex regular icosahedron is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol
Schläfli symbol
3, 5 , containing 20 triangular faces, with 5 faces meeting around each vertex. Its 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
Schläfli symbol
is 3, 5/2 . Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges. Its dual polyhedron is the great stellated dodecahedron (5/2, 3), having three regular star pentagonal faces around each vertex. Stellated icosahedra[edit] Stellation
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 resulting 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

The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.

Pyritohedral
Pyritohedral
symmetry[edit]

Pyritohedral
Pyritohedral
and tetrahedral symmetries

Coxeter diagrams (pyritohedral) (tetrahedral)

Schläfli symbol s 3,4 sr 3,3 or

s

3

3

displaystyle s begin Bmatrix 3\3end 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

Net

A regular icosahedron can be distorted or marked up as a lower pyritohedral symmetry,[2] and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron, and pseudo-icosahedron. 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
Pyritohedral symmetry
has the symbol (3*2), [3+,4], with order 24. Tetrahedral symmetry
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. These symmetries offer Coxeter diagrams: and respectively, each representing the lower symmetry to the regular icosahedron , (*532), [5,3] icosahedral symmetry of order 120.

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.

Cartesian coordinates[edit]

Construction from the vertices of a truncated octahedron, showing internal rectangles.

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.[2]

Jessen's icosahedron[edit]

The regular icosahedron and Jessen's icosahedron.

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 icosahedra[edit]

Rhombic icosahedron

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. Pyramid
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:[3]

J22 J35 J36 J59 J60 J92

Gyroelongated triangular cupola

Elongated triangular orthobicupola

Elongated triangular gyrobicupola

Parabiaugmented dodecahedron

Metabiaugmented dodecahedron

Triangular hebesphenorotunda

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

See also[edit]

600-cell

References[edit]

^ Jones, Daniel (2003) [1917], Peter Roach, James Hartmann and Jane Setter, eds., English Pronouncing Dictionary, Cambridge: Cambridge University Press, ISBN 3-12-539683-2 CS1 maint: Uses editors parameter (link) ^ a b John Baez (September 11, 2011). "Fool's Gold".  ^ Icosahedron
Icosahedron
on Mathworld.

v t e

Polyhedra

Listed by number of faces

1–10 faces

Monohedron Dihedron Trihedron Tetrahedron Pentahedron Hexahedron Heptahedron Octahedron Enneahedron Decahedron

11–20 faces

Hendecahedron Dodecahedron Tridecahedron Tetradecahedron Pentadecahedron Hexadecahedron Heptadecahedron Octadecahedron Enneadecahedron Icosahedron

Others

Triacontahedron Hexecontahedron Enneacontahedron Skew apeirohedrons

v t e

Convex polyhedra

Platonic solids (regular)

tetrahedron cube octahedron dodecahedron icosahedron

Archimedean solids (semiregular or uniform)

truncated tetrahedron cuboctahedron truncated cube truncated octahedron rhombicuboctahedron truncated cuboctahedron snub cube icosidodecahedron truncated dodecahedron truncated icosahedron rhombicosidodecahedron truncated icosidodecahedron snub dodecahedron

Catalan solids (duals of Archimedean)

triakis tetrahedron rhombic dodecahedron triakis octahedron tetrakis hexahedron deltoidal icositetrahedron disdyakis dodecahedron pentagonal icositetrahedron rhombic triacontahedron triakis icosahedron pentakis dodecahedron deltoidal hexecontahedron disdyakis triacontahedron pentagonal hexecontahedron

Dihedral regular

dihedron hosohedron

Dihedral uniform

prisms antiprisms

duals:

bipyramids trapezohedra

Dihedral others

pyramids truncated trapezohedra gyroelongated bipyramid cupola bicupola pyramidal frusta bifrustum rotunda birotunda

Degenerate polyhedra are