Convex regular icosahedron
In geometry, an icosahedron ( or [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" () or "icosahedrons".
There are many kinds of icosahedra, with some being more symmetrical than others. The most well known is the regular convex or Platonic icosahedron.
Contents
- 1 Regular icosahedra
- 1.1 Convex regular icosahedron
- 1.2 Great icosahedron
- 2 Stellated icosahedra
- 3 Pyritohedral symmetry
- 3.1 Cartesian coordinates
- 3.2 Jessen's icosahedron
- 4 Other icosahedra
- 4.1 Rhombic icosahedron
- 4.2 Pyramid and prism symmetries
- 4.3 Johnson solids
- 5 Icosahedral resistance network
- 6 References
Regular icosahedra
Two kinds of regular icosahedra
Convex regular icosahedron |
Great icosahedron |
The most symmetrical are the two kinds of regular icosahedra, while the nonconvex form is called a great icosahedron. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry.
Convex regular icosahedron
Main article: Regular icosahedron
The convex regular icosahedron is the more common and is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.
The dual polyhedron is the regular dodecahedron {5, 3} having three regular pentagonal faces around each vertex.
Great icosahedron
Main article: Great icosahedron
The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra. Its 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.
The dual polyhedron is the great stellated dodecahedron (5/2, 3), having three regular star pentagonal faces around each vertex.
Stellated icosahedra
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 |
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| The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry. |
Pyritohedral symmetry
| Pyritohedral and tetrahedral symmetries |
| Coxeter diagrams |
(pyritohedral)
(tetrahedral) |
| Schläfli symbol |
s{3,4}
sr{3,3} or |
| 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 |
Construction from the vertices of a truncated octahedron, showing internal rectangles.
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), [3+,4], 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.
These symmetries offer Coxeter diagrams: and respectively, each representing the lower symmetry to the regular icosahedron , (*532), [5,3] icosahedral symmetry of order 120.
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| 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
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
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
Rhombic icosahedron
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 and prism symmetries
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
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 |
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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 |
Icosahedral resistance network
If each edge of a triangular icosahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 0.5 ohms, and that between adjacent vertices is 11/30 ohms.[4]
References
- ^ Jones, Daniel (2003) [1917], Peter Roach, James Hartmann and Jane Setter, eds., English Pronouncing Dictionary, Cambridge: Cambridge University Press, ISBN 3-12-539683-2
- ^ John Baez (September 11, 2011). "Fool's Gold".
- ^ Icosahedron on Mathworld.
- ^ Klein, Douglas J. (2002), "Resistance-Distance Sum Rules" (PDF), Croatica Chemica Acta 75 (2): 633–649, retrieved 2006-09-15
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Polyhedra
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Listed by number of faces
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| 1–10 faces |
- Monohedron
- Dihedron
- Trihedron
- Tetrahedron
- Pentahedron
- Hexahedron
- Heptahedron
- Octahedron
- Enneahedron
- Decahedron
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| 11–20 faces |
- Hendecahedron
- Dodecahedron
- Tridecahedron
- Tetradecahedron
- Pentadecahedron
- Hexadecahedron
- Heptadecahedron
- Octadecahedron
- Enneadecahedron
- Icosahedron
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| Others |
- Triacontahedron
- Hexecontahedron
- Enneacontahedron
- Apeirohedron
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Convex polyhedra
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| Platonic solids (regular) |
- tetrahedron
- cube
- octahedron
- dodecahedron
- icosahedron
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Archimedean solids
(semiregular or uniform)
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- truncated tetrahedron
- cuboctahedron
- truncated cube
- truncated octahedron
- rhombicuboctahedron
- truncated cuboctahedron
- snub cube
- icosidodecahedron
- truncated dodecahedron
- truncated icosahedron
- rhombicosidodecahedron
- truncated icosidodecahedron
- snub dodecahedron
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Catalan solids
(duals of Archimedean)
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- 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
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| Dihedral regular |
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| Dihedral uniform |
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| Dihedral others |
- pyramids
- truncated trapezohedra
- gyroelongated bipyramid
- cupola
- bicupola
- pyramidal frusta
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Degenerate polyhedra are in italics.
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