Icosahedron

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Convex regular icosahedron Icosahedron.svg
Convex regular icosahedron
A tensegrity icosahedron Icosahedral tensegrity structure.png
A tensegrity icosahedron

In geometry, an icosahedron ( /ˌkɒsəˈhdrən,-kə-,-k-/ or /ˌkɒsəˈhdrən/ [1] ) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and from Ancient Greek ἕδρα (hédra) ' seat'. The plural can be either "icosahedra" ( /-drə/ ) or "icosahedrons".

Contents

There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.

Regular icosahedra

Two kinds of regular icosahedra
Icosahedron.png
Convex regular icosahedron
Great icosahedron.png
Great icosahedron

There are two objects, one convex and one nonconvex, 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

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 {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

A detail of Spinoza monument in Amsterdam Icosahedron-spinoza.jpg
A detail of Spinoza monument in Amsterdam

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.

Its 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
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.svg Seventeenth stellation of icosahedron.png
Stellation diagram of icosahedron.svg Small triambic icosahedron stellation facets.svg Great triambic icosahedron stellation facets.svg Compound of five octahedra stellation facets.svg Compound of five tetrahedra stellation facets.svg Compound of ten tetrahedra stellation facets.svg Great icosahedron stellation facets.svg Excavated dodecahedron stellation facets.svg Echidnahedron stellation facets.svg
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 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
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
Icosahedron in cuboctahedron.png Icosahedron in cuboctahedron net.png
A regular icosahedron is topologically identical to a cuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry. There exists a kinematic transformation between cuboctahedron and icosahedron.

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 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: 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 respectively, 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.

Cartesian coordinates

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

Jessen's icosahedron Jessen's icosahedron.svg
Jessen's icosahedron

In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently so that the figure is non-convex and 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.

Cuboctahedron

Progressions between an octahedron, pseudoicosahedron, and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid. A3-P5-P3.gif
Progressions between an octahedron, pseudoicosahedron, and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.

A regular icosahedron is topologically identical to a cuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, and double cover octahedron. Cyclical kinematic transformations among the members of this family exist.

Other icosahedra

Rhombic icosahedron Rhombic icosahedron.png
Rhombic icosahedron

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:

Johnson solids

Several Johnson solids are icosahedra: [3]

J22J35J36J59J60J92
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

See also

Related Research Articles

Cuboctahedron Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

Regular icosahedron One of the five Platonic solids

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

Octahedron Polyhedron with 8 triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

Rhombicuboctahedron Archimedean solid with 26 faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

Stellation Extending the elements of a polytope to form a new figure

In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

Rhombic dodecahedron Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

Rhombic triacontahedron Catalan solid with 30 faces

In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

Great icosahedron Kepler-Poinsot polyhedron with 20 faces

In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,52} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

Great icosidodecahedron

In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,52}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).

Chamfered dodecahedron Goldberg polyhedron with 42 faces

In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

Jessens icosahedron Right-angled non-convex polyhedron

Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it in 1967. In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by Adrien Douady under the name six-beakedshaddock; later authors have applied variants of this name more specifically to Jessen's icosahedron.

Excavated dodecahedron

In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. Its exterior surface represents the Ef1g1 stellation of the icosahedron. It appears in Magnus Wenninger's book Polyhedron Models as model 28, the third stellation of icosahedron.

Tetrahedrally diminished dodecahedron Family of derived polyhedra

In geometry, a tetrahedrally diminished dodecahedron is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces.

Chamfer (geometry) Geometric operation which truncates the edges of polyhedra

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.

The skeleton of a cuboctahedron, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have structural rigidity and consequently its vertices can be repositioned by folding at edges and face diagonals. The cuboctahedron's kinematics is noteworthy in that its vertices can be repositioned to the vertex positions of the regular icosahedron, the Jessen's icosahedron, and the regular octahedron, in accordance with the pyritohedral symmetry of the icosahedron.

References

  1. Jones, Daniel (2003) [1917], Peter Roach; James Hartmann; Jane Setter (eds.), English Pronouncing Dictionary, Cambridge: Cambridge University Press, ISBN   3-12-539683-2
  2. 1 2 John Baez (September 11, 2011). "Fool's Gold".
  3. Icosahedron on Mathworld.