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
3.1 Cartesian coordinates 3.2 Jessen's icosahedron 4 Other icosahedra 4.1 Rhombic icosahedron
4.2
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
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.
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.
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.
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".
^
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 |