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These symmetries offer Coxeter diagrams: and respectively, each representing the lower symmetry to the regular icosahedron, (*532), [5,3] icosahedral symmetry of order 120. Cartesian coordinates Construction from the vertices of a truncated octahedron , showing internal rectangles.
It is called the icosahedral pyramidal group and is the 3D icosahedral group, [5,3]. A regular dodecahedral pyramid can have this symmetry, with Schläfli symbol: ( ) ∨ {5,3}. A chiral half subgroup is [(5,3) +,2,1 +] = [5,3,1] + = [5,3] +, (= ), order 60, (Du Val #31' (I/C 1;I/C 1), Conway + 1 / 60 [IxI]). This is the 3D chiral icosahedral ...
The five-fold, three-fold, and two-fold are labeled in blue, red, and magenta respectively. The mirror planes are the cyan great circle . The regular icosahedron has six five-fold rotation axes passing through the two opposite vertices, ten three-fold axes rotating a triangular face, and fifteen two-fold axes passing through any of its edges.
Icosahedral symmetry fundamental domains A soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry. Rotations and reflections form the symmetry group of a great icosahedron. In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron.
John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion .
The truncated icosahedral graph. According to Steinitz's theorem, the skeleton of a truncated icosahedron, like that of any convex polyhedron, can be represented as a polyhedral graph, meaning a planar graph (one that can be drawn without crossing edges) and 3-vertex-connected graph (remaining connected whenever two of its vertices are removed ...
The complete icosahedron is formed from all the cells in the stellation, but only the outermost regions, labelled "13" in the diagram, are visible. The stellation of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as ...
It has icosahedral symmetry (I h) and the same vertex arrangement as a rhombic triacontahedron. This can be seen as the three-dimensional equivalent of the compound of two pentagons ({10/2} "decagram"); this series continues into the fourth dimension as the compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic ...