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Tridiminished icosahedron

63rd Johnson solid (8 faces) From Wikipedia, the free encyclopedia

Tridiminished icosahedron
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In geometry, the tridiminished icosahedron is a Johnson solid that is constructed by removing three pentagonal pyramids from a regular icosahedron.

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Construction

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The tridiminished icosahedron can be constructed by removing three regular pentagonal pyramid from a regular icosahedron.[1] The aftereffect of such construction leaves five equilateral triangles and three regular pentagons.[2] Since all of its faces are regular polygons and the resulting polyhedron remains convex, the tridiminished icosahedron is a Johnson solid, and it is enumerated as the sixty-third Johnson solid .[3] This construction is similar to other Johnson solids as in gyroelongated pentagonal pyramid and metabidiminished icosahedron.[1]

One can construct the vertices of a tridiminished icosahedron with the following Cartesian coordinates: where , obtained from the equation of a golden ratio .[4]

The tridiminished icosahedron is a non-composite polyhedron. That is, no plane intersects its surface only in edges, so that it cannot be thereby divided into two or more regular or Johnson polyhedra.[5]

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Properties

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The surface area of a tridiminished icosahedron is the sum of all polygonal faces' area: five equilateral triangles and three regular pentagons. Its volume can be ascertained by subtracting the volume of a regular icosahedron from the volume of three pentagonal pyramids. Given that is the edge length of a tridiminished icosahedron, they are:[2]

A tridiminished icosahedron has three kinds of dihedral angles. These angles are between two triangles: 138.1°, triangle to pentagon: 100.8°, and two pentagons: 63.4°.[6]

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As a 4-polytope cell

The tridiminished icosahedron is a cell of a snub 24-cell, a four-dimensional polytope consisting of 120 regular tetrahedra and 24 icosahedra vertex figures.[7]

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References

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