The double-twisted rhomboicosododecahedron [1] is one of Johnson's polyhedra ( J 74 , according to Zalgaller - 2 M 6 + M 13 + M 6 ).
| Twisted oblique rhombicosododecahedron | |||
|---|---|---|---|
 ( 3D model ) | |||
| Type of | Johnson's polyhedron | ||
| The properties | convex | ||
| Combinatorics | |||
| Items |
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| Facets | 20 triangles 30 squares 12 pentagons | ||
| Vertex configuration | 5x4 (3.4 2 .5) 4x2 + 8x4 (3.4.5.4) | ||
Scan
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| Classification | |||
| Designations | J 74 , 2 M 6 + M 13 + M 6 | ||
| Symmetry group | C 2v | ||
Composed of 62 faces: 20 regular triangles , 30 squares and 12 regular pentagons . Among the pentagonal 4 faces are surrounded by five square, 6 faces - four square and triangular, the remaining 2 - three square and two triangular; among the square faces 1 is surrounded by two pentagonal and two square, 11 - two pentagonal and two triangular, 8 - two pentagonal, square and triangular, the remaining 10 - pentagonal, square and two triangular; among the triangular faces, 10 are surrounded by three square ones, the other 10 are pentagonal and two square.
It has 120 edges of the same length. 50 edges are located between the pentagonal and square faces, 10 edges are between the pentagonal and triangular, 10 edges are between two square, the remaining 50 are between square and triangular.
A double skew twisted rhomboicosododecahedron has 60 vertices. In each pentagonal, two square and triangular faces converge.
The twisted oblique rhombicosododecahedron can be obtained from the rhomboicosododecahedron by choosing two parts in it - any two non-opposite and non-intersecting five-sloping domes ( J 5 ) - and each rotating 36 ° around its axis of symmetry. The volume and surface area will not change; the described and half-inscribed spheres of the obtained polyhedron also coincide with the described and half-inscribed spheres of the original rhombicosododecahedron.
Metric
If a double oblique twisted rhomboicosododecahedron has an edge of length , its surface area and volume are expressed as
The radius of the described sphere (passing through all the vertices of the polyhedron) will be equal to
radius of a half-inscribed sphere (touching all edges in their midpoints)
Notes
- ↑ Zalgaller V. A. Convex polyhedra with regular edges / Zap. scientific sem. LOMI, 1967. - V. 2. - Page. 23.
Links
- Weisstein, Eric W. Double-oblique twisted rhombicosododecahedron on the Wolfram MathWorld website.