The five-fold straight birotund [1] is one of Johnson's polyhedra ( J 34 , according to Zalgaller - 2M 9 ).
| Five Slope Direct Birotond | |||
|---|---|---|---|
 ( 3D model ) | |||
| Type of | Johnson's polyhedron | ||
| The properties | convex | ||
| Combinatorics | |||
| Items |
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| Facets | 20 triangles 12 pentagons | ||
| Vertex configuration | 10 (3 2 .5 2 ) 2x10 (3.5.3.5) | ||
Scan
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| Classification | |||
| Designations | J 34 , 2M 9 | ||
| Symmetry group | D 5h | ||
Composed of 32 faces: 20 regular triangles and 12 regular pentagons . Among the pentagonal faces 2 are surrounded by five triangular, the remaining 10 are pentagonal and four triangular; among the triangular faces, 10 are surrounded by three pentagonal, the other 10 by two pentagonal and triangular.
It has 60 edges of the same length. 5 edges are located between two pentagonal faces, 50 edges are between pentagonal and triangular, 5 edges are between two triangular.
The five-slope straight biotond has 30 peaks. Two pentagonal and two triangular faces converge in each.
A five-slope direct biotond can be obtained from the icosododecahedron by dividing it into two halves, each of which is a five-slope rotunda ( J 6 ), and rotating one of them 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 initial icosododecahedron.
Metric
If the five-slope straight biotond 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. 21.
Links
- Weisstein, Eric W. Five-Slope Direct Biotond (English) at Wolfram MathWorld .