Disdyakis dodecahedron

The Hexakisoktaeder ( from Greek ἑξάκις hexakis " six times " and octahedron " octahedron " ) or Disdyakisdodekaeder ( δίς dis " twice," δυάκις dyakis " twice " and dodecahedron " Zwölfflächner ") is a convex polyhedron composed of 48 irregular triangles and the bodies the Catalan counts. It is dual to the Kuboktaederstumpf and has 26 vertices and 72 edges.

  • 2.1 Regular
  • 2.2 Rhombisch 2.2.1 General
  • 2.2.2 Specially

Formation

Rhombic dodecahedron as a basis

Will the 12 boundary surfaces of a rhombic dodecahedron ( edge length ) pyramids with edge lengths and placed, creating a Hexakisoktaeder, if the following condition is satisfied:

  • For the above-mentioned minimum value of the placed pyramids have depth 0, so that only the rhombic dodecahedron is left with the edge length.
  • The special Hexakisoktaeder with the same dihedral angles at the edges and arises when is.
  • If the maximum value previously mentioned, the Hexakisoktaeder degenerates into a Deltoidalikositetraeder with the edge lengths and.
  • Exceeds the maximum value, the polyhedron is not convex.

Kuboktaederstumpf as a basis

By connecting the midpoints of three edges that meet at each corner of the room of the truncated cuboctahedron, creating a triangle whose perimeter inscribed circle of the triangle, the boundary surface of the Hexakisoktaeders is, at the same time. In this particular type, all dihedral angles equal ( ≈ 155 ° ), and there is a uniform sphere radius edges.

Let d be the length of the edges Kuboktaederstumpfs, the resulting side lengths of the triangle are determined by

Formulas

In the following denote the longest edge of each Hexakisoktaeders ().

Regular

The basis is the truncated cuboctahedron ( dual Archimedean bodies).

Rhomboidal

The basis is the rhombic dodecahedron ( edge length ).

Generally

Specifically

Occurrence

  • The Hexakisoktaeder occurs in nature as crystal form. It is the general surface shape of the crystal class m3m hexakisoktaedrischen.
  • To apply the Hexakisoktaeder comes as dice ( W48 ).
Pyramid (geometry) Deltoidal icositetrahedron Midsphere Inscribed sphere Point group Wolfram Research Triakis tetrahedron Triakis octahedron Pentagonal icositetrahedron Pentakis dodecahedron Triakis icosahedron Deltoidal hexecontahedron Pentagonal hexecontahedron
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