Tetrakis hexahedron
Encyclopedia
In geometry
, a tetrakis hexahedron is a Catalan solid
. Its dual is the truncated octahedron
, an Archimedean solid
. It can be seen as a cube
with square pyramid
s covering each square face; that is, it is the Kleetope
of the cube.
It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron
.
) formations of tetrahexahedrons are observed in copper
and fluorite
systems.
Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamer
s.
A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron
, with the rhombic faces divided into two triangles.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a tetrakis hexahedron is a Catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
. Its dual is the truncated octahedron
Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
, an Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
. It can be seen as a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
with square pyramid
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
s covering each square face; that is, it is the Kleetope
Kleetope
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid...
of the cube.
It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
.
Uses
Naturally occurring (crystalCrystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
) formations of tetrahexahedrons are observed in copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...
and fluorite
Fluorite
Fluorite is a halide mineral composed of calcium fluoride, CaF2. It is an isometric mineral with a cubic habit, though octahedral and more complex isometric forms are not uncommon...
systems.
Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamer
Gamer
Historically, the term "gamer" usually referred to someone who played role-playing games and wargames. Since they became very popular, the term has included players of video games...
s.
A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...
, with the rhombic faces divided into two triangles.
See also
- Disdyakis triacontahedronDisdyakis triacontahedronIn geometry, a disdyakis triacontahedron, or hexakis icosahedron is a Catalan solid and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons...
- Disdyakis dodecahedronDisdyakis dodecahedronIn geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons...
- Bisected hexagonal tiling
- Compound of three octahedraCompound of three octahedraIn mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized...
External links
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- VRMLVRMLVRML is a standard file format for representing 3-dimensional interactive vector graphics, designed particularly with the World Wide Web in mind...
model - Conway Notation for Polyhedra Try: "dtO"
- VRML
- Tetrakis Hexahedron -- Interactive Polyhedron model
- The Uniform Polyhedra