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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube) is a
Catalan solid The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...
. Its dual is the
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
, an
Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
. It can be called a disdyakis hexahedron or hexakis tetrahedron as the dual of an
omnitruncated In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each Flag (geometry), flag of the original polytope and a Facet (geometry), facet for each face of any dimension of the original pol ...
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, and as the
barycentric subdivision In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is a canonical method to refining them. Therefore, the barycentric subdivision is an important tool ...
of a tetrahedron.


As a Kleetope

The name "tetrakis" is used for the Kleetopes of polyhedra with square faces. Hence, the tetrakis hexahedron can be considered as a
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
with
square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...
s covering each square face, the Kleetope of the cube. The resulting construction can be either convex or non-convex, depending on the square pyramids' height. For the convex result, it comprises twenty-four isosceles triangles. A non-convex form of this shape, with
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
faces, has the same surface geometry as the
regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...
, and a paper octahedron model can be re-folded into this shape. This form of the tetrakis hexahedron was illustrated by
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
in
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
's ''
Divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'' (1509). Denoting the edge length of the base cube by , the height of each pyramid summit above the cube is . The inclination of each triangular face of the pyramid versus the cube face is \arctan\tfrac \approx 26.565^\circ . One edge of the isosceles triangles has length , the other two have length which follows by applying the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
to height and base length. This yields an altitude of \tfrac in the triangle (). Its
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
is \tfrac, and the internal angles are \arccos\tfrac \approx 48.1897^\circ and the complementary 180^\circ - 2\arccos\tfrac \approx 83.6206^\circ. The volume of the pyramid is so the total volume of the six pyramids and the cube in the hexahedron is This non-convex form of the tetrakis hexahedron can be folded along the square faces of the inner cube as a net for a four-dimensional cubic pyramid.


As a Catalan solid

The tetrakis hexahedron is a
Catalan solid The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...
, the
dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...
of a
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
. The truncated octahedron is an
Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
, constructed by cutting all of a
regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...
's vertices, so the resulting polyhedron has six squares and eight hexagons. The tetrakis hexahedron has the same symmetry as the truncated octahedron, the
octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
. See p. 247.
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
for the 14 vertices of a tetrakis hexahedron centered at the origin, are the points \bigl( , 0, 0 \bigr),\ \bigl( 0, \pm\tfrac32, 0 \bigr),\ \bigl( 0, 0, \pm\tfrac32 \bigr),\ \bigl(, \pm1, \pm1 \bigr). The length of the shorter edges of this tetrakis hexahedron equals 3/2 and that of the longer edges equals 2. The faces are acute isosceles triangles. The larger angle of these equals \arccos\tfrac19 \approx 83.62^ and the two smaller ones equal \arccos\tfrac23 \approx 48.19^.


Applications

Naturally occurring (
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
) formations of tetrahexahedra are observed in
copper Copper is a chemical element; it has symbol Cu (from Latin ) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkish-orang ...
and
fluorite Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2. It belongs to the halide minerals. It crystallizes in isometric cubic habit, although octahedral and more complex isometric forms are not uncommon. The Mohs scal ...
systems. Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by
gamer A gamer is someone who plays interactive games, either video games, tabletop role-playing games, skill-based card games, or any combination thereof, and who often plays for extended periods of time. Originally a hobby, gaming has evolved in ...
s. A
24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...
viewed under a vertex-first
perspective projection Linear or point-projection perspective () is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, generally on a flat surface, of ...
has a surface topology of a tetrakis hexahedron and the geometric proportions of the
rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...
, with the rhombic faces divided into two triangles. The tetrakis hexahedron appears as one of the simplest examples in
building A building or edifice is an enclosed Structure#Load-bearing, structure with a roof, walls and window, windows, usually standing permanently in one place, such as a house or factory. Buildings come in a variety of sizes, shapes, and functions, a ...
theory. Consider the Riemannian symmetric space associated to the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
''SL''4(R). Its Tits boundary has the structure of a spherical building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical
simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
(chambers) can be obtained by taking the radial projection of a tetrakis hexahedron.


Symmetry

With
tetrahedral symmetry image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that co ...
, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from six
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
s on a sphere. It can also be seen by a cube with its square faces triangulated by their vertices and face centers, and a tetrahedron with its faces divided by vertices, mid-edges, and a central point.


See also

* Disdyakis triacontahedron * Disdyakis dodecahedron * Kisrhombille tiling * Compound of three octahedra * Deltoidal icositetrahedron, another 24-face Catalan solid.


References

* (Section 3-9) * (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron) *''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,

(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Tetrakis hexahedron)


External links

*
Virtual Reality Polyhedra
www.georgehart.com: The Encyclopedia of Polyhedra ** VRMLbr>model


Try: "dtO" or "kC"
Tetrakis Hexahedron
– Interactive Polyhedron model
The Uniform Polyhedra
{{Polyhedron navigator Catalan solids