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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a hexagonal trapezohedron or deltohedron is the fourth in an infinite series of trapezohedra which are dual polyhedra to the antiprisms. It has twelve faces which are congruent kites. It can be described by the Conway notationbr>
It is an isohedral (face-transitive) figure, meaning that all its faces are the same. More specifically, all faces are not merely congruent but also ''transitive'', i.e. lie within the same ''
symmetry orbit In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
''. Convex isohedral polyhedra are the shapes that will make fair dice.


Symmetry

The symmetry a hexagonal trapezohedron is D6d of order 24. The
rotation group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
is D6 of order 12.


Variations

One degree of freedom within D6 symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, and these become
bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does ...
s. Crystal arrangements of atoms can repeat in space with hexagonal trapezohedral cells.3 2 and Hexagonal-trapezohedric Class, 6 2 2
/ref> If the kites surrounding the two peaks are of different shapes, it can only have C6v symmetry, order 12. These can be called ''unequal trapezohedra''. The dual is an ''unequal antiprism'', with the top and bottom polygons of different radii. If it twisted and unequal its symmetry is reduced to cyclic symmetry, C6 symmetry, order 6.


Spherical tiling

The hexagonal trapezohedron also exists as a
spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...
, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator. :


Related polyhedra


References


External links

*
Virtual Reality Polyhedra
The Encyclopedia of Polyhedra **
VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...
mode
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{{Polyhedron navigator Polyhedra