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 ...

, a hexagonal trapezohedron or deltohedron is the fourth in an infinite series of

trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...

which are

dual polyhedra 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. ...

to the

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s. It has twelve faces which are

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

kites A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...

. It can be described by the Conway notationbr>
It is an

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

(face-transitive) figure, meaning that all its

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

are the same. More specifically, all faces are not merely congruent but also ''transitive'', i.e. lie within the same '' symmetry orbit''.

Convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

isohedral polyhedra are the shapes that will make

fair dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating Statistical randomness, random values, commonly as part of tabletop games, including List of dice game ...

Symmetry

The symmetry a hexagonal trapezohedron is D_6d 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 bipyramids. 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 C_6v symmetry, order 12. These can be called ''unequal trapezohedra''. The dual is an ''unequal

'', 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 c ...

, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator. : Spherical hexagonal trapezohedron

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