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 ...

, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is 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 oth ...

of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a higher symmetry, its faces are ''kites'' (also called ''trapezoids'', or ''deltoids''). The "" part of the name does not refer to faces here, but to two arrangements of each vertices around an axis of symmetry. The dual antiprism has two actual faces. An trapezohedron can be dissected into two equal

pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...

and an antiprism.

Terminology

These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles. ''Twisted'' ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra'' (with six, eight, and twelve ''twisted'' congruent kite faces) exist as crystals; in crystallography (describing the crystal habits of

mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. (2 ...

s), they are just called ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra''. They have no plane of symmetry, and no center of inversion symmetry;^, but they have a center of symmetry: the intersection point of their symmetry axes. The trigonal trapezohedron has one 3-fold symmetry axis, perpendicular to three 2-fold symmetry axes. The tetragonal trapezohedron has one 4-fold symmetry axis, perpendicular to four 2-fold symmetry axes of two kinds. The hexagonal trapezohedron has one 6-fold symmetry axis, perpendicular to six 2-fold symmetry axes of two kinds. Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2
/ref> Also in crystallography, the word ''trapezohedron'' is often used for the polyhedron with 24 congruent non-twisted kite faces properly known as a '' (deltoidal) icositetrahedron'', which has eighteen order-4 vertices and eight order-3 vertices. This is not to be confused with the ''dodecagonal trapezohedron'', which also has 24 congruent kite faces, but two order-12 apices (i.e. poles) and two rings of twelve order-3 vertices each. Still in crystallography, the ''deltoid dodecahedron'' has 12 congruent non-twisted kite faces, six order-4 vertices and eight order-3 vertices (the '' rhombic dodecahedron'' is a special case). This is not to be confused with the '' hexagonal trapezohedron'', which also has 12 congruent kite faces, but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.

Forms

An -trapezohedron is defined by a regular zig-zag skew -gon base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of

adjacent Adjacent or adjacency may refer to: * Adjacent (graph theory), two vertices that are the endpoints of an edge in a graph * Adjacent (music), a conjunct step to a note which is next in the scale See also * Adjacent angles, two angles that share ...

basal edges to one apex. An -trapezohedron has two apical vertices on its polar axis, and basal vertices in two regular -gonal rings. It has congruent kite faces, and it is isohedral. Special cases: * . A degenerate form of trapezohedron: a geometric

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

with 6 vertices, 8 edges, and 4 degenerate kite faces that are degenerated into triangles. Its dual is a degenerate form of antiprism: also a tetrahedron. * . The dual of a ''triangular antiprism'': the kites are rhombi (or squares); hence these trapezohedra are also zonohedra. They are called

rhombohedra In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be use ...

. They are cubes scaled in the direction of a body diagonal. They are also the parallelepipeds with congruent rhombic faces. Gyroelongated triangular bipyramid

** A special case of a rhombohedron is one in which the rhombi forming the faces have angles of and . It can be decomposed into two equal regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra. * . The pentagonal trapezohedron is the only polyhedron other than the

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

s commonly used as a

die Die, as a verb, refers to death, the cessation of life. Die may also refer to: Games * Die, singular of dice, small throwable objects used for producing random numbers Manufacturing * Die (integrated circuit), a rectangular piece of a semicondu ...

in roleplaying games such as '' Dungeons & Dragons''. Being convex and face-transitive, it makes fair dice. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Typically, two dice of different colors are used for the two digits to represent numbers from to .

Symmetry

The symmetry group of an -gonal trapezohedron is , of order , except in the case of : a cube has the larger symmetry group of order , which has four versions of as subgroups. The rotation group of an -trapezohedron is , of order , except in the case of : a cube has the larger rotation group of order , which has four versions of as subgroups. Note: Every -trapezohedron with a regular zig-zag skew -gon base and congruent non-twisted kite faces has the same (dihedral) symmetry group as the dual-uniform -trapezohedron, for . One degree of freedom within symmetry from D_''n''d (order 4''n'') to D_''n'' (order 2''n'') changes the congruent kites into congruent quadrilaterals with three edge lengths, called ''twisted kites'', and the ''n''-trapezohedron is called a ''twisted trapezohedron''. (In the limit, one edge of each quadrilateral goes to zero length, and the ''n''-trapezohedron becomes an ''n''- bipyramid.) If the kites surrounding the two peaks are not twisted but are of two different shapes, the ''n''-trapezohedron can only have C_''n''v (cyclic with vertical mirrors) symmetry, order 2''n'', and is called an ''unequal'' or ''asymmetric trapezohedron''. Its dual is an ''unequal n- antiprism'', with the top and bottom ''n''-gons of different radii. If the kites are twisted and are of two different shapes, the ''n''-trapezohedron can only have C_''n'' (cyclic) symmetry, order ''n'', and is called an ''unequal twisted trapezohedron''.

Star trapezohedron

A star -trapezohedron (where ) is defined by a regular zig-zag skew star -gon base, two symmetric apices with no degree of freedom right above and right below the base, and quadrilateral faces connecting each pair of

basal edges to one apex. A star -trapezohedron has two apical vertices on its polar axis, and basal vertices in two regular -gonal rings. It has congruent kite faces, and it is isohedral. Such a star -trapezohedron is a ''self-intersecting'', ''crossed'', or ''non-convex'' form. It exists for any regular zig-zag skew star -gon base (where ). But if , then , so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if , then , so the dual star antiprism must be flat, thus degenerate, to be uniform. A dual-uniform star -trapezohedron has Coxeter-Dynkin diagram .

References

* Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms *

External links

HCR's Formula for n-gonal trapezohedron/deltohedron
from Academia.edu * *
Virtual Reality Polyhedra
The Encyclopedia of Polyhedra ** VRML model
(George Hart)<3><4><5><6><7><9><10>

Try: "dA''n''", where ''n''=3,4,5... Example: "dA5" is a pentagonal trapezohedron.
Paper model tetragonal (square) trapezohedron
{{Polyhedron navigator Polyhedra fr:Antidiamant