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 Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...

(describing the

crystal habit In mineralogy, crystal habit is the characteristic external shape of an individual crystal or crystal group. The habit of a crystal is dependent on its crystallographic form and growth conditions, which generally creates irregularities due to l ...

s 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. ( ...

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 A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space. For an object, any unique centre and, more ...

: 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 In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...

'' is a special case). This is not to be confused with the ''

hexagonal trapezohedron In geometry, 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 congruence (geometry), congruent kite (geometry), kites. It can be ...

'', 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 The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex ...

with no

degree of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

right above and right below the base, and

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

faces connecting each pair of adjacent 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 A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...

faces, and it is

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 ...

. 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 the o ...

with 6 vertices, 8 edges, and 4 degenerate

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 In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...

. 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

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. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

s scaled in the direction of a body diagonal. They are also the

parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...

s 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 In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra. * . The

pentagonal trapezohedron In geometry, a pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites. It can ...

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 edges c ...

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 ...

roleplaying games A role-playing game (sometimes spelled roleplaying game, RPG) is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, either through literal ac ...

such as ''

Dungeons & Dragons ''Dungeons & Dragons'' (commonly abbreviated as ''D&D'' or ''DnD'') is a fantasy tabletop role-playing game (RPG) originally designed by Gary Gygax and Dave Arneson. The game was first published in 1974 by TSR (company)#Tactical Studies Rules ...

''. Being

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 polytope ...

and

face-transitive 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 ...

, it makes

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 ...

. 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 In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

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 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. ...

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

with no

right above and right below the base, and

faces connecting each pair of adjacent 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

faces, and it is

. 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

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Virtual Reality Polyhedra
The Encyclopedia of Polyhedra **

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