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 ...

, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron Remarks: the faces of a deltohedron are deltoids; a (non-twisted) kite or deltoid can be dissected into two

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

s or "deltas" (Δ), base-to-base. 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 other ...

of an

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

. The faces of an 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 modu ...

and symmetrically staggered; they are called ''twisted kites''. With a higher symmetry, its faces are ''kites'' (sometimes 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 () is a Nonbuilding structure, structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a Pyramid (geometry), pyramid in the geometric sense. The base of a pyramid ca ...

and an

Terminology

These figures, sometimes called deltohedra, are not to be confused with deltahedra, whose faces are equilateral triangles. ''Twisted'' ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra'' (with six, eight, and twelve ''twisted''

kite faces) exist as crystals; in

crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

(describing the

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

s of

mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid substance with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. (2011): Mi ...

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

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 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 ...

'' 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 with no

degree of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinites ...

right above and right below the base, and

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

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

kite A kite is a tethered heavier than air flight, heavier-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. Kites often have ...

faces, and it is isohedral.

Special cases

* . A degenerate form of trapezohedron: a geometric figure with 6 vertices, 8 edges, and 4 degenerate

faces that are visually identical to triangles. As such, the trapezohedron itself is visually identical to the regular

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 ...

. Its dual is a degenerate form of

that also resembles the regular 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 point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

. They are called

rhombohedra In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombus, rhombi. It can be used to define the rhombohedral lattice system, a Ho ...

. They are

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 ...

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. Three equiva ...

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 (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

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

pentagonal trapezohedron In geometry, a pentagonal trapezohedron is the third in the infinite family of trapezohedra, face-transitive polyhedra. Its dual polyhedron is the pentagonal antiprism. As a decahedron it has ten faces which are congruent kites. It can be dec ...

is the only polyhedron other than the

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s commonly used as a die in roleplaying games such as ''

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

''. 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 polytop ...

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 Face (geometry), faces are the same. More specifically, all faces must be not ...

, 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 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 amb ...

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 (order ) to (order ) changes the congruent kites into congruent quadrilaterals with three edge lengths, called ''twisted kites'', and the -trapezohedron is called a ''twisted trapezohedron''. (In the limit, one edge of each quadrilateral goes to zero length, and the -trapezohedron becomes an -

bipyramid In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...

.) If the kites surrounding the two peaks are not twisted but are of two different shapes, the -trapezohedron can only have (cyclic with vertical mirrors) symmetry, order , and is called an ''unequal'' or ''asymmetric trapezohedron''. Its dual is an ''unequal -

'', with the top and bottom -gons of different radii. If the kites are twisted and are of two different shapes, the -trapezohedron can only have (cyclic) symmetry, order , 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

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

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

* *
Paper model tetragonal (square) trapezohedron
{{Polyhedron navigator Polyhedra fr:Antidiamant