These figures, sometimes called deltohedra,[1] are not to be confused with deltahedra,[2] whose faces are equilateral triangles.
Twistedtrigonal, tetragonal, and hexagonal trapezohedra (with six, eight, and twelve twistedcongruent kite faces) exist as crystals; in crystallography (describing the crystal habits of minerals), they are just called trigonal, tetragonal, and hexagonal trapezohedra. They have no plane of symmetry, and no center of inversion symmetry;Template:Sfn,Template:Sfn 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.Template:Sfn 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.Template:Sfn
Crystal arrangements of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.[3]
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,Template:Sfn 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 dodecahedronTemplate:Sfn 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,Template:Sfn but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.
An Template:Mvar-trapezohedron has two apical vertices on its polar axis, and 2nScript error: No such module "Check for unknown parameters". basal vertices in two regular Template:Mvar-gonal rings. It has 2nScript error: No such module "Check for unknown parameters".congruentkite faces, and it is isohedral.
n = 2Script error: No such module "Check for unknown parameters".. A degenerate trapezohedron: a geometric figure with 6 vertices, 8 edges, and 4 degenerate kite faces that are visually identical to triangles. As such, the trapezohedron itself is visually identical to the regular tetrahedron. Its dual is a degenerate form of antiprism that also resembles the regular tetrahedron.
n = 3Script error: No such module "Check for unknown parameters".. The dual of a triangular antiprism: the kites are rhombi (or squares); hence these trapezohedra are also zonohedra. They are called rhombohedra. They are cubes scaled in the direction of a body diagonal. They are also the parallelepipeds with congruent rhombic faces.File:Gyroelongated triangular bipyramid.pngA 60°Script error: No such module "Check for unknown parameters". rhombohedron, dissected into a central regular octahedron and two regular tetrahedra
A special case of a rhombohedron is one in which the rhombi forming the faces have angles of 60°Script error: No such module "Check for unknown parameters". and 120°Script error: No such module "Check for unknown parameters".. 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.
n = 5Script error: No such module "Check for unknown parameters".. The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die 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 00Script error: No such module "Check for unknown parameters". to 99Script error: No such module "Check for unknown parameters"..
Symmetry
The symmetry group of an Template:Mvar-gonal trapezohedron is Dnd = DnvScript error: No such module "Check for unknown parameters"., of order 4nScript error: No such module "Check for unknown parameters"., except in the case of n = 3Script error: No such module "Check for unknown parameters".: a cube has the larger symmetry group OdScript error: No such module "Check for unknown parameters". of order 48 = 4×(4×3)Script error: No such module "Check for unknown parameters"., which has four versions of D3dScript error: No such module "Check for unknown parameters". as subgroups.
The rotation group of an Template:Mvar-trapezohedron is DnScript error: No such module "Check for unknown parameters"., of order 2nScript error: No such module "Check for unknown parameters"., except in the case of n = 3Script error: No such module "Check for unknown parameters".: a cube has the larger rotation group OScript error: No such module "Check for unknown parameters". of order 24 = 4×(2×3)Script error: No such module "Check for unknown parameters"., which has four versions of D3Script error: No such module "Check for unknown parameters". as subgroups.
Note: Every Template:Mvar-trapezohedron with a regular zig-zag skew2nScript error: No such module "Check for unknown parameters".-gon base and 2nScript error: No such module "Check for unknown parameters". congruent non-twisted kite faces has the same (dihedral) symmetry group as the dual-uniformTemplate:Mvar-trapezohedron, for n ≥ 4Script error: No such module "Check for unknown parameters"..
One degree of freedom within symmetry from DndScript error: No such module "Check for unknown parameters". (order 4nScript error: No such module "Check for unknown parameters".) to DnScript error: No such module "Check for unknown parameters". (order 2nScript error: No such module "Check for unknown parameters".) changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the Template:Mvar-trapezohedron is called a twisted trapezohedron. (In the limit, one edge of each quadrilateral goes to zero length, and the Template:Mvar-trapezohedron becomes an Template:Mvar-bipyramid.)
If the kites surrounding the two peaks are not twisted but are of two different shapes, the Template:Mvar-trapezohedron can only have CnvScript error: No such module "Check for unknown parameters". (cyclic with vertical mirrors) symmetry, order 2nScript error: No such module "Check for unknown parameters"., and is called an unequal or asymmetric trapezohedron. Its dual is an unequal Template:Mvar-antiprism, with the top and bottom Template:Mvar-gons of different radii.
If the kites are twisted and are of two different shapes, the Template:Mvar-trapezohedron can only have CnScript error: No such module "Check for unknown parameters". (cyclic) symmetry, order Template:Mvar, and is called an unequal twisted trapezohedron.
Example: variations with hexagonal trapezohedra (n = 6)
A star p/qScript error: No such module "Check for unknown parameters".-trapezohedron has two apical vertices on its polar axis, and 2pScript error: No such module "Check for unknown parameters". basal vertices in two regular Template:Mvar-gonal rings. It has 2pScript error: No such module "Check for unknown parameters".congruentkite faces, and it is isohedral.
Such a star p/qScript error: No such module "Check for unknown parameters".-trapezohedron is a self-intersecting, crossed, or non-convex form. It exists for any regular zig-zag skew star 2p/qScript error: No such module "Check for unknown parameters".-gon base (where 2 ≤ q < 1pScript error: No such module "Check for unknown parameters".).
But if Template:Sfrac < Template:SfracScript error: No such module "Check for unknown parameters"., then (p − q)Template:Sfrac < Template:SfracTemplate:SfracScript error: No such module "Check for unknown parameters"., so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if Template:Sfrac = Template:SfracScript error: No such module "Check for unknown parameters"., then (p − q)Template:Sfrac = Template:SfracTemplate:SfracScript error: No such module "Check for unknown parameters"., so the dual star antiprism must be flat, thus degenerate, to be uniform.
The Haunter of the Dark, a short story by H.P. Lovecraft in which a fictional ancient artifact known as The Shining Trapezohedron plays a crucial role.
References
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↑ abScript error: No such module "citation/CS1". Remarks: the faces of a deltohedron are deltoids; a (non-twisted) kite or deltoid can be dissected into two isosceles triangles or "deltas" (Δ), base-to-base.
