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.
Template:Math. A degenerate form of 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.
One degree of freedom within symmetry from Template:Math (order Template:Math) to Template:Math (order Template:Math) 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 Template:Math (cyclic with vertical mirrors) symmetry, order Template:Math, 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 Template:Math (cyclic) symmetry, order Template:Mvar, and is called an unequal twisted trapezohedron.
Example: variations with hexagonal trapezohedra (n = 6)
Such a star Template:Math-trapezohedron is a self-intersecting, crossed, or non-convex form. It exists for any regular zig-zag skew star Template:Math-gon base (where Template:Math).
But if Template:Math, then Template:Math, so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if Template:Math, then Template:Math, 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.
↑ 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.
