Editing Trapezohedron (section)

==Terminology==
These figures, sometimes called delt'''o'''hedra,<ref name=":1" /> are not to be confused with [[Deltahedron|delt'''a'''hedra]],<ref name=":2" /> whose faces are equilateral triangles.

[[#Symmetry|''Twisted'']] ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra'' (with six, eight, and twelve ''twisted'' [[Congruence (geometry)|congruent]] kite faces) exist as crystals; in [[crystallography]] (describing the [[crystal habit]]s of [[mineral]]s), they are just called ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra''. They have no plane of symmetry, and no center of inversion symmetry;{{sfn|Spencer|1911|p=581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Rhombohedral Division'', TRAPEZOHEDRAL CLASS, FIG. 74}}<sup>,</sup>{{sfn|Spencer|1911|p=577, or p. 599 on Wikisource, CRYSTALLOGRAPHY, 2. TETRAGONAL SYSTEM, TRAPEZOHEDRAL CLASS}} 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.{{sfn|Spencer|1911|p=581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Rhombohedral Division'', TRAPEZOHEDRAL CLASS, FIG. 74}} 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.{{sfn|Spencer|1911|p=582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Hexagonal Division'', TRAPEZOHEDRAL CLASS}}

[[Crystal system|Crystal arrangements]] of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.<ref name=":0">[http://www.metafysica.nl/turing/promorph_crystals_2.html 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 [[Congruence (geometry)|congruent]] non-twisted kite faces properly known as a ''[[deltoidal icositetrahedron]]'',{{sfn|Spencer|1911|p=574, or p. 596 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, HOLOSYMMETRIC CLASS, FIG. 17}} 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''{{sfn|Spencer|1911|p=575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIG. 27}} 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,{{sfn|Spencer|1911|p=582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Hexagonal Division'', TRAPEZOHEDRAL CLASS}} but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.