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The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces. The triangular-prism-first orthographic projection of the octahedral prism into 3D space has a hexagonal prismic envelope. The two octahedral cells project onto the two hexagonal faces.
The infinite series of axial or prismatic groups have an index n, which can be any integer; in each series, the nth symmetry group contains n-fold rotational symmetry about an axis, i.e., symmetry with respect to a rotation by an angle 360°/n. n=1 covers the cases of no rotational symmetry at all.
The prismatic groups are denoted by D nh. These groups are characterized by i) an n-fold proper rotation axis C n; ii) n 2-fold proper rotation axes C 2 normal to C n; iii) a mirror plane σ h normal to C n and containing the C 2 s. The D 1h group is the same as the C 2v group in the pyramidal groups section.
The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group B 2 or [4,3], as well as a Coxeter diagram: . There are 48 triangles, visible in the faces of the disdyakis dodecahedron, and in the alternately colored triangles on a sphere: #
A prismatic polytope is a Cartesian product of two polytopes of lower dimension; familiar examples are the 3-dimensional prisms, which are products of a polygon and a line segment. The prismatic uniform 4-polytopes consist of two infinite families: Polyhedral prisms: products of a line segment and a uniform polyhedron.
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name.
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, [1] orbifold notation, [2] and order.
A polychoric group is one of five symmetry groups of the 4-dimensional regular polytopes. There are also three polyhedral prismatic groups, and an infinite set of duoprismatic groups. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes. The dihedral angles between the mirrors determine order of dihedral symmetry.