User:Tomruen/complex polytopes

Regular and uniform complex polytopes

Complex Polygons (C2)
The complex reflection group is p[q]r, order $$g = 8/q \cdot (1/p+2/q+1/r-1)^{-2}$$ has, configuration matrix: $$\begin{bmatrix}\begin{matrix}g/r & r \\ p & g/p\end{matrix}\end{bmatrix}$$

= (Order 2p2 and p2) - Related to p-p duoprisms

(order pq) - related to p-q duoprism

= (order 18 and 9) - related to 3-3 duoprism

(order 6) - related to triangular prism

(Order 18) - related 3-3 duopyramid

(Order 18)

Möbius–Kantor polygon =, (order 24)

= (order 48 and 24)

Complex polyhedra (C3)
There are 9 unique regular and uniform complex polyhedra from 14 Wythoff constructions (ringed patterns) in the L3 and M3 Shephard groups. These polyhedra can be seen a complex analogues of tetrahedral symmetry and octahedral symmetry of the regular tetrahedron, cube, and octahedron.

Hessian polyhedron
= - analogous to real tetrahedron

Rectified Hessian polyhedron
= - analogous to real octahedron

Truncated Hessian polyhedron
=  -  analogous to real truncated tetrahedron

Cantellated Hessian polyhedron
= - analogous to real cuboctahedron

Cantitruncated Hessian polyhedron
= - analogous to real truncated octahedron

Double Hessian polyhedron
Double Hessian polyhedron - analogous to real cube

Truncated double Hessian polyhedron
- analogous to real truncated cube

Cantellated double Hessian polyhedron
- analogous to real rhombicuboctahedron

Cantitruncated double Hessian polyhedron
- analogous to real truncated cuboctahedron

Witting polytope (C4)
Witting polytope - - Real representation 421 polytope

- Honeycomb of Witting polytope: L5 is order 155520N - Real representation 521 honeycomb