infinite regular polyhedra

The infinite regular polyhedra  are periodic structures which have all the properties of the platonic solids but being finite. Their faces are thus identical regular polygons and their vertices are symmetrical and identical (the same number of faces meet at each vertex).
Such a structure may be built using an elementary module whose number of tunnels defines the genius of the polyhedron.

Here are three examples discovered in 1926 by John Flinders Petrie and H.S.M. Coxeter.

the most simplest is not rigid
lattice of cubes with two opposite faces removed
6 squares at each vertex - module : 12 squares - genus 3

lattice of truncated octahedra with their square faces removed
4 hexagons at each vertex - module : 8 hexagons - genus 3

lattice of truncated tetrahedra with their triangular faces removed
6 hexagons at each vertex - module : 4 hexagons - genus 3

In the sixties J.R. Gott and A.F. Wells published lists of pseudo-polyhedra  whose vertices are only identical (without being symmetrical). Here is my favorite:

lattice of icosahedra linked with octahedral tubes
9 triangles at each vertex - module : one icosahedron and 4 octahedra - genus 4


references: Infinite regular polyhedra: Vladimir Bulatov  -  superliminal.com

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French 		convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects November 2013
updated 03-12-2013