“Infinite Sponge Polyhedra’ are generated by tessellation (mapping) of continuous toroidal or hyperbolic (2-D manifold) surfaces. Among the multiply infinite number of such ‘maps – tessellations’, many, satisfying symmetry constraints, can develop into periodic and even uniform or regular Infinite Sponge Polyhedra, meaning: to have one type of vertex configurations, identical edges and regular, planar
polygonal faces.

Besides coxeter’s pioneering discovery of the regular skew polyhedra (1928), and later mention in A.F.Wells publications (1962), their conception and imagery was glimpsed in the Doctoral thesis-‘Spatial Arrangements and Polyhedra with Curved Surfaces…..’ (Burt, 1966), later to be developed and expanded considerably, to lay the ground for the development of the 1.P.L (Infinite Polyhedral
Lattice) Space Frames (1969) and subsequently leading to the publication of ‘Infinite Polyhedra’ (Wachman, Burt, Kleinman – 1974; 2005). which was attempted as an exhaustive search of all toroidal and hyperbolic regular-polygonal plane-faceted, univertex polyhedra. In regard to this publication it is worth to mention a remark by leading geometer-mathematician Branko Grunbaum (2008):”NO other source has anything approaching the richness of this collection”. They could be conveniently described (when concerned with the uniform-regular cases, as loose packing of finite polyhedra and as continuous polyhedral envelopes, subdividing space into two complementary sub spaces. As in the case of the periodic hyperbolic surfaces, the most revealing aspect of their phenomenology is in the relationships
prevalent between the envelopes, subdividing space into two complementary subspaces, their dual (complementary-reciprocal) tunnel networks and their symmetry groups. From its 1974 publication, the notion and the realm of the Infinite Polyhedra were researched (topologically-geometrically- symmetrically) and expanded considerably, and were pivotal in the development of the ‘Periodic Table of the Polyhedral Universe’ (Burt, 1996) and beyond. By relaxing the definition of the polyhedral feature, to include non- planar polygonal faces and curved edges, the extent of the phenomenon was blown to ‘galactic’ proportions, with the primary parameters of genus, Za & Val. Of the translation unit reaching to infinity.