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At sixes and sevens, and eights, and nines: schwarzites p 3, p = 7, 8, 9

“…The kgn structure appears to have first been identified by Fowler et al [20,21] as derived from the famous regular 7 3 polyhedron of Felix Klein. This is a tiling of a finite genus 3 surface by 24 regular heptagons.…”

The Polyhedral State of Molecular Matter

“…In this case the Euler theorem must be extended to include the number of cells, c: Equation (13). (13) As an example in the case of a body-centred tetrahedron one has v ϭ 5, e ϭ 10, f ϭ 10, c ϭ 4, resulting in the Euler characteristic χ ϭ 1, as indicated in Table 3 for polyhedra with the topology of a solid sphere.…”

Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group

“…A geometrical model of the heptakisoctahedral group is the so-called Klein tessellation containing 56 vertices (all of degree 3), 84 edges, and 24 equivalent heptagonal faces embedded in a surface of genus 3 (figure 2) [4,9,10]. The automorphism group of the Klein tessellation is 7 O.…”

“…The key finite subgroup of SO (7) for the study of the atomic f shell would seem to be the heptakisoctahedral [4] or didodecahedral [5] group, 7 O, of order 168. This group can be constructed from a finite field of order 7 in a manner exactly analogous to the construction of the icosahedral group from a finite field of order 5 [6].…”