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January 30th, 2012, 09:42 AM   #1
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Question about polyhedra without diagonals

I am not a mathematician but a physicist (from Holland), so excuse my formulation. I have a question about the relation between complete graphs and Euclidean metric.

This I learnt from literature: Two polyhedra are known without diagonals in Euclidean space R^3: the tetrahedron and the Császár polyhedron. The graph of the tetrahedron is a complete planar graph K4 and the Császár polyhedron is a complete graph K7. A manipulation of the Euler characteristic is: g={(v-3)(v-4)}/12 with v the number of vertices, is the genus of the surface onto which the complete graph must be embedded. The tetrahedron v=4 gives g=0 and the Császár polyhedron v=7 gives g=1. For the next number of vertices in which the the genus is an integer is v=12;it is known that this polyhedron (v=12) has no realisation in R^3.

Q1: as K4 is the 'last' (K5 is not planar) complete graph in R^2, is K7 the 'last' complete graph in R^3?
Q2: does this implicate something about the structure of Euclidean space? Or is Euclidean metric defined by those graphs?
Q: where can I find more literature about Q1 and Q2. It is quite a search when I cannot define the the right terms.

Thank you in advance!
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