My Math Forum  

Go Back   My Math Forum > College Math Forum > Applied Math

Applied Math Applied Math Forum

LinkBack Thread Tools Display Modes
January 30th, 2012, 09:42 AM   #1
Joined: Jan 2012

Posts: 1
Thanks: 0

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!
Maurice is offline  

  My Math Forum > College Math Forum > Applied Math

diagonals, polyhedra, question

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
diagonals caters Algebra 5 February 28th, 2014 05:58 PM
More Finite Diagonals! johnr Number Theory 2 April 29th, 2013 05:00 AM
How many differently polyhedra can be obtained? mathLover Algebra 4 April 17th, 2012 02:02 AM
Polyhedra problem Crouch Algebra 1 June 19th, 2011 11:33 AM
Polyhedra Algebra 7 November 21st, 2010 04:59 AM

Copyright © 2019 My Math Forum. All rights reserved.