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Symmetries and Transformations of n-Cubes and the Nimber-Simplex Graph

Abstract

The objective: This project is a continuation of last year's project which mapped finite groups under Nim addition to n-dimensional Simplexes, creating the Nimber-Simplex graph. The goals of this year's project are first, to apply finite group theory to regular convex polytopes, second, to determine the symmetry group of an n-cube using the Nimber-Simplex graph, and third, to illustrate a reversible transformation between the Nimber-Simplex graph and an n-cube.

Methods/Materials

In this project, two isomorphisms are defined: one between the Nimsum group 2^n-1 and the Cartesian product C(2)^n, the other between the symmetry group of a Simplex-(n-1) and the permutation group S(n). The symmetry group of an n-dimensional hypercube is determined by mapping the group C(2)^n to the vertices of the n-cube as well as to diagonal matrices representing reflection operations. Permutations of coordinate axes, P(n), are shown to be isomorphic to S(n). The group of symmetries of the n-cube, G(n), is then a semidirect product of the normal subgroup N(n), representing the reflection symmetries, and the subgroup P(n). That is, G(n)=N(n)xP(n). A reversible transformation between the Nimber-Simplex graph in (n-1) dimensions and an n-dimensional hypercube is demonstrated. Materials used in this project include Zometool, an hp Deskjet printer, and a Dell PC running Microsoft Windows 98 and Word 97.

Results

The original ideas developed in this project include the definition of the Nimber-Simplex graph, the isomorphism between the Nimsum group 2^n-1 and C(2)^n, and mapping C(2)^n to vertices of an n-cube as well as the reflection matrices of the n-cube. These two results were used to prove that the Nimber-Simplex graph in (n-1) dimensions determines the symmetry group of an n-cube and that the Nimber-Simplex graph unfolds into an n-cube. Finally, it was discovered that this year's project connects all regular convex polytopes in n>4 dimensions!

Conclusions/Discussion

The first hypothesis that the symmetry group of an n-cube is a semidirect product of the symmetry group of a Simplex-(n-1) and the Nimsum group 2^n-1 was proven. The second hypothesis that there is a reversible transformation between the Nimber-Simplex graph in (n-1) dimensions and an n-cube was also proven. Since n-cubes and n-dimensional cross-polytopes are dual, the Nimber-Simplex graph relates all regular convex polytopes in n>4 dimensions.

This Mathematical project determines the symmetry group of an n-cube using the Nimber-Simplex graph, demonstrates a reversible transformation between the Nimber-Simplex Graph and an n-cube, and relates all regular convex polytopes in >4 dimensions.

Science Fair Project done By Rebecca E. Jacobs