Shape Computation Lab

Cube Orbits

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01. The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube (1-12).

02. The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube (13-24).

03. The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube (25-36).

04. The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube (37-48).

05. The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube (1-12).

06. The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube (13-24).

07. The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube (25-36).

08. The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube (37-48).

09. The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube (1-12).

10. The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube (13-24).

11. The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube (25-36).

12. The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube (37-48).

Jerry Hsu and Athanassios Economou

2005

President’s Undergraduate Research Award (PURA) Award, Georgia Tech

Keywords: Cube; Hexahedral group; Permutation groups; Cycle index; Symmetry; Configuration;

The cube has always been one of the most ubiquitous shapes in three-dimensional spatial design. Aspects of the mathematical structure of the cube have been explored with different tools drawn from Euclidean geometry, descriptive geometry, group theory, permutations and combinatorics. Here one of the most interesting visual representations of the structure of the cube is given in terms of the cycle index of the permutation groups of the vertices, edges and faces of the shape under the symmetry group of the cube. All permutations are given in a consistent projection depicting the symmetry transformation that induces them as the geometrical loci of points that remain invariant under the transformation (typically axes of rotation and rotor reflection, and planes of reflections) and the cycles of permutations of the vertices, edges and faces as closed rings circumscribing them.