Color symmetry and colored polyhedra

Symmetry of Polytopes and Polyhedra

regular polytopes are combinatorial structures that generalize the familiar regular polytopes. The terminology adopted is patterned after the classical theory. Many symmetric figures discussed in earlier sections could be treated (and their structure clarified) in this more general framework. Much of the research in this area is quite recent. For a comprehensive account see McMullen and Schulte...

Polyhedra, complexes, nets and symmetry.

Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zigzag or helical. A polyhedron or complex is regular if its geometr...

Determining the Symmetry of Polyhedra

The Roundest Polyhedra with Symmetry Constraints

Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been a...

Effect of rotational symmetry on colored lattices.

A colored lattice L(c) has a geometrical lattice L. A subgroup lattice L' of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L' in L is given by Delta, the determinant of the matrix (t(jk)) that converts L into L'. This is the order of the factor group {L/L'}, and is also the number n of colors present. The crystal systems-i.e., the com...