Pellicer, Daniel; Matematični kolokvij april 2010

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Combinatorial structure of chiral polyhedra in the Euclidean space

Daniel Pellicer

Instituto de Matemáticas Unidad Morelia, Mehika

29. april 2010

In the XX century the concept of polyhedron was modified to allow non-convex structures. As before, regular polyhedra are those allowing maximal symmetry by reflections. On the other hand, chiral polyhedra are those admitting maximal symmetry by rotations, but no symmetry by reflections. With these definitions there are 48 discrete regular polyhedra in the Euclidean space.

Symmetry can also be understood in the combinatorial way without involving any geometry. An automorphism of a polyhedron as an abstract structure is therefore a bijection of the vertices, edges and faces that preserves the incidence.

In 2005 Schulte classified all geometrically chiral polyhedra in the Euclidean space in six families and provided a detailed geometric description of them. It remained to determine whether these polyhedra are regular or chiral as combinatorial structures.

We shall determine which of the geometrically chiral polyhedra are also abstractly chiral. For those who are regular we shall show a geometric explanation of their chiral realization.

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