All polytopes are coset geometries: characterizing automorphism groups of k-orbit abstract polytopes

Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from their automorphism groups. This is also known to be true for 2- and 3- orbit 3-polytopes. In this paper we show that every abstract $n$-polytope can be constructed as a coset geometry. This construction is done by giving a characterization, in terms of generators, relations and intersection conditions, of the automorphism group of a $k$-orbit polytope with given symmetry type graph. Furthermore, we use these results to show that for all $k\neq 2$, there exist $k$-orbit $n$-polytopes with Boolean automorphism groups, for all $n\geq 3$.Comment: 41 pages, 6 figure

Composite planar coverings of graphs

AbstractWe shall prove that a connected graph G is projective-planar if and only if it has a 2n-fold planar connected covering obtained as a composition of an n-fold covering and a double covering for some n⩾1 and show that every planar regular covering of a nonplanar graph is such a composite covering

AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automorphisms simply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications

SMARANDACHE MULTI-SPACE THEORY, Second Edition

We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multi-space came into being by purely logic. Another is the mathematical combinatorics motivated by a combinatorial speculation, i.e., a mathematical science can be reconstructed from or made by combinatorialization. Both of them contribute sciences for consistency of research with that human progress in 21st century