2,342 research outputs found

    Faces of platonic solids in all dimensions

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    This paper considers Platonic solids/polytopes in the real Euclidean space R^n of dimension 3 <= n < infinity. The Platonic solids/polytopes are described together with their faces of dimensions 0 <= d <= n-1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types An, Bn, Cn, F4 and of non-crystallographic Coxeter groups H3, H4. Our method consists in recursively decorating the appropriate Coxeter-Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic.Comment: 11 pages, 1 figur

    Multipartite quantum correlations: symplectic and algebraic geometry approach

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    We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as identification of states with `the same correlations properties' i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question `what can be transformed into what \textit{via} available means?'. Exactly such an interpretations, i.e, in terms of mutual transformability can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa

    Polyhedral Gauss Sums, and polytopes with symmetry

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    We define certain natural finite sums of nn'th roots of unity, called GP(n)G_P(n), that are associated to each convex integer polytope PP, and which generalize the classical 11-dimensional Gauss sum G(n)G(n) defined over Z/nZ\mathbb Z/ {n \mathbb Z}, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group W\mathcal{W}, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let G\mathcal G be the group generated by W\mathcal{W} as well as all integer translations in Zd\mathbb Z^d. We prove that if PP multi-tiles Rd\mathbb R^d under the action of G\mathcal G, then we have the closed form GP(n)=vol(P)G(n)dG_P(n) = \text{vol}(P) G(n)^d. Conversely, we also prove that if PP is a lattice tetrahedron in R3\mathbb R^3, of volume 1/61/6, such that GP(n)=vol(P)G(n)dG_P(n) = \text{vol}(P) G(n)^d, for n∈{1,2,3,4}n \in \{ 1,2,3,4 \}, then there is an element gg in G\mathcal G such that g(P)g(P) is the fundamental tetrahedron with vertices (0,0,0)(0,0,0), (1,0,0)(1, 0, 0), (1,1,0)(1,1,0), (1,1,1)(1,1,1).Comment: 18 pages, 2 figure
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