2,342 research outputs found
Faces of platonic solids in all dimensions
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
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
We define certain natural finite sums of 'th roots of unity, called
, that are associated to each convex integer polytope , and which
generalize the classical -dimensional Gauss sum defined over , to higher dimensional abelian groups and integer polytopes.
We consider the finite Weyl group , generated by the reflections
with respect to the coordinate hyperplanes, as well as all permutations of the
coordinates; further, we let be the group generated by
as well as all integer translations in . We prove
that if multi-tiles under the action of , then we
have the closed form . Conversely, we also prove
that if is a lattice tetrahedron in , of volume , such
that , for , then there is
an element in such that is the fundamental tetrahedron
with vertices , , , .Comment: 18 pages, 2 figure
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