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Register a new userFaces of platonic solids in all dimensions
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Authors
Marzena Szajewska
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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.
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A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surfaces of Euler characteristics $-1$ $&$ $-2$ which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and antiprisms are vertex-transitive maps on $mathbb{S}^2$. Here we show that there is exactly one semi-equivelar map on $mathbb{S}^2$ which is not vertex-transitive. More precisely, we show that a semi-equivelar map on $mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedean solid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As a consequence, we show that all the semi-equivelar maps on $mathbb{RP}^2$ are vertex-transitive. Moreover, every semi-equivelar map on $mathbb{S}^2$ can be geometrized, i.e., every semi-equivelar map on $mathbb{S}^2$ is isomorphic to a semi-regular tiling of $mathbb{S}^2$. In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on $mathbb{S}^2$. Here, we present self-contained combinatorial proofs of all our results.
The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are considered. CSLs in 3 dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest to also look at CSLs in dimensions $d>3$. Here, we discuss the CSLs of the root lattice $A_4$ and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoint. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be linked to certain ideals and compute their indices, their multiplicities and encapsulate all this in generating functions in terms of Dirichlet series. In addition, we sketch how these results can be generalised for 4--dimensional $Z$--modules by discussing the icosian ring.
This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Capacity theoretic methods are used to produce dimension bounds for images of sets under Holder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$alpha$ fractional Brownian motion in terms of capacity theoretic dimension profiles. As a corollary, this establishes continuity of the profiles for all Borel sets, allowing us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Holder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets in the setting of projections.
In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p in (1, 1.58]$. We conjecture that the family also exists for $p in (1.58, log_2 3 = 1.5849625ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.
We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconers seminal results on homogeneous self-affine sets.
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