In search for a perfect shape of polyhedra: Buffon transformation

For an arbitrary polygon consider a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buffon (1707-1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine $B$-regular polyhedra. The main result is the proof of existence of star-shaped affine $B$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro

Complexity as a Sclae-Space for the Medial Axis Transform

The medial axis skeleton is a thin line graph that preserves the topology of a region. The skeleton has often been cited as a useful representation for shape description, region interpretation, and object recognition. Unfortunately, the computation of the skeleton is extremely sensitive to variations in the bounding contour. In this paper, we describe a robust method for computing the medial axis skeleton across a variety of scales. The resulting scale-space is parametric with the complexity of the skeleton, where the complexity is defined as the number of branches in the skeleton

Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases

Using a recent strategy to encode the space of flat connections on a three-manifold with string-like defects into the space of flat connections on a so-called 2d Heegaard surface, we propose a novel way to define gauge invariant bases for (3+1)d lattice gauge theories and gauge models of topological phases. In particular, this method reconstructs the spin network basis and yields a novel dual spin network basis. While the spin network basis allows to interpret states in terms of electric excitations, on top of a vacuum sharply peaked on a vanishing electric field, the dual spin network basis describes magnetic (or curvature) excitations, on top of a vacuum sharply peaked on a vanishing magnetic field (or flat connection). This technique is also applicable for manifolds with boundaries. We distinguish in particular a dual pair of boundary conditions, namely of electric type and of magnetic type. This can be used to consider a generalization of Ocneanu's tube algebra in order to reveal the algebraic structure of the excitations associated with certain 3d manifolds.Comment: 45 page