Problem 2 (Richard Kenyon). Let M be a closed polyhedral surface homeomorphic to S which is entirely composed of equal regular pentagons. If M is immersed in 3-space, is it necessarily the boundary of a union of solid dodecahedra that are glued together at common facets? The pentagonal faces may intersect each other (and the “union of solid dodecahedra” must be defined appropriately) but two di...

Compositional heterogeneity in shaped, bimetallic nanocrystals offers additional variables to manoeuvre the functionality of the nanocrystal. However, understanding how to manipulate anisotropic elemental distributions in a nanocrystal is a great challenge in reaching higher tiers of nanocatalyst design. Here, we present the evolutionary trajectory of phase segregation in Pt-Ni rhombic dodecahe...

We deene an analogue of the Gibbs{Curie energy for quasi{crystals. We show that there exists a Wull{type shape optimizing this energy, which is always a convex polytope. This way we can model the triacontahedron and the dodecahedron as shapes of quasi{crystals, according to reality. We also provide a shorter proof for the well{known formula for the number of points of a non{periodic set in a la...

A self-assembling robotic system capable of approximating arbitrary three dimensional shapes utilizing repeated rhombic dodecahedron shaped modules is presented. This shape allows very simple rotational motions about its edges. Using this shape and the motion constraints it imposes, a distributed algorithm using localized information is created and simulated. The algorithm causes each module to...

We analyse the anisotropy of the cosmic microwave background (CMB) for the Poincaré dodecahedron which is an example for a multi-connected spherical universe. We compare the temperature correlation function and the angular power spectrum for the Poincaré dodecahedral universe with the first-year WMAP data and find that this multi-connected universe can explain the surprisingly low CMB anisotrop...

We report the results of a very high statistics Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear σ model. We find a significant discrepancy between the continuum extrapolation of our data and the form factor prediction of Balog and Niedermaier, inspired by the Zamolodchikovs’ S-matrix ansatz. On the other hand our results for the O(3) and the dodecahedron model ar...