Regular lattice polytopes and root systems

Regular Polytopes, Root Lattices, and Quasicrystals*

The icosahedral quasicrystals of five-fold symmetry in two, three, and four dimensions are related to the corresponding regular polytopes exhibiting five-fold symmetry, namely the regular pentagon (H2 reflection group), the regular icosahedron 3,5 (H3 reflection group), and the regular four-dimensional polytope 3,3,5 (H4 reflection group). These quasicrystals exhibiting five-fold symmetry can b...

2 00 6 Classification of Lattice - Regular Lattice Convex Polytopes

In this paper for any dimension n we give a complete list of lattice convex polytopes in R that are regular with respect to the group of affine transformations preserving the lattice.

Regular polytopes

Lattice polytopes cut out by root systems and the Koszul property

We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.

polymake and Lattice Polytopes

The polymake software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes to the polymake core, which will be discussed briefly.