Koebe Realizations

Given a $3$-dimensional polytope $P$, this code computes a numerical approximation of a realization of $P$ with edges tangent to the unit sphere and with barycenter centered at the origin. The algorithm is described in Variational principles for circle patterns and Koebe's theorem.

This implementation is motivated by the paper Algebraic Degrees of 3-Dimensional Polytopes.

Installation

using Pkg
Pkg.add(url="https://github.com/marabelotti/KoebeRealizations.jl.git")

Constructing the realization

There are two ways of giving the polytope.

Via vertex-facet incidence matrix

The first one is as the vertex-facet incidence matrix

using KoebeRealizations

cube = Bool[
    1 1 1 1 0 0 0 0
    1 1 0 0 1 1 0 0
    0 1 1 0 0 1 1 0
    0 0 1 1 0 0 1 1
    1 0 0 1 1 0 0 1
    0 0 0 0 1 1 1 1
]
koebe_realization(cube)

And this returns the matrix whose rows are the vertices of the realization.

8×3 Matrix{Float64}:
  0.316228  -0.707107   0.948683
 -0.948684  -0.707106   0.316228
 -0.316228  -0.707107  -0.948683
  0.948683  -0.707107  -0.316228
  0.316228   0.707107   0.948683
 -0.948683   0.707107   0.316228
 -0.316228   0.707107  -0.948683
  0.948683   0.707107  -0.316228

Via polytope

The input can also be a polytope from Polymake.jl. In this case the function also returns a polymake polytope.

using KoebeRealizations, Polymake

john73 = Polymake.polytope.johnson_solid(73)
K = koebe_realization(john73)
visual(K, FacetColor="0.54 0.17 0.89", VertexColor= "0.54 0.17 0.89", FacetTransparency=0.5)

Visualizing the circle packing

We can naturally associate to the realization a circle packing on the sphere by considering, for every vertex, the circle passing through the tangency points of the edges containing that vertex.

This can be visualized, together with its dual, as follows:

using KoebeRealizations

john73 = Polymake.polytope.johnson_solid(73)
plot_circle_packing(john73)

Authors

Mara Belotti and Sascha Timme