PDS

Résolution numérique de l'équation des ondes sur l'espace dodécaédrique de Poincaré.

(Wave Computation on the Poincaré dodecahedral space. Class. Quantum Grav. 30 (2013) 235010.)

The Poincaré dodecahedral space (PDS) is a plausible multi-connected universe model with constant positive spatial curvature.
This 3-manifold without boundary is the quotient of the unit 3-sphere S3 under the action of the binary icosahedral group I∗ of isometries of S3, with I∗ acting by left multiplication.
PDS is also represent by a fundamental domain F ⊂ S3 and an equivalence relation ∼ such that S3/I∗ = F/ ∼ .
F is a regular spherical dodecahedron (dual of a regular icosahedron), and ∼ is obtained by identifying the opposite pentagonal faces of F after rotating by π/5 in the clockwise direction around the axis orthogonal to the face. 120 such spherical dodecahedra tile the 3-sphere in the pattern of a regular 120-cell.

This is the diagramm of vertices of F, and a diagramm of the vertices of the 120-cell. Staight lines are drawn to better see the pentagons.

In fact PDS is endowed with the spherical metric induced by this one of S3, and this picture is a visualisation of a part of S3 tilled by the 120 dodecahedra.
That is you can see a projection in R³ of a fundamental domain (dark blue) and six of its image by elements of the binary icosahedral group.

Then, we compute the solutions of the wave equation in the time domain, by using a variational method and a discretization with finite elements. There are few movies of solutions:

Time evolution with a centered initial data:

On some faces:

On a cutting plane x=0:

Time evolution with a non centered initial data:

On some faces:

On other faces:

On a cutting plane z=0:

On a cutting plane x=0:.

Time evolution with another initial data:

On some faces:

On a cutting plane:

On two cutting planes:

The support of the two first initial data is far from $\partial \mathcal{F}_v$, therefore they respect obviously the constraint of the equivalent points. It is interesting to consider the last initial data involving several equivalent points.