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

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 R

Time evolution with a centered initial data:

On some faces: •t=0 to t=1 On a cutting plane x=0: •t=0 to t=1 |

Time evolution with a non centered initial data:

On some faces: • t=0 to t=0.5 •t=0.5 to t=1 On other faces: •t=0 to t=0.5 • t=0.5 to t=1 On a cutting plane z=0: • t=0 to t=0.5 •t=0.5 to t=1 On a cutting plane x=0: •t=0 to 0.5 • t=0.5 to t=1. |

Time evolution with another initial data:

On some faces: • t=0 to t=5 •t=5 to t=10 On a cutting plane: • t=0 to t=5 •t=5 to t=10 On two cutting planes: •t=0 to 0.5 |