The Computing Logo

The manifold is remarkable on (at least) three accounts.

First of all, it is highly symmetrical - its isometry group is D6. The portion of packing shown illustrates the C6 symmetry; the C2 extension coming from interchanging the two cusps. The complete packing is a hexagonal tiling by copies of the logo.

Secondly, the manifold contains an embedded surface. This surface is a connected sum of three projective planes and the embedding is one sided. The limit set of the surface group separates the banyan root like cluster of horoballs from the moat like clust er of horoballs.

Thirdly, there are several other two cusped, seven tetrahedron hyperbolic manifolds which seem to be close cousins of this manifold. The horoball packings and volumes are close, but the symmetry and surface are broken.

A more detailed image

For aficionados of hyperbolic three-manifolds

In the first (2M), we see the horoball packing with the Ford skeleton of v3551. The viewpoint is from a neighborhood of one cusp, looking in through a horotorus across the thick part of the manifold. The parallelogram i s a fundamental domain for the torus subgroup of the fundamental group of v3551, coming from the cusp.

In the second (905K) we see the same view except now the horoballs, instead of being shaded to look like balls, are colored either light or dark according as wether they come from the first or second cusp.

The third (3.25M)is a combination of the previous two in vivid color.

The Ford skeleton is the dual of an ideal triangulation, and can be thought of as the result of letting equal volume horotori about the cusps expand, keeping equal volumes, until they meet, like a soap foam. It is a standard spine of v3551, and by deletin g the large 24 sided two-cell we obtain a one sided, nonorientable surface of Euler characteristic -1 embedded in v3551.

The cousins of v3551 are the census manifolds v3501, v3530, v3537, v3547, v3549, v3550, and v3551.