---
tag: FBVEG8HVTP
kind: example
name: Seifert–Weber space
msc: [57K32]
instance_of: [GHVW1Z35BS]
sources:
  - url: https://en.wikipedia.org/wiki/Seifert%E2%80%93Weber_space
same_as:
  wikidata: Q7446658
  wikipedia: Seifert–Weber space
---

## Example

Take a solid regular dodecahedron.
Glue each face to the opposite face after a turn of $3/10$ of a full turn, that is, by the angle $3\pi/5$.
The result is the Seifert–Weber space $W$, a closed orientable [[3-manifold|R20TB2KEQQ]].

Under these gluings, the $30$ edges of the dodecahedron fall into $6$ classes of $5$ edges, and all $20$ vertices become one point.
The cell structure has $1$ vertex, $6$ edges, $6$ faces and $1$ cell.
So its Euler characteristic is $1 - 6 + 6 - 1 = 0$, as it must be for a closed $3$-manifold.
In hyperbolic space $\mathbb{H}^3$ there is a regular dodecahedron with dihedral angles $2\pi/5$.
With this dodecahedron the $5$ angles around each edge class add up to $2\pi$, and the gluing maps are isometries.
So $W$ is a closed [[hyperbolic 3-manifold|GHVW1Z35BS]].

## Remarks

Its first homology group is $(\Z/5)^3$.
Gluing opposite faces with a turn of $1/10$ instead gives the [[Poincaré homology sphere|NHERCK7C43]], and a turn of $1/2$ gives the [[real projective space|1DDYB6VVSR]] $\mathbb{RP}^3$.
Seifert and Weber described the space in 1933; it was one of the first known closed hyperbolic $3$-manifolds.

## Explanations

### L1 human

The Seifert–Weber space is made from a dodecahedron, the solid with twelve pentagonal faces, by gluing opposite faces with a twist.
In ordinary space the corners do not fit together, but in hyperbolic space a large enough dodecahedron has smaller angles, and one size makes them exactly right.
So a hyperbolic creature living inside would see a perfect tiling of its space by dodecahedra.