---
tag: YH170WBQ6M
kind: concept
name: Triangulation
msc: [57Q15, 55U10]
sources:
  - url: https://en.wikipedia.org/wiki/Triangulation_(topology)
same_as:
  wikidata: Q7840154
  wikipedia: Triangulation (topology)
---

## Summary

A triangulation of a topological space is a homeomorphism from the geometric realization of a simplicial complex onto the space.
It cuts the space into vertices, edges, triangles and their higher-dimensional analogues, glued along faces.
Triangulations turn topological questions into combinatorial ones, which is how simplicial homology and the Euler characteristic were first computed.
Whether every manifold can be triangulated, and whether triangulations are essentially unique, are classical questions of geometric topology.

## Explanations

### L1 human

A geodesic dome or a soccer ball made of panels is a sphere cut into flat pieces.
A triangulation does the same with triangles, and in higher dimensions with tetrahedra and their analogues.
The surface of a tetrahedron is a triangulation of the sphere with $4$ vertices, $6$ edges and $4$ triangles.
The count $4 - 6 + 4 = 2$ is the Euler characteristic of the sphere, and every triangulation of the sphere gives the same number.