TOPICS
Search

Three-Torus

The three-torus is the topological space T^3=S^1×S^1×S^1, the Cartesian product of three circles. It is a compact three-dimensional manifold without boundary. Equivalently, it is the quotient R^3/Z^3, obtained by identifying opposite faces of a cube (Kühnel and Lassmann 1984). It differs from the two-dimensional torus S^1×S^1. The triple torus instead denotes a two-dimensional surface of genus 3.

A simplicial triangulation here is a finite simplicial complex whose underlying topological space is homeomorphic to T^3. The vertices counted are those of that complex, rather than corners identified in a quotient description. Kühnel and Lassmann (1984) constructed a simplicial triangulation of T^3 with 15 vertices. Lim (2026) proved that no such triangulation has 13 vertices. Together with the existing lower bound, this leaves the minimum number of vertices equal to either 14 or 15. The 14-vertex case remains open.

See also

Manifold, Torus, Triangulation

Explore with Wolfram|Alpha

References

Kühnel, W. and Lassmann, G. "The Rhombidodecahedral Tessellation of 3-Space and a Particular 15-Vertex Triangulation of the 3-Dimensional Torus." Manuscripta Math. 49, 61-77, 1984. https://doi.org/10.1007/BF01174871.Lim, G. "No 13-Vertex Triangulation of the Three-Torus." 10 Nov 2026. https://arxiv.org/abs/2611.14367v1.

Cite this as:

Weisstein, Eric W. "Three-Torus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Three-Torus.html

Subject classifications