# how will the minimal triangulation of a torus look like?

how how will the minimal triangulation of a torus look like?

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Let $\omega = e^{2 \pi i/3}$. Let $L$ be the planar lattice generated by $1$ and $\omega$. Let $M$ be the sublattice generated by $2-\omega$ and $\omega (2 - \omega)$. Tile $\mathbb{R}^2$ by equilateral triangles with vertices at $L$, and quotient by $M$. This is a triangulation of the torus with $7$ vertices (exercise!). It is minimal because any triangulation of the torus must have a vertex of degree $6$, so you can't get use fewer vertices.
As a bonus this shows the embedding of $K_7$ in the torus, and the dual graph is the strongly regular Heawood graph (= incidence graph of the Fano plane) and tiles the torus with $7$ pairwise adjacent hexagons. –  Noam D. Elkies Apr 9 '12 at 20:23
Moreover, this $7$-vertex triangulation can be realized as an embedded simplicial complex in $3$-space, so you can actually see it. (I made one out of construction paper during an idle undergraduate afternoon.) For more information about this, see, for example, eg-models.de/models/Classical_Models/2001.02.069/_preview.html . –  Robert Bryant Apr 9 '12 at 23:38