The minimal triangulation of the torus, a remarkable Belyi pair and octonions

The heroes of the talk have been known since the XIX century – the complete graph K_7 and the dual Heawood graph (the incidence graph of the Fano plane), embedded into the torus. The brief historical overview will be presented.
The talk will be based on the recent paper by Bruno Sevennec, we follow him in the visualization of octonion multiplication. The corresponding Belyi pair is beautiful and clarifies the arithmetic of the modular curve X_1(7); besides it is a convenient model of the speaker and Voevodsy's theory of combinatorial jacobians.
The embeddings of other complete graphs will also be mentioned.