On this subject, there's a lovely symmetrical picture of this: Create a rosette of 7 congruent regular hexagons: * a * f b * e * * c * d * * i * * * * * i * * h * * * * * * h * * g * * * * * g * * f * a * * e * b d * c * Now identify the 18 exposed edges, by translations, in pairs. The result is a highly symmetrical torus, that is clearly tiled by 7 congruent smaller tori, each of which touches the other 6, showing that ch(T^2) >= 7. Hmm: QUESTION: What is the chromatic number of the unit distance graph of this torus? (Assuming that the distance between the centers of any two hexagons is exactly 1.) --Dan On 2013-04-03, at 6:44 PM, Cris Moore wrote:
we can draw a K_7 on the torus. In fact, I remember a Martin Gardner column with a (non-convex) polyhedron, with the topology of a torus, where each side shared an edge with the other 6!