A nice example of seemingly trivial structure that hides highly nontrivial structure is that of a projective space. Such a space consists of "points', "lines", and "planes" with the obvious properties: there is a unique line through any two points, any two planes meet in a unique line, three points not on a line lie on a unique plane, and so on.

Surprisingly, any such space has an underlying skew field which coordinatizes the space so that lines and planes have linear equations. This due to the fact that the Desargues theorem holds in any projective space. Hilbert (1899) showed (in a highly roundabout way) that one can then define sum and product of points, and use the Desargues theorem to prove their skew field properties.