Projective spaces provide an example (counter to the usual trend) where the jump from dimension 2 to 3 actually brings greater simplicity. In any projective space of dimension 3 the Desargues theorem holds, which implies that space can be coordinatized by a skew field.

In dimension 2 (projective planes) the Desargues theorem need not hold. As a result, projective planes cannot be founded on any familiar algebraic structure and they are very hard to classify.