a self-dual configuration

The principle of duality turns any projective figure into a dual figure by swapping the roles of points and lines. Usually the dual looks different from the original. But some figures are so symmetric that, after the swap, you get back a figure of exactly the same shape — the same count of points and lines, joined in the same pattern. Such a figure is called self-dual, and these are among the most elegant objects in projective geometry.

To make 'same shape' precise, geometers record a configuration by its type, written (p_a, l_b): it has p points and l lines, with each point lying on a of the lines and each line passing through b of the points. The dual swaps points with lines, so the dual configuration has type (l_b, p_a). A configuration is self-dual when this dual type matches the original — which requires at least p = l and a = b, plus a genuine matching of the incidence pattern, not merely equal counts. The triangle is the simplest example: 3 points and 3 lines, type (3_2, 3_2), and dualising sends it to a figure of the same type — vertices become sides and sides become vertices, the triangle maps to itself. The Desargues configuration is a richer case: 10 points and 10 lines, each point on 3 lines, each line through 3 points, type (10_3, 10_3), and it is self-dual precisely because Desargues' theorem and its converse are dual statements.

Self-dual configurations matter because they reveal where duality is not just a tool but a built-in symmetry of the figure itself, and they are exactly the settings where a theorem coincides with its own dual (so you prove two results in one stroke). One honest caution: matching the type numbers (p_a, l_b) is necessary but not sufficient for self-duality — you also need an actual duality (a correlation, often a polarity from a conic) that carries the configuration's points to its lines respecting all incidences. Equal counts can be a coincidence; a true self-dual configuration comes with the map that realises the symmetry.