Playing with the Pappus configuration

english

2026

volume 26, issue 2, article p2.01

collinearity   concurrence   configuration   incidence   Pappus layout   Pappus line

Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)

Faced with the enormity of Hexagrammum Mysticum, trying to understand it and wishing to discover in it something genuinely new, I decided to first try my hand at something smaller and easier to handle, to get some practice. To this end I have chosen the Pappus layout of two lines with three points on each, a degenerate variation of the Pascal’s sextet of coconic points which is the construction base of Hexagramum Mysticum. I permuted the points on one of the lines, and generated the six Pappus three-point lines corresponding to the permuted hexagons. Six three-point lines was too little material even for a Hexagrammum Brevis, so I adopted into the configuration the nine lines connecting pairs of points on the opposite lines of a Pappus layout, and for good measure added the two points of concurrence of two tercets of Pappus lines. The resulting Frankenstein Creature configuration, sewed together from three so different parts, surprisingly turned out to be the highly symmetric Adler configuration, of type (20, 15). And this was only the warm-up. After some slightly weird configuration mining I dug up thirty-three configurations of type (21), the inward Pappus configurations (as I named them), that are partitioned into four classes of isomorphic combinatorial configurations. Besides that I found also a cross-eyed cousin to the Desargues configuration, of type (10), and then, using a special geometric realization of this configuration, composed a configuration of type (28, 21).

27.10.2026