Pascal’s Triangle of Configurations

Abstract

We introduce an infinite class of configurations which we call Desargues–Cayley–Danzer configurations. The term is motivated by the fact that they generalize the classical (10 3) Desargues configuration and Danzer’s (35 4) configuration; moreover, their construction goes back to Cayley. We show that these configurations can be arranged in a triangular array which resembles the classical Pascal triangle also in the sense that it can be recursively generated. As an interesting consequence, we show that all these configurations are connected to incidence theorems, like in the classical case of Desargues. We also show that these configurations can be represented not only by points and lines, but points and circles, too.