Danzer's configuration

In mathematics, Danzer's configuration is a self-dual configuration of 35 lines and 35 points, having 4 points on each line and 4 lines through each point. It is named after the German geometer Ludwig Danzer and was popularised by Branko Grünbaum.^[1] The Levi graph of the configuration is the Kronecker cover of the odd graph O₄,^[2] and is isomorphic to the middle layer graph of the seven-dimensional hypercube graph Q₇. The middle layer graph of an odd-dimensional hypercube graph Q_2n+1(n,n+1) is a subgraph whose vertex set consists of all binary strings of length 2n + 1 that have exactly n or n + 1 entries equal to 1, with an edge between any two vertices for which the corresponding binary strings differ in exactly one bit. Every middle layer graph is Hamiltonian.^[3]

Danzer's configuration DCD(4) is the fourth term of an infinite series of $({\tbinom {2n-1}{n}}_{n})$ configurations DCD(n), where DCD(1) is the trivial configuration (1₁), DCD(2) is the trilateral (3₂) and DCD(3) is the Desargues configuration (10₃). In ^[4] configurations DCD(n) were further generalized to the unbalanced $({\tbinom {n}{d}}_{d},{\tbinom {n}{d-1}}_{n-d+1})$ configuration DCD(n,d) by introducing parameter d with connection DCD(n) = DCD(2n-1,n). DCD stands for Desargues-Cayley-Danzer. Each DCD(2n,d) configuration is a subconfiguration of the $(2_{2n+1}^{2n})$ Clifford configuration. While each DCD(n,d) admits a realisation as a geometric point-line configuration, the Clifford configuration can only be realised as a point-circle configuration and depicts the Clifford's circle theorems.