Pascal's triangle of configurations
Let $n\geq 3$ be an integer and consider the $d$-dimensional real projective space $\mathbb{P}^{d}$ with $2 \leq d \leq n-1$. Choose $n$ hyperplanes in $\mathbb{P}^{d}$ in general position. Now, every $d$ hyperplanes of this arrangement meet in a point, and every $d-1$ hyperplanes meet in a line. Thus, we have altogether $\binom{n}{d}$ points and $\binom{n}{d-1}$ lines. The incidence between the points and lines is determined by the containment between the corresponding subsets, and it is clear that each point is incident with $d$ of the $\binom{n}{ d-1}$ lines and each line is incident with $n-d+1$ of the $\binom{n}{d}$ points. Taking a suitable projection, we obtain a planar point-line configuration of type \[ \left( \binom{n}{d}_{d} , \binom{n}{d-1} _{n-d+1} \right). \] We call the configuration obtained by the above construction a Desargues-Cayley-Danzer configuration. We denote such a configuration (i.e., its realization) by $\mathrm{DCD}(n,d)$. One can show, for instance, that for all $n > 4$ the configuration $\mathrm{DCD}(n,d)$ is movable, which means that its geometric realizations do not form a single class under projective transformations. Let us recall that by the incidence sum of configurations $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ one means the configuration $\mathcal{C}$ which is the disjoint union of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ together with a specified set $I \subseteq \mathcal{P}_{1} \times \mathcal{L}_{2} \cup \mathcal{P}_{2} \times \mathcal{L}_{1}$ of incident point-line pairs, where $\mathcal{P}_{i}$ denotes the point set and $\mathcal{L}_{i}$ denotes the set of lines for each $\mathcal{C}_{i}$. We denote it by $\mathcal{C}_{1} \oplus_{I} \mathcal{C}_{2}$. Theorem A. For all integers $n\geq 3$ and $d \in (2,n-1)$, the configuration $\mathrm{DCD}(n,d)$ is the incident sum of the form $\mathcal{C}_{1} \oplus_{I} \mathcal{C}_{2}$ such that \begin{itemize} \item[(i)] $\mathcal{C}_{1} = \mathrm{DCD}(n-1, d-1)$, $\mathcal{C}_{2} = \mathrm{DCD}(n-1,d)$; \item[(ii)] the set $I$ of new incidences can be given as follows: for each line in $\mathcal{C}_{2}$, the number of points incident with it increases by one, the new points on these lines are precisely the points of $\mathcal{C}_{1}$, and hence $|I| = \binom{n-1}{d-1}$. \end{itemize} In the last section, the author notices that every configuration $\mathrm{DCD}(n,d)$ can be also realized geometrically as a configuration of points and circles. Moreover, all the circles in such a configuration are of equal size, and one calls a point-circle configuration with this particular property isometric. For the entire collection see [Zbl 1400.52002].
scientific article; zbMATH DE number 7049366