I have several questions relating the definition and property of configuration in geometry:
- What are the intuitions and purposes behind the fact that each point is incident to the same number of lines and each line is incident to the same number of points?
- Does a configuration completely describe the underlying structure? That is, if $A$ and $B$ are collection of points and lines satisfying a configuration, then there exists a map $\phi: A \to B$ that preserves incident structure.
- How symmetric is a configuration? Using definition from above, what extra constraints can I give to $\phi,$ such as choosing arbitrary $a \in A$ and $b \in B$ then require that $\phi(a) = b.$
Other than triangle, quadrangle and quadrilateral, I feel like the concept of configuration doesn't really do anything. I often see texts refer to Desargues configuration or Pappus configuration, but then just leave it at that.