CONFIGURATIONS OF HIGHER ORDERS

Abstract

This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order 2 - referred to in other sources as 3-designs. We then proceed to investigate a further extension to the notion of points and $k$-planes ($k$-dimensional hyperplanes) which we refer to as configurations of order $k$. We present a number of general examples such as stacked configurations of order $k$ - intuitively layering lower order configurations - and product configurations of order $k$. We discuss many analogues of standard configurations such as dual configurations, isomorphisms, graphical representations, and when a configuration is geometric. We focus mostly on configurations of order 2 and specifically compute the number of possible symmetric configurations of order 2 when each plane contains 3 points for small values on $n$ - the total number of points in the configuration.

Received: August 4, 2022;
Accepted: September 9, 2022;