Counting plane arrangements via oriented matroids

TL;DR

The paper surveys how oriented matroids encode the combinatorial structure of plane and hyperplane arrangements, focusing on the relationship between sign-vector descriptions and geometric faces. It explains the foundational axioms, affine enhancements via a marked element, and the suspension/cone construction that connects affine and essential arrangements, then discusses counting problems and known counterexamples to representability and stretchability. Leveraging results from Finschi and Goodman–Pollack, it highlights how many affine arrangements arise from oriented matroids and where realizability fails (e.g., the Pappus arrangement for $9$ lines and nonstretchable pseudolines/pseudoplanes). The authors illustrate the $n=4$ and $n=5$ cases in $\mathbb{R}^3$, enumerating $27$ rank-3 arrangements for $n=5$ and showing how invariants distinguish face-combinatorial types, thereby clarifying the landscape of plane arrangements and identifying open questions in realizability and enumeration.

Abstract

Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets: the planes can intersect or not. Here we review how oriented matroids can be used to try to capture the combinatorial aspect, giving a way to encode with finite sets all the ways that $n$ planes can interact. We mention how the one-to-one correspondence breaks down in 2 dimensions for 9 lines, and in 3D for 8 planes. We include illustrations of all the types of plane arrangements using $n=4$ and 5.

Figures (13)

• Figure 1: Here are the five ways that three planes can intersect. The top row shows the non-central cases in which there is no simultaneous solution to the three (affine) linear equations. The top left is the trivial arrangement (rank 1), the bottom left is essential (full rank, $r=3$), and the other three are rank 2.
• Figure 2: On the left is a line arrangement with its faces labeled: chambers $X, Y, Z, W,$ edges $P, Q, R, S,$ and a point $A.$ On the right the same faces are shown with their associated sign vectors; the small double arrows point to the plus side of each line. These will be discussed in Example \ref{['newsmole']} and in the following section on oriented matroids.
• Figure 3: Eight non-trivial affine arrangements of four lines. At bottom left is the only central arrangement. At bottom right is the only arrangement in general position.
• Figure 4: The top row shows the construction of a plane arrangement by adding one (highlighted) plane at a time, ending with the essential arrangement of five planes below right. The affine arrangement of four lines shown at left corresponds to the marked arrangement where the horizontal plane (5) is marked. Thus it can be seen as a top down view, or cross section on the positive side of plane 5. Chambers include $X,Q,Y,Z$; faces $S$ and $P$ are 2D but are seen as edges from above, and faces $R$ and $W$ are edges but seen as points from above. From the line arrangement we can recover the 3D version by taking the suspension. Compositions and restrictions of the labeled faces in 3D are seen in Example \ref{['examplo']}.
• Figure 5: The Pappus arrangement on the left, and a nonstretchable pseudoline arrangement on the right.

Theorems & Definitions (3)

• Example 1
• Example 2
• Definition 3