Geometric explanation for near-symmetry of k-chain region counts

Determine a geometric explanation for why the maximum number of regions formed by n k-chains is essentially the same as that formed by k n-chains, apart from the difference in the numbers of infinite regions.

Background

The paper derives the formula a_{kC}(n) = k²n²/2 − 3kn/2 + 2n + 1 for the maximum number of regions formed by n k-chains. Interchanging k and n changes the value only by 2|k−n|, which the authors identify with the difference in the numbers of infinite regions in the corresponding optimal graphs.

Although this numerical near-symmetry is explained algebraically, the authors examine and reject a simple one-to-one correspondence between optimal arrangements. They therefore leave unresolved the geometric reason for the phenomenon.