Experimental Computation of Eight-Line Arrangements Generated by All Possible Transversals on Real Projective Plane for Image Production

Abstract

: In this report, a problem of classifying eight-line arrangements on the real projective plane is discussed . By using the combinatorial geometric method of finding transversals for all given segments , all possible eight-line arrangements have been generated from the three sets of 14 kinds of samples of seven-line arrangements, which are different each other . As a result, 135 kinds of eight-line arrangements have been observed.

A Relation between the Weyl Group W(e8) and Eight-Line Arrangements on a Real Projective Plane

“…We are now going to identify the space P 0 (2, 8) with a subset of R 4,5,6,7,8) are constants but the remaining R ijk are polynomials of x, y. Moreover we take the polynomials T ij (1 ≤ i < j ≤ 8) and T j (1 ≤ j ≤ 8) defined in [8].…”

“…Before stating the next lemma, we recall the definition of the adjacent relation among polygons (cf. [6], [7]). Let A(H) be an n-line arrangement, where H = (l 1 , l 2 , .…”

“…In fact, it is proved by direct computation that there are eleven kinds of adjacent relations among polygons for seven-line arrangements (cf. [7]). This fact is in accord with what was described in Grünbaum's book [9], Chapter 18, namely, it is shown by Cummings and White (cf. [2], [3], [15]) that there are eleven different classes of non-equivalent seven-line arrangements in a real projective plane.…”

“…In this paper, we shall study a relationship between eight-line arrangements on a real projective plane and the root system of type E 8 on the same analogy of seven-line case. We have already studied simple eight-line arrangements in [4], [5], [6], [7], [8] and the references there. The theme treated in this paper is, among other things, related with the conjecture in [4].…”