Department of Mathematics

An \((n,\, 4)\) configuration is a collection of \( n \) points and lines where each line contains exactly 4 points, and every 4 points lie on exactly one line. Unlike the case of \((n,\, 3)\)s, not much is known about geometrically realizable \((n,\, 4)\)s. Here, by the phrase "geometrically realizable" it is meant that points and lines are in the real plane. For example, it is known that there are no geometric configurations \((n,\, 4)\) for \( n ≤ 17 \). What one would like to have in this context is a visually pleasing or a geometrically understandable construction of the 4-blocks.

Here we present an arithmetic procedure to embed the cyclic \((n,\, 4)\) configurations into groups such that whenever four points \( \lbrace P,\, Q,\, R,\, S\rbrace \) form a block in the given \((n,\, 4)\) configuration, then \( P+Q+R+S = 0 \) in the group. Using this group law, we obtain a new geometric realization of the \((n,\, 4)\)s over the real plane where the four points in a block are concyclic. In this paper we prove that the three cyclic \((n,\, 4)\)-configurations \( C4(15,\,1,\,4,\,6) \), \( C4(16,\,1,\,4,\,6) \), and \( C4(17,\,1,\,3,\,9)\) are circle-realizable in the real plane.

References:

1. Jürgen Bokowski and Lars Schewe, There are no realizable \( (15, 4) \) and \( (16, 4) \) configurations, Rev. Roumaine Math.Pures Appl. 50, no. 5-6, 483-493 (2005).
2. Grünbaum, Branko, Configurations of Points and Lines, Graduate Studies in Mathematics vol 103, AMS, (2009).
3. Grünbaum, Branko, Which \((n,\, 4)\) configuration exist? Geombinatorics 9, no.4, 164--169, (2000).
4. N.S. Mendelsohn, R. Padmanabhan, B.Wolk, Designs embeddable in cubic curves, Note di Matematica, vol VII, 113-148, (1987).
5. Lanyon, Jeffrey, Concrete Realizations of Configurations Master's Thesis, University of Manitoba, 2019.