A visual representation of the Steiner triple systems of order 13

Authors

  • Marco Pavone Universit√† degli Studi di Palermo, Italy

DOI:

https://doi.org/10.26493/2590-9770.1564.2b8

Keywords:

Steiner triple system, STS, regular hexagram, STS(13), configuration

Abstract

Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15.

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Published

2023-01-31

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Section

Articles