On rotational tessellations and copses

Abstract

This paper is a self-contained sequal to Miller (1970), entitled ‘Periodic forests of stunted trees’. It is concerned with ‘Copses’ and ‘Tessellations’ based on an infinite background of nodes at the vertices of a plane tessellation of unit equilateral triangles, forming either a finite larger equilateral triangle for a copse, or an infinite doubly periodic tessellation otherwise. In any such background the unit triangles form two sets, of opposing orientations. We label the nodes individually with a 1 (called live nodes ) or a 0 (called vacant nodes) in such a way that for the nodes on each triangle of one set with the same orientation the sum of the labels equals 0 (mod 2); the sum round unit triangles of the other orientation is not restricted. The tessellations are obtained by joining by an edge every pair of adjacent live nodes. The purpose of the paper is to study which copses and tessellations exist, and to enumerate them, and to show how they may be constructed and listed. In Miller (1970) this was done for forests, slightly different from tessellations but with an identical theoretical approach. In the present paper we are particularly interested in copses and tessellations with rotational symmetry about each of a lattice of symmetry centres, either with or without reflexive symmetry as well. A copse is determined by a particular vector of node labels along one of its edges: the symmetry studied in the present paper corresponds to having identical vectors along all three edges of the copse. We find a basis for ‘permissible vectors’ yielding such symmetry for each size of copse. A tessellation is determined by an infinite vector of labels-periodic in this investigation-along a straight line of adjacent nodes. This in turn is generated as a series of coefficients by a ‘generating fraction’ of which the denominator is a | generating polynomial’ of finite degree depending on the period. The vector defining a copse of size k + 1, and the minimum polynomial of degree k , generating a tessellation of period n (which will be such that n I2 k -11) turn out to be eigen-vectors (respectively row- and column-vectors) of the same Pascal matrix (consisting of a triangle of binomial coefficients, modulo 2, together with a right bottom half all zeros). These are fully studied. Finally it is known that the product of all irreducible polynomials with coefficients in GF(2) and of degree dividing k is just t 2k + 1. It is shown in this paper that, in a similar fashion, all (suitably-defined) ‘primary’ reflexive polynomials of degree dividing 2 k , themselves divide t 2k+1 +1, and that all 'primary’ rotational polynomials of degree dividing 3 k in a similar way divide either t 2k+1 + 1 or t 2k+1 + t + 1. It is also established that the ‘primary’ polynomials of each type, i.e. all, reflexive, rotational and triangular (both rotational and reflexive) each have the same enumerating function for the respective degrees k , 2 k , 3 k ,and 6 k . We also find that there is only one irreducible triangular polynomial, namely t 2 + t + 1. The various types of copses and polynomials have been enumerated for a number of values of k or n , and likewise rotational and triangular tessellations for those values of n for which they exist. A substantial selection of these tables is given in the paper. A very large number of diagrams have been drawn, and a substantial, and I hope representative, selection reproduced herein.