An intriguing family of tile patterns can be generated by applying the self-replicating geometry of a pair of isosceles triangles called the golden triangles. The patterns have an aesthetic appeal and have geometric properties which might be investigated further. The family includes patterns which were discovered by Roger Penrose in the 1970s, and which received much attention when they were proposed as a model for quasicrystals. The use of golden triangles to obtain tile patterns stems from the work of R.M. Robinson cited in [2, p540] and extends the investigations of [3]. The method employed in obtaining the patterns is experimental mathematics; geometrically defined experiments are performed using a computer and the results examined for their properties.

A certain type of transformation of a set of numbers may be represented as the multiplication of a vector by a square matrix. Repetition of the operation is equivalent to multiplying the original vector by a power of the matrix. Properties of powers of matrices are thus of considerable importance, and many general properties have been established for some time.

The following, written at the reguest of the Editor of the Gazette, gives an account of the mathematical part of the remarks made by the lecturer at the Annual Meeting of the Association last January. There are added some words dealing with geometry on a sphere, in regard to which some questions were raised at the meeting.