Linear Structural Patterns Based on an Array of Numbers Derived from the Double-Entry Multiplication Table

Abstract

I became interested in the double-entry multiplication table (Fig. 1, left) in 1972 when a student in my sculpture workshop at the Academia de San Carlos (Mexico City) showed me a single-digit form of it (Fig. 1, right) with the numbers to be multiplied limited to integers in the range 1 to 9. Except for the 3 x 3 array in the upper left-hand corner, it does not look like the ordinary double-entry multiplication table because products, when composed of two digits (i.e.from 10 to 81), have been replaced by a onedigit number. Two-digit products are reduced to one-digit numbers by the following procedure: for the multiplication 4 x 6 = 24. one adds 2 and 4 to get the digit 6: for 4 x 7 = 28, one adds 2 and 8 to get 10, and then one adds 1 and 0 to get 1. The first part of my investigation of designs coincides with a study carried out by a group of English investigators [1]. My inquiries, however, have taken a course that allows me to find more design possibilities and then to find projections of these in three dimensions. This last step leads to two procedures: one is discussed below; the other, in which I convert the square into a cube and proceed to locate numbers within it to produce a spatial geometry, will be discussed in a forthcoming article.