Computer generated color patterns from mathematical functions

In 1954 the author purchased a tablecloth at the International Congress of Mathematicians~ Amsterdam, which used the distribution of the Gaussian primes to develop a two-color block design. This design created by the Dutch Mathematician Balth Van der Pol inspired the author to examine other applications of mathematics to the generation of color patterns. The following year he presented a paper on "Functional Design or Colorful Mathematics" to the Alabama Academy of Science describing various ways in which mathematical functions could produce multi-color block designs. Several hand-created designs using the various methods were displayed. The method used involved the assignment of two coordinates to each block formed in a square grid. A centrally located "origin" square would be given coordinates (0,0), whereas coordinates (x,y) would be associated with the block x units to the right and y units above the coordinate square, etc. Working with integer valued functions of integers x and y and using a convenient modular arithmetic, one of the integers 0,1,2,...,n-l, where n is the modulus, could be determined for each square in the array. By associating a different color with each of those n integers a block color diagram could be formed for a wide variety of functions, but the computations and the hand coloring became rather tedious for designs with a large number of blocks and a large modulus. Recently, the author acquired access to an Intecolor 8001 intelligent terminal which can function independently or on line to our CDC CYBER systems to produce color patterns of the above type electronically. With aid of student programmers, software has been developed to reproduce color patterns originally done by hand, and to handle much more complex functions where hand computations and coloring would be impractical. As an illustrations of a simple function of the type discussed whose design is familiar to all, we cite: