
Each square in the spiral represents a different positive integer, starting with n = 1 towards the center and increasing in the outward direction. The color assignment for the squares is based on the values of the abundancy index of the corresponding numbers. The abundancy index of a number is defined to be the sum of all divisors of the number divided by the number. So, for example, the abundancy index of 17 is (1+17)/17 = 18/17, and the abundancy index of 6 is (1+2+3+6)/6 = 2. There are many open problems in mathematics relating to the abundancy index, and many of these problems date back to antiquity. The Greeks are believed to have been the first people to study perfect numbers  those numbers having abundancy index equal to 2 (meaning that 6 is perfect). They wondered whether or not all perfect numbers are even, a question that still remains unanswered today. Numbers having abundancy index less than 2 are known as deficient and those with abundancy index greater than 2 are known as abundant numbers. In my painting, I color deficient numbers in shades of green and yellowgreen and abundant numbers in shades of blue and violet. The resulting pattern is complicated, reflecting the difficulty of the underlying mathematics. 

How did I determine the color scale? I wrote a computer program that allowed me to experiment with different choices of palate. A computer can color squares so much faster than I can, and I was not excited about painting 450+ little squares without knowing in advance that I would like the resulting pattern. In the end, I chose to restrict my palate by excluding the warm colors. You will find more of my mathematical paintings here. 
Back to the Kenyon Homepage  Back to the Math Homepage  Back to JAH's Homepage 