# Harriet Fell's Mathematical Images © Harriet Fell 2012

#### From Math for Tanabata

This image and others are part of the Art Exhibit at the 2013 Joint Mathematics Meetings.

You can make your own images like this using Math4Tanabata. © Harriet Fell 1994

#### Coloring of the Unit Disk ala Dubuc and Malik

Dubuc and Malik (Convex hull of powers of complex number, trinomial equations and the Farey sequences, Num. Algorithms, 2 #1, pp. 1-32) note that the convex hull of the powers 1, z, z2, z3, . . . of a complex number z, in the interior of the unit disk, is a polygon. They define the color of the number to be the number of vertices of this polygon. © Harriet Fell 1994

#### Coloring of the Unit Disk Based on an Almost Colinear Condition

Given epsilon > 0, define the color of the complex number z to be the smallest positive integer n such that |(z - 1).im / (z - 1).re - (zn - 1).im / (zn - 1).re| < epsilon. That is, the slope of the line through z and 1 differs from the slope of the line through zn and 1 by less than epsilon. For this image, epsilon = 0.02 and the highest power of z considered is 24.