Harriet Fell's Mathematical Images

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.

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.

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Harriet J. Fell
College of Computer Science
Northeastern University, Boston, MA 02115
Phone: (617) 373-2198
fell@ccs.neu.edu

The URL for this document is: http://www.ccs.neu.edu/home/fell/mathGraphics.html