Finding Zeros with Domain Coloring and Winding Numbers
Research Mentor(s)
Stephanie Treneer
Description
Many applications of mathematics require solving for the zeros of a real 2-dimensional transformation. Any such transformation can be plotted using its domain coloring. Specifically, every point on the plane takes the hue of where that point’s transformation lands on a color wheel centered at the origin. This creates a colorful visualization for otherwise cumbersome transformations and provides a framework for identifying which points are mapped to the origin. Given any closed boundary on this domain, its winding number is representative of how many loops of the color spectrum are completed when moving clockwise along the boundary. If the winding number is a nonzero integer, the boundary can be repeatedly halved as it converges onto a zero of the transformation. This process can be replicated both visually and mathematically using the Wolfram Language.
Document Type
Event
Start Date
May 2022
End Date
May 2022
Location
SMATE Library (Bellingham, Wash.)
Department
Mathematics
Genre/Form
student projects; posters
Type
Image
Rights
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.
Language
English
Format
application/pdf
Finding Zeros with Domain Coloring and Winding Numbers
SMATE Library (Bellingham, Wash.)
Many applications of mathematics require solving for the zeros of a real 2-dimensional transformation. Any such transformation can be plotted using its domain coloring. Specifically, every point on the plane takes the hue of where that point’s transformation lands on a color wheel centered at the origin. This creates a colorful visualization for otherwise cumbersome transformations and provides a framework for identifying which points are mapped to the origin. Given any closed boundary on this domain, its winding number is representative of how many loops of the color spectrum are completed when moving clockwise along the boundary. If the winding number is a nonzero integer, the boundary can be repeatedly halved as it converges onto a zero of the transformation. This process can be replicated both visually and mathematically using the Wolfram Language.