For this research project a clear and concise definition of the Hairy Ball Theorem, also known as the Hedgehog Theorem, will be considered. This theory addresses the way combed vectors can be visualized by thinking of the hair on an individual’s head or the spines of a rolled-up hedgehog, and how there will always be a zero vector or a cowlick. Look into how the Theorem might interact with higher dimensions or with other shapes in 3-D. Covering how Hopf Fibrations might explain why the Hairy Ball Theorem always holds true. After looking at the proof the question will be posed, “How would manipulated vectors on a shape other than a sphere, specifically a torus or donut shape, behave?” This article is meant to get creative mathematical juices going and encourage us to question, “What if”.
Berthold, Corvette and Berthold, Corvette, "The Hairy Ball Theorem and Other Hairy Situations" (2021). Academic Excellence Showcase Proceedings. 305.
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