Products of Chords in a Circle (math visualization)

In this video, we describe a classic problem about finding the product of the lengths of the chords obtained by placing n equally spaced points on the unit  circle and connecting one of the points to each of the others. The answer is surprising and one proof relies on a limiting argument using complex numbers and the nth roots of unity.

Once we have this interesting fact, we then show how to couple the fact with the law of cosines to get some interesting trigonometric identities for strange  products of sines, cosines, and tangents.

Finally, we finish with a puzzle about what happens when you do ask the same question about an ellipse. We suggest one particularly interesting ellipse to investigate.

#math #mathvideo #manim #circle #chords #visualproof #trigonometry #sine #cosine #tangent #rootsofunity #complexanalysis #complexnumbers #products #geometry

This animation is based on a famous problem that has been discussed in numerous places. In particular, a great reference for this problem and related ones (with a fantastic bibliography) is the source Chords of an Ellipse, Lucas Polynomials, and Cubic Equations in Issue 8 of the 2020 American Mathematical Monthly (https://doi.org/10.1080/00029890.2020...) by Ben Blum-Smith and Japheth Wood. You can also find the source here: https://arxiv.org/abs/1810.00492.

06:09 note that at the end there are rounding error typos for the challenge problem. The first mistake is at 90, which should be 91. The rest have rounding errors most likely.


For more about using manim, see https://www.manim.community/.

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Music in this video:
I Am Running Down the Long Hallway of Viewmont Elementary by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/...
Source: http://chriszabriskie.com/honor/
Artist: http://chriszabriskie.com/