How many revolutions on this Ellipse? @drpeyam

Uploaded June 2022 | Updated September 2026, 1 week ago

How many revolutions on this Pythagorean ellipse? Suppose an ellipse rolls over a sine curve in such a way that one revolution of the ellipse equals to one period of the curve, how are the axes of the ellipse related to the amplitude. The answer is surprisingly related to the Pythagorean theorem. This is a really cool Putnam problem that we solve using calculus and multivariable calculus, more precisely by calculating arclength two ways with integrals and using a u substitution. It’s a must see for calculus and geometry lovers.

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$Anti Pythagoras 2: the Line, Ellipse, and Infinite Triplets Anti Pythagoras 2: the Line, the Ellipse, and the Infinity. How to generate infinitely many anti Pythagorean triplets of the form b^2 = a^2 + c^2 - ac The triplets relate the sides of a triangle where one angle is 60 degrees. It is a variation of the Pythagorean theorem from 8th grade math. It has to do with intersecting an ellipse with a line of rational slope and performing some clever algebra like expanding quadratic polynomials and quadratic forms and comparing numerator and denominators of fractions with integers. Then we give an example of a triangle with such a triplet. Special thanks goes to Ian Fowler who has introduced me to this problem. Ian would like to give a big shout out to Emidio Iacobucci and Frank Cirone for their part in the adventure. 0:00 Introduction 1:40 Rational Slopes 2:40 Finding Nemo 6:00 The triplets are born Anti Pythagorean theorem: https://youtu.be/TMR1oYkCKEs YT channel: https://youtube.com/drpeyam TikTok channel: https://www.tiktok.com/@drpeyam Instagram: https://www.instagram.com/peyamstagram/ Twitter: https://twitter.com/drpeyam Teespring merch: https://teespring.com/stores/dr-peyam$

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