If you trace a point on the circle rolling inside another circle with four times its radius you get a star-shaped* curve called the astroid. ✨ [Wiki bit.ly/2U0A0oW] #50FamousCurves
*I know, I know, stars aren't actually spikey (bit.ly/2EkZrw3).
Drawing an astroid with a do-nothing machine. #50FamousCurves
By the way, do-nothing machines can be used to draw ellipses as well.
Fun fact: The envelope of ellipses having the same centre and sum of major and minor axes is an astroid. #50FamousCurves
The area of an astroid is one and a half times the area of its inscribed circle. So the area of the astroid given by x⅔+y⅔=a⅔ is A=(3/2)×(a/2)²π=3πa²/8. #50FamousCurves
Fun fact: You see astroids used every time you hop on/off a bus with doors sliding along perpendicular tracks. This door mechanism saves 62.5% of floor space if compared to the usual hinged door. #50FamousCurves
Fun fact: The traces of 4×4 unitary matrices with determinant 1 form an astroid in the complex plane.
Earlier this year a friend* and I've solved a long-standing problem which, in part, meant finding the eigenvectors of this matrix. In this thread, I'll review our result and bits of 170 years of history
*J.F. van Diejen, Talca
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The title of our paper is "Elliptic Kac–Sylvester Matrix from Difference Lamé Equation" and it was recently published in the mathematical physics journal Annales Henri Poincaré.
Just to "name-drop" some of the characters that will appear in the story: Sylvester (duh), Jacobi, Boltzmann, two Ehrenfests, Schrödinger and Kac (obvs).
(I'll expand the thread over several days so please be patient.)
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"Here's a photo of my boy, Peter. He doesn't yet know what the continuum is, but he doesn't know what fascism is either." - George Szekeres' message to Paul Erdős.
Peter Szekeres was born in Shanghai, where his parents George Szekeres and Esther Klein escaped from Nazi persecution in 1938.
Happy Ending Theorem: any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral.
Erdős gave this name to the theorem, because it led to the marriage of Szekeres and Klein
As a run-up to the "Introduction to Integrability" series (see my pinned tweet), I decided to share some interesting bits from the history of integrable systems.
Let's start at the beginning, shall we? So Newton...
#1 It all started with Newton solving the gravitational 2-body problem and deriving Kepler's laws of planetary motion as a result. I would argue that this was possible, because the Kepler problem is (super)integrable. [1/2]
This roughly means that there are many conversed physical quantities like energy, angular momentum, and the Laplace-Runge-Lenz vector. These conservation laws restrict the motion and allow for explicit analytic solutions of otherwise difficult equations. [2/2]