Take two gears with 3:1 ratio and a handle of length 3 on the smaller gear. The end of the handle traces a circle and the corresponding point on the larger gear traces a trifolium. What do you think happens with other gear ratios? [Wiki bit.ly/2Sadb5o] #50FamousCurves
The trifolium belongs to the family of rose curves defined by the polar equation r=a·sin(kφ) associated with gear ratio k:1. If k is odd, there are k petals, but if k is even, you get 2k petals. Here is the case when k=2. #50FamousCurves
Table of rose curves r=sin(kφ) with k=n/d (gear ratio n:d) and n,d=1,2,...,7. #50FamousCurves
Fun fact: "Folium" is Latin for "leaf". So the name "trifolium" means "three leaves". #50FamousCurves
How to draw a trifolium? 1) Take two circles with 3:1 radius ratio. 2) Put a handle of length 2 on the smaller circle. 3) Roll the smaller circle around on the inside of the larger circle.
The end of the handle will trace a trifolium. #50FamousCurves
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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]