/cu24

this talk!

September 13, 2024

Polar/angular coordinates!

Rectangular coordinates!

Rectangular coordinates, sorta

Probability that a random point in an \(n\)-dimensional cube lies inside the inscribed sphere

\mathbb{P}_n = \frac{\pi^{n/2}}{2^n \Gamma(n/2+1)}

Polar coordinates?

Polar coordinates?

Polar coordinates, non-uniformly

Spherical coordinates?

2D standard Gaussian points

Normalized 2D standard Gaussian points

Distance distribution for 2D Gaussian

Distance distribution for 5D Gaussian

Distance distribution for 20D Gaussian

Cylindrical coordinates?

Archimedes’ Theorem:

The red and blue areas are equal!

Cylindrical coordinates!

W. S. B. Woolhouse, *Educational Times* **18** (1865), p. 189

J. J. Sylvester, *Educational Times* **18** (1865), p. 68

W. S. B. Woolhouse, *The Lady's and Gentleman's Diary* **158** (1861), p. 76

Probability \(p\)

\(p\)

\(p\)

\(p\)

\(p\)

Uh oh!

**Proposition **[Portnoy]

If the vertices of the triangle are chosen from the standard Gaussian on \(\mathbb{R}^2\), then

\(\mathbb{P}(\text{obtuse}) = \frac{3}{4}\)

Choose three vertices uniformly in the disk:

\(\mathbb{P}(\text{obtuse})=\frac{9}{8}-\frac{4}{\pi^2}\approx 0.7197\)

Choose three vertices uniformly in the square:

\(\mathbb{P}(\text{obtuse})=\frac{97}{150}-\frac{\pi}{40}\approx 0.7252\)

Sidelengths \((a,b,c)\) uniquely determine a triangle (SSS).

Obtuseness is scale-invariant, so pick a perimeter \(P\) and we have \(a+b+c=P\).

\(b+c<a\)

\(a+b<c\)

\(a+c<b\)

\(\mathbb{P}(\text{obtuse})=9-12\ln 2 \approx 0.68\)

\(b^2+c^2=a^2\)

\(a^2+b^2=c^2\)

\(a^2+c^2=b^2\)

Suppose \(AB\) is the longest side. Then

\(\mathbb{P}(\text{obtuse})=\frac{\pi/8}{\pi/3-\sqrt{3}/4} \approx 0.64\)

But if \(AB\) is the *second* longest side,

\(\mathbb{P}(\text{obtuse}) = \frac{\pi/2}{\pi/3+\sqrt{3}/2} \approx 0.82\)

reentrant

J.J. Sylvester,
*Educational Times*, April 1864

J.J. Sylvester,
*Phil. Trans. R. Soc. London*
**154** (1864), p. 654, footnote 64(b)

W.S.B. Woolhouse,
*Mathematical Questions with Their Solutions *
**VII** (1867), p. 81

A. De Morgan,
*Trans. Cambridge Phil. Soc.*
**XI** (1871), pp. 147–148

W.S.B. Woolhouse,
*Mathematical Questions with Their Solutions*
**VI** (1866), p. 52

C.M. Ingleby,
*Mathematical Questions with Their Solutions *
**V** (1865), p. 82

G.C. De Morgan,
*Mathematical Questions with Their Solutions *
**V** (1865), p. 109

W.S.B. Woolhouse,
*Mathematical Questions with Their Solutions *
**VI** (1866), p. 52

W.S.B. Woolhouse,
*Mathematical Questions with Their Solutions*
**VIII** (1868), p. 105

\(\mathbb{P}(\text{reflex})=\frac{1}{3}\)

\(\mathbb{P}(\text{reflex})=\frac{35}{12\pi^2}\approx 0.296\)

**Theorem [Blaschke, 1917]**

\(\frac{35}{12\pi^2}\leq\mathbb{P}(\text{reflex})\leq\frac{1}{3}\)

J.M. Wilson,
*Mathematical Questions with Their Solutions*
**V** (1866), p. 81

W.A. Whitworth,
*Mathematical Questions with Their Solutions*
**VIII** (1868), p. 36

Report on J.J. Sylvester’s presentation of his paper “On a Special Class of Questions on the Theory of Probabilities” to the British Association for the Advancement of Science, 1865

Protonated P2VP

Random flight model

Knotted DNA

Random polygon model

Action-angle coordinates

Polytope determined by triangle inequalities

**Theorem** [with Cantarella]

Sampling action-angle coordinates uniformly is equivalent to sampling equilateral polygons uniformly.

**Theorem** [with Cantarella and Schumacher]

We can generate random equilateral \(n\)-gons in expected time \(\Theta(n^2)\).

**Frisch–Wasserman–Delbrück Conjecture**

In any reasonable model of random knots parametrized by a “size,” the probability of knotting goes to 1 as the size goes to infinity.

Random polygon

Random diagram

Random petal knot

**Frisch–Wasserman–Delbrück Conjecture**

In any reasonable model of random knots parametrized by a “size,” the probability of knotting goes to 1 as the size goes to infinity.

Random polygon

**Theorem** [Diao]

For random equilateral \(n\)-gons, the probability of knotting is

\(\geq 1-e^{-n^{\epsilon}}\) for some \(\epsilon > 0\).

\mathbb{P}(\text{unknot}) \approx e^{-n/251}\left[1-0.51\sqrt{n}-\frac{1.2}{n}\right]

Synthetic chemists can now produce simple topological polymers in usable quantities.

\(\theta\)-curves in solution at the Tezuka lab

What are good coordinates for random embeddings of graphs?

shonkwiler.org/cu24

The symplectic geometry of closed equilateral random walks in 3-space

Jason Cantarella and Clayton Shonkwiler

*Annals of Applied Probability* **26** (2016), no. 1, 549–596

New stick number bounds from random sampling of confined polygons

Thomas D. Eddy and Clayton Shonkwiler

*Experimental Mathematics* **31** (2022), no. 4, 1373–1395

Random triangle and polygons in the plane

Jason Cantarella, Tom Needham, Clayton Shonkwiler, and Gavin Stewart

*The American Mathematical Monthly* **126** (2019), no. 2, 113–134

A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

Jason Cantarella, Henrik Schumacher, and Clayton Shonkwiler

*Journal of Physics A: Mathematical and Theoretical* **57** (2024), no. 28, 28205