Random points in a disk

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

Random points in a disk

Polar coordinates?

Random points in a disk

Random points on a sphere

Spherical coordinates?

Random points on a sphere

Earlier versions

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

Random Points on an Infinite Plane

A probabilist’s answer

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:

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

Restricted domain?

Choose three vertices uniformly in the square:

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

Side Lengths?!

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

Carroll’s answer

Suppose $AB$ is the longest side. Then

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

Sylvester and Cayley’s solution

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’s solution

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

C.M. Ingleby’s solution

G. C. De Morgan’s solution

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

W.S.B. Woolhouse’s solution

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

W.S.B. Woolhouse’s solutions

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

Quadrilaterals in convex bodies

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

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

J.M. Wilson’s solution

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

Wassermann et al. Science 229 (1985), 171–174

Ring Polymers

Knotted DNA

Symmetries $\Rightarrow$ Coordinates

Action Coordinates

Polytope determined by triangle inequalities

Action-Angle Coordinates are Good

Theorem [with Cantarella]

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

What is the Probability of Knotting?

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.

What is the Probability of Knotting?

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 $\epsilon > 0$.