/linz2019

This talk!

- How to generate random triangles?
- What are optimal paths between triangles?
- How does this generalize to \(n\)-gons?

*Wallpaper*, joint with Anne Ligon Harding

*Isometries*

Frame from *Nucleation*

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

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

Let \(e_1, \ldots , e_n\) be the edges of a planar \(n\)-gon with total perimeter 2. Choose \(z_1, \ldots , z_n\) so that \(z_k^2 = e_k\). Let \(z_k = u_k + i v_k\).

The polygon is closed \(\Leftrightarrow e_1 + \ldots + e_n = 0\)

\(\sum e_k =\sum z_k^2 = \left(\sum u_k^2 - \sum v_k^2\right) + 2i \sum u_k v_k\)

The polygon is closed \(\Leftrightarrow \|\vec{u}\|=\|\vec{v}\|\) and \(\vec{u} \bot \vec{v}\)

Since \(\sum |e_k| = \sum u_k^2 + \sum v_k^2 = \|\vec{u}\|^2 + \|\vec{v}\|^2\), we see that \((\vec{u},\vec{v})\) is an orthonormal pair of vectors in \(n\)-dimensional space; the collection of such things is called the *Stiefel manifold* \(\mathrm{St}_2(\mathbb{R}^n)\).

`StiefelSample[n_]:=Orthogonalize[RandomVariate[NormalDistribution[],{2,n}]]`

560 random quadrilaterals

A random 100,000-gon

`Convexify[edges_] := SortBy[edges, PositiveArg[Complex @@ #] &];`

Permuting edges is an isometry of the Stiefel manifold, so this produces a uniform random sample of convex \(n\)-gons.

Random convex 20-gons

*A to Z*, in the art exhibition

\(n\)-gons in \(\mathbb{R}^3\) are parametrized by points in \(\mathrm{St}_2(\mathbb{C}^n)\), and we can do exactly the same sorts of things…

36 random 16-gons

*Re-Tie*

More precisely, points in \(\operatorname{St}_2(\mathbb{C}^n)\) map to *framed polygons*.

*Framing*

Funding: Simons Foundation

This talk: math.graphics/linz2019

Random triangles and polygons in the plane

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

*The American Mathematical Monthly* **126** (2019), 113–134

Probability theory of random polygons from the quaternionic viewpoint

Jason Cantarella, Tetsuo Deguchi, and Clayton Shonkwiler

*Communications on Pure and Applied Mathematics* **67** (2014), 1658–1699