Clayton Shonkwiler PRO
Mathematician and artist
/linz2019
This talk!
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
By Clayton Shonkwiler