Clayton Shonkwiler PRO
Mathematician and artist
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians
/mlm18
this talk!
Jason Cantarella
U. of Georgia
Tom Needham
Ohio State
Gavin Stewart
NYU
Laney Bowden
CSU
Andrea Haynes
CSU
Aaron Shukert
CSU
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
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,
\(\mathbb{P}(\text{obtuse}) = \frac{\pi/2}{\pi/3+\sqrt{3}/2} \approx 0.82\)
Proposition [Portnoy]: If the distribution of \((x_1,y_1,x_2,y_2,x_3,y_3)\in\mathbb{R}^6\) is spherically symmetric (for example, a standard multivariate Gaussian), then
\(\mathbb{P}(\text{obtuse}) = \frac{3}{4}\)
Consider the vertices \((x_1,y_1),(x_2,y_2),(x_3,y_3)\) as determining a single point in \(\mathbb{R}^6\).
For example, when the vertices of the triangle are chosen from independent, identically-distributed Gaussians on \(\mathbb{R}^2\).
Probability \(p\)
\(p\)
\(p\)
\(p\)
\(p\)
Uh oh!
There is no way to choose points randomly in the plane so that the probability that a point lies in a region depends only on the size of the region.
Fancier Version
The uniform measure on the plane is not a probability measure.
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:
\(\mathbb{P}(\text{obtuse})=\frac{97}{150}-\frac{\pi}{40}\approx 0.7252\)
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
Are these questions really about choosing random points, or are they actually about choosing random polygons?
How would you choose a polygon “at random”?
— Stephen Portnoy, Statistical Science 9 (1994), 279–284
The space of all \(n\)-gons should be a nice space \(P_n\) with a transitive symmetry group. We should use the invariant probability measure on \(P_n\). Then we should compute the measure of the subset of \(n\)-gons satisfying our favorite condition.
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 \(z_k = a_k + i b_k\).
The polygon is closed \(\Leftrightarrow e_1 + \ldots + e_n = 0\)
\(\sum e_k =\sum z_k^2 = \left(\sum a_k^2 - \sum b_k^2\right) + 2i \sum a_k b_k\)
The polygon is closed \(\Leftrightarrow \|a\|=\|b\|\) and \(a \bot b\)
Since \(\sum |e_k| = \sum a_k^2 + \sum b_k^2 = \|a\|^2 + \|b\|^2\), we see that \((a,b) \in V_2(\mathbb{R}^n)\), the Stiefel manifold of 2-frames in \(\mathbb{R}^n\).
Proposition: Rotating \((a,b)\) in the plane it spans rotates the corresponding \(n\)-gon twice as fast.
Corollary [Hausmann–Knutson]
The Grassmannian \(G_2(\mathbb{R}^n)\) of 2-dimensional linear subspaces of \(\mathbb{R}^n\) covers the space of planar \(n\)-gons of perimeter 2.
Definition [w/ Cantarella & Deguchi]
The symmetric measure on \(n\)-gons of perimeter 2 up to translation and rotation is the pushforward of Haar measure on \(G_2(\mathbb{R}^n)\).
Therefore, \(O(n)\) acts transitively on \(n\)-gons and preserves the symmetric measure.
A triangle corresponds to an element of \(G_2(\mathbb{R}^3)\), which is a plane in \(\mathbb{R}^3\) with ON basis \((a,b)\)...
or, equivalently, the line through
\(p=a\times b\)
so
Law of Cosines
Perimeter \(2\)
The rotations are natural transformations of the sphere, and the corresponding action on triangles is natural.
\(c=1-z^2\) fixed
\(z\) fixed
\(C(\theta) = (\frac{z^2+1}{2}\cos 2\theta, z \sin 2\theta)\)
The equal-area-in-equal-time parametrization of the ellipse
The right triangles are exactly those satisfying
\(a^2+b^2=c^2\) & permutations
Since \(a=1-x^2\), etc., the right triangles are determined by the quartic
\((1-x^2)^2+(1-y^2)^2=(1-z^2)^2\) & permutations
\(x^2 + x^2y^2 + y^2 = 1\), etc.
\(\mathbb{P}(\text{obtuse})=\frac{1}{4\pi}\text{Area} = \frac{24}{4\pi} \int_R d\theta dz\)
But now \(C\) has the parametrization
And the integral reduces to
By Stokes’ Theorem
\(\frac{6}{\pi} \int_R d\theta dz=\frac{6}{\pi}\int_{\partial R}z d\theta = \frac{6}{\pi}\left(\int_{z=0} zd\theta + \int_C zd\theta \right)\)
\(\left(\sqrt{\frac{1-y^2}{1+y^2}},y,y\sqrt{\frac{1-y^2}{1+y^2}}\right)\)
\(\frac{6}{\pi} \int_0^1 \left(\frac{2y}{1+y^4}-\frac{y}{1+y^2}\right)dy\)
Theorem [with Cantarella, Needham, Stewart]
With respect to the symmetric measure on triangles, the probability that a random triangle is obtuse is
\(\frac{3}{2}-\frac{3\ln 2}{\pi}\approx0.838\)
convex
reflex/reentrant
self-intersecting
Modern Reformulation: What is the probability that all vertices of a random quadrilateral lie on its convex hull?
A polygon corresponds to \(\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \\ \vdots & \vdots \\ a_n & v_n \end{bmatrix}\), where \((a_k + i b_k)^2 = e_k\).
\((a_k,b_k)\mapsto(-a_k,-b_k)\) doesn’t change the polygon, so \((\mathbb{Z}/2\mathbb{Z})^n\) acts trivially on polygons.
\(S_n \le SO(n)\) permutes rows and hence edges.
The hyperoctahedral group \(B_n = (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n = S_2 \wr S_n\) acts by isometries on \(G_2(\mathbb{R}^n)\) and permutes edges.
\(|B_n| = 2^n n!\); e.g., \(|B_3| = 2^3 3! = 48\).
\(S_n \le SO(n)\) permutes rows and hence edges.
\(S_n \le SO(n)\) permutes rows and hence edges.
Proposition [with Cantarella, Needham, Stewart]
Upshot: We can count how many elements of the permutation orbit of a single element of \(\mathcal{D}_4\) are convex, reflex, and self-intersecting.
The (empirical) flag mean of \(\mathcal{D}_4\).
The flag mean of
is (the plane spanned by) the first two left singular vectors of the matrix
Theorem [with Cantarella, Needham, Stewart]
Convex, reflex, and self-intersecting quadrilaterals are all equiprobable.
Theorem [with Cantarella, Needham, Stewart]
With respect to any measure on \(n\)-gon space which is invariant under permuting edges, the fraction of convex \(n\)-gons is exactly \(\frac{2}{(n-1)!}\).
Symmetric triangles
Isosceles
triangles
Degenerate
triangles
Theorem [with Bowden, Haynes, Shukert]
The least symmetric triangle has side length ratio
Theorem [with Bowden, Haynes, Shukert]
The least symmetric obtuse triangle has side length ratio
Theorem [with Bowden, Haynes, Shukert]
The least symmetric acute triangle has side length ratio
There is a version of this story for \(n\)-gons in space as well.
Polygons in space provide a foundational theoretical and computational model for ring polymers like bacterial DNA.
Wassermann et al., Science 229, 171--174
Lyubchenko, Micron 42, 196--206
Polygons in space correspond to planes in complex \(n\)-dimensional space \(\mathbb{C}^n\), which we also understand very well.
to
?
Example Theorem [w/ Cantarella, Grosberg, Kusner]
The expected total curvature of a random space \(n\)-gon is exactly
\(\frac{\pi}{2}n + \frac{\pi}{4} \frac{2n}{2n-3}\)
The Grassmannian theory deals with random polygonal loops; find the analogous theory for more complicated polygonal graphs.
A synthetic \(K_{3,3}\) (Tezuka Lab, Tokyo)
Novel polymer topologies have novel material properties.
Funding: Simons Foundation
Probability Theory of Random Polygons from the Quaternionic Perspective
J. Cantarella, T. Deguchi, and C. Shonkwiler
Communications on Pure and Applied Mathematics 67
(2014), 1658–1699
Random Triangles and Polygons in the Plane
J. Cantarella, T. Needham, C. Shonkwiler, and G. Stewart
The American Mathematical Monthly, to appear
Spherical Geometry and the Least Symmetric Triangle
L. Bowden, A. Haynes, C. Shonkwiler, and A. Shukert
Geometriae Dedicata (2018), https://doi.org/10.1007/s10711-018-0327-4
The Expected Total Curvature of Random Polygons
J. Cantarella, A. Grosberg, R. Kusner, and C. Shonkwiler
American Journal of Mathematics 137 (2015), 411–438
By Clayton Shonkwiler
An Introduction to Geometric Probability, Shape Spaces, Group Actions, and Grassmannians