Computer Visualization in Experimental MathematicsJoshua Holder, Xiaomin Li, Doris Wang, Jinlin Xu, Daniel Carmody, George Francis, Karthik Vasu

Introduction

Motivations•Make mathematics more accessible by providing publicly available interactive visu-

alizations of complex topics.•Use interactive visualizations to postulate or reject conjectures.

Webpagehttps://mathviz19.pages.math.illinois.edu/webpage/Use the QR code in the top right of the poster to access the webpage containing theinteractive visualizations.

Regular eversions of the sphere

Background:

• In 1957, Stephen Smale proved that, allowing for self-intersections, it is possible to turna sphere S2 ⊂ R3 inside out without creating a tear or crease ([1]). There is a regular ho-motopy between the standard immersion ι : S2 ↪→ R3 and the immersion a ◦ ι : S2 ↪→ R3

where a is the map x 7→ −x.

• Because π2(R3) is trivial, any two maps S2 → R3 are homotopic. The surprising aspectof this result is that the homotopy H : [0, 1]× S2→ R3 can be chosen such that Ht is animmersion for all t.

• Thurston’s sphere eversion was famously animated by the Geometry Center ([2]).

Goal: Make models of Morin-Apery sphere eversions more accessible by coding them asinteractive javascript programs. This allows any user to experiment with sphere eversionswithout having to manually download and compile archaic C(++) code.

Figure 1: A sphere (left) and cylinder (right) undergoing eversions.

Billiards and covering spaces

Background•Given a square billiard table with four circular pockets of fixed radius ε, how long does it

take for a ball shot at some angle to fall into a pocket?• The billiard table can be viewed as an orbifold: it’s a quotient of R2 by a group of reflec-

tions.• The path of a billiard ball can be realized as an orbifold path, and hence has a piece-

wise lift to R2. This lift can in fact be chosen to be a continuous path in R2.• Because R2 is the universal cover of the torus, a path in R2 yields a path on the torus.

It follows that any billiard trajectory yields a path on the torus.Goal: Visualize the path on the torus corresponding to a billiard trajectory and use this todevelop intuition for the relationship between the angle of the initial shot and the length ofa billiard path.

Figure 2: The structure of the interactive visualization (left), and a sample billiard trajec-tory (right).

Higher dimensional Koch surfaces

Background:• The Koch snowflake is a classical fractal constructed iteratively by adding triangular

extrusions to equilateral triangles.•Replace triangles with tetrahedra to obtain a higher dimensional Koch snowflake.• Iterating Koch extrusions on a tetrahedron seems to yield a cube.•Given a regular polyhedron as input, does the corresponding Koch surface yield another

regular polyhedron?Goal: By animating the Koch surfaces on regular polyhedra, understand the relationshipsbetween regular polyhedra given by iterating the Koch construction.

Figure 3: First three iterations of Koch extrusions on a tetrahedron.

Figure 4: Higher iterations of Koch extrusions on an octahedron, tetrahedron, and icosa-hedron.

Confidence surfaces for linear regression

Background:

• The Gauss-Markov theorem tells us that, given a random vector which is modeled asas an unknown linear combination y = Xβ + ε of known data X and a mean zero,homoscedastic, random error ε = [ε1 . . . εn]

T with diagonal covariance matrix, the bestlinear unbiased estimator (BLUE) of the model parameter β is given by the ordinaryleast squares parameter βOLS = (XTX)−1XTy.

• If we make the additional assumption that ε ∼ N (0, σ2) is normally distributed, thenconditionally on X, βOLS ∼ N (β, (XTX)−1σ2).

•Now fix an x, and define y = xβOLS. We can construct a test statistic for the null hy-pothesis E[y | x] = y0 as y−y0

σxwhere σx = σ2xT (XTX)−1x where σ is the adjusted

sample variance of βOLS. This test statistic has a Student’s t-distribution, and by fixinga confidence level α, we can get bounding surfaces.

Goal: Develop an animation to help statistics students understand confidence surfacesfor linear regression.

Figure 5: Some sample distributions and the corresponding regression planes and confi-dence surfaces.

Future Directions

• Experiment with different ways of visually presenting the mathematical content in eachproject.

• (Sphere eversions) Incorporate other sphere eversions (i.e. Thurston’s eversion) andmerge all the eversions into a single webpage.

• (Billiards) Compare billiard paths to geodesics in the usual metric on the torus (thepullback metric from an embedding in R3).

• (Koch extrusions) Figure out a reasonable definition of Koch extrusion for the dodeca-hedron. Investigate further the limit of the Koch extrusion process.

• (Confidence surfaces) Compare confidence surfaces obtained from assuming normalityof the residuals to confidence surfaces obtained by bootstrapping.

References

[1] Smale, Stephen. ”A classification of immersions of the two-sphere”, Trans. of the Amer. Math. Soc., 90 (2): 281-290, 1958.[2] http://www.geom.uiuc.edu/

Support for this project was provided by the Illinois Geometry Lab and the Department of Mathematics at the University of Illinois at Urbana-Champaign.

IGL Poster Session Fall 2019