There are many topics in math that I have learned best myself by trying to code up some visuals. In some of these cases, the results have been pretty enough (and the topics compelling enough) that I want to try and share! The topics below are kind of all over the place, with the common thread being I've managed to produce a something I'm happy enough with for each. I'm always on the lookout for new topics, so feel free to share ideas with me.

Seifert Fibrations of $\mathbb{S}^3$

The three sphere is a circle bundle over the two sphere, giving a fibration of $\mathbb{S}^3$ as well as a system of toroidal coordiantes. Furthermore, this fibration is but the simplest in an infinite sequence of Seifert fibrations of $\mathbb{S}^3$ which in general fill the sphere with $(p,q)$ torus knots.

27 Lines on a Cubic

Every (complex) cubic surface contains exactly 27 (projective) lines. One way of starting to approach this fact (which was introduced to me at a summer school of Benson Farb and Jesse Wolfson) involves a construction taking in a single line on a cubic and producing more, by considering the pencil of planes containing that line in projetive space.

Braid Monodromy of Complex Polynomials

The space $\Sigma$ of complex polynomials of degree $n$ with distinct roots has as its fundamental group the $n$-strand Braid group. Thus subvarieties of $\Sigma$ come equipped with braid monodromy, or homomorphisms from their fundamental groups into the braid groups.

Convex $\mathbb{R}\mathsf{P}^2$ Structures

Hyperbolic triangle groups are rigid, a fact easily believed once one remembers that a hyperbolic triangle is completely determined by its angles. However, these groups admit nontrivial deformations into $\mathsf{SL}(3;\mathbb{R})$, geometrically describing convex real projective structures.

Flat Tori and the Hopf Map

The preimage of a simple closed curve on the two sphere under the Hopf map is an intrisically Euclidean torus in $\mathbb{S}^3$. In fact Ulrich Pinkall showed that all Euclidean tori may be constructed in this way, and gives an explicit formula relating the moduli of such a torus to the lenght/enclosed area of the cuve on $\mathbb{S}^2$, which we briefly discuss here.

Taylor Series

The taylor series of an analytic function has a radius of convergence, but often the corresponding cirlce is hidden from view when considering only real valued functions. Here, we use domain coloring of $\mathbb{C}$ to visualize various functions and their series expansions.

Inside the Thurston Geometries

In three dimensions, there are eight important homogeneous geometries. Named after Thurston (their importance arises from their occurance in his Geometrization Conjecture), these geometries include the three constant curvature spaces $\mathbb{H}^3,\mathbb{S}^3,\mathbb{E}^3$, the product spaces $\mathbb{H}^2\times\mathbb{R},\mathbb{S}^2\times\mathbb{R}$, and three additional geometries $\mathsf{Nil},\mathsf{Solv}$ and $\mathsf{SL}_2(\mathbb{R})$. The project below documents our most current work on producing geometrically correct, real time virtual reality renderings of the intrinsic geometry of these eight worlds. This work is the continuation of a long line of projects, beginning with renders using the inverse exponential map. This portion includes original work of Henry Segerman, Vi Hart and Andrea Hawksley and myself, Kenny Kim and Dennis Adderton visualizing constant curvature spaxes via the inverse exponential map, and further work of Segerman, Hart, Hawksley together with Sabetta Matsumoto extending this to $\mathbb{H}^2\times\mathbb{R}$. The modern incarnation, based on raymarching comes from work of Segerman and Michael Woodard on $\mathbb{H}^3$, and their collaboration with Roice Nelson on an original version of $\mathbb{S}^3$ and $\mathbb{E}^3$.

Seeing in Other Geometries

Understanding what it would be like to live in a different geometry requries first thinking carefully about what it means to see. In reality, light from a lightsoruce travels along straight lines (geodesics) until it hits an object, and we see that object if the light then is not absorbed, but rather bounces off (angle of incidence equals angle of reflection) and travels along another geodesic to our eye, where it is focused onto a sort of screen: our retina. Thus, if we are able to model quickly and accurately movement along geodesics in a space, we can make accurate visuals of what it would look like, if we were actually in that space. GPU processing allows these computations to run massively in parallel, for each pixel, and provides real time rendering capability.

Euclidean Geometry

The cubical tiling of Euclidean space provides a familiar starting point to exploring 3 dimensional geometries. This scene is ray marched, meaning rays travel along straight lines (geodesics) from the screen into the scene to determine what is rendered.

Hyperbolic Geometry

Ideal cubes, meeting six to an edge, tile hyperbolic space. Like the Euclidean view above, these images are generated via raymarching along geodesics, providing an accurate view of the intrinsic geometry of hyperbolic space.

Spherical Geometry

Instead of meeting six to an edge (hyperbolic) or four to an edge (euclidean), when cubes meet three to an edge, the configuration is forced to close up and forms the hypercube, a tiling of the 3-sphere.


The geometry $\mathbb{H}^2\times\mathbb{E}$ is our first non-isotropic geometry, having two directions that are a hyperbolic plane, orthogonal to a Euclidean line. The tiling here is by cubes, meeting six around an edge parallel to the $\mathbb{E}$ direction, and four to an edge for edges contained in $\mathbb{H}^2$


The other product geometry is spherical in two directions, orthogonal to a Euclidean line. We have tiled this space by cubes, meeting three at a time along edges parallel to $\mathbb{E}$ (that is, forming the standard cube tiling of $\mathbb{S}^2$ by squares), and four at a time for edges contained in $\mathbb{S}^2$.

Immersive Geometry in the Allosphere

The Allosphere is a 30ft diameter spherical screen enclosing a wide walkway which can accomodate 30+ visitors for simultaneous fully immersive virtual reality experiences. Designed and operated by Jo Ann Kuchera-Morin of UCSB's Media Arts and Technology (MAT) department (site), it has been used for both artistic and data-visualization purposes. In collaboration with MAT graduate students Kenny Kim and Dennis Adderton, I have worked with the Allosphere to simulate the interior of geometric 3-manifolds and give visual lectures on various topics in low dimensional topology.

The projective models of hyperbolic and spherical geometry can be utilized to produce perspectivally-correct views of these spaces, showing the viewer accurately what such a world would look like from the inside. From the convergence/divergence of geodesics to parallax in curved space, this allows one to experience some of the fundamental concepts of Riemannian geometry.

The Three Sphere

A good introduction to the geometry of the three sphere is the regular polytopes in dimension four. Beginning from stereographic and perspective projections of the hypercube, we move through the other polytopes with viewers, focusing on the geometry of the 120 cell and its decomposition into rings of dodecahedra. We have also worked to produce accurate visualizations of the Hopf and Seifert fibrations. These have been used to give introductory lectures on low dimensional topology, as well as talks on quaternions, complex projective space and the trefoil knot complement.

Hyperbolic Space

Similar work in hyperbolic space allows the visualization of the interior of hyperbolic 3-manifolds. The first images below stem from the figure eight knot complement. The first shows the shows a Cayley graph of the figure-eight knot group embedded in $\mathbb{H}^3$, and the the second shows the preimage of a single object (a tetrahedron) in the manifold under the covering map; rendered in the Klein ball model. The third shows an intrinsic view of the same situation: a single tetrahedron in the figure-eight knot complement (the multiple images arise from light traveling around nontrivial loops in the space). The final image instead comes from an infinite volume manifold, with limit set the Apollonian gasket on the ideal boundary sphere.