Surfaces

Algebraic surfaces rendered in a custom path tracer I wrote.

The Thurston Geometries

Lattices of spheres in the eight Thurston geometries.

Knots and Links

The Figure 8 Knot.

The figure 8 knot embedded in the 3 sphere. Rendered by computing a normal tube of small radius about the knot with respect to the 3-sphere's metric, then stereographically projecting into R3 from a point on the knot.

The Whitehead Link.

The Whitehead Link embedded in the 3 sphere. Rendered by computing a normal tube of small radius about the knot with respect to the 3-sphere's metric, then stereographically projecting into R3.

Torus Knots

A program to draw the various torus knots in the 3-sphere: you may enter the parameters (p,q). Rendered by computing a normal tube of small radius about the knot with respect to the 3-sphere's metric, then stereographically projecting into R3.

3-Manifold Geometry

Geodesic Spheres in Sol Geometry.

The geodesic ball of radius $r$ at a point $p$ is the set of all points which can be connected to $p$ by a geodesic of length less than or equal to $r$. The geodesic sphere is the boundary of this set; which we illustrate here for Sol.

Geodesics in Sol Geometry.

Non-rotating black hole with an object in orbit (here drawn as the Earth/Moon system for recognizability). Carefully analyzing the appearence of the background stars and the orbiting object shows the extreme gravitational lensing.

Geodesics in Nil Geometry.

Various geodesics originating from a point in Nil geometry. As this animation makes apparent, standing at a certain distance away from an object, one will see multiple images of the object in their vision, due to the convergence of various geodesics leaving your eye with initially distinct tangent directions.

Spheres in Nil Geometry.

Applying the exponential map to spheres in the tangent space to Nil results generically in self-intersecting singular surfaces, due to the non-uniqueness of geodesics. The exterior portion of this surface is the geodesic sphere of radius $r$ in Nil, which is singular at at most two points.

Geometry of Surfaces

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 length/enclosed area of the cuve on $\mathbb{S}^2$.

A Willmore Torus in S3

Using Pinkall's construction from above, we can seek a hopf torus which additionally minimizes mean curvature in S3, via a differential equation on the space of curves on S2. This shows a tours near such a minimum, visualizes via a perspective (not conformal) projection of R4.

A Closed Geodesic on the Modular Surface

A closed geodesic on the moduli space of tori, illustrated not through conformal embeddings into R3 as above, but ratherr by drawing the associated loop of Weierstrass p-functions.

A transition of conemanifold structures on S2

A geometric transition from the (2,3,7) hyperbolic trianagle tiling to the (2,3,6) euclidean one, computed as the developing maps for a unit area geometric structures on the (2,3,x) spherical conemanifold.

Topology

The Rectangular Peg Problem.

This program attempts to visualize some portions of the argument by Joshua Green and Andrew Lobb that every aspect ratio of rectangle can be inscribed in any smooth Jordan curve. Their paper describing this construction is available here: https://arxiv.org/abs/2005.09193. The "About Menu" describes the controls in terms of the paper.

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.

A boundary slope of the figure 8 knot complement.

An (arbitrarily chosen) boundary slope for the figure 8 knot complement, rendered intrinsically in the incomplete Euclidean metric it inhertis under stereographic projection from a point on the knot.

Rotations of the Hypercube.

A hypercube tiling of S3 with totally geodesic faces, rotating under a 1-parameter subgroup of SO(4).

Lorentzian Geometry

Lightlike Trajectories around Schwarzschild black hole.

The trajectory of lightlike curves around a non-rotating black hole in Schwarzschild coordinates, with time projected off (as the Schwarzschild solution is static, we may do this in a well-defined way).

Schwarzschild black hole.

Non-rotating black hole with an object in orbit (here drawn as the Earth/Moon system for recognizability). Carefully analyzing the appearence of the background stars and the orbiting object shows the extreme gravitational lensing.

Accertion disk around Schwarzschild black hole.

Annular disk around a non-rotating black hole, to show effect of gravitational lensing. (Note: This simulation is not physically accurate, as a rigid disk would likely not be able to sit motionless so close to the event horizon. The disk is for illustrative purposes only to make apparent the trajectories of various light rays).

Algebraic Geometry

Finding 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.

27 Lines on a Cubic Suface

The lines on a cubic surface, all of which are real, animated as the choice of affine patch for RP3 is smoothly varied.

Cubic Sufaces II

A nice animation of a cubic surface as teh affine patch is slowly changed, that arose from work on the animation above.

Elliptic Fibrations

The real points of an elliptically fibered surface, animating the fibration by elliptic curves and their degenerations.

Orientation Double Cover of a Cubic Surface

By a theorem of Klein, real cubic surfaces with all real lines are diffeomorphic to the connect sum of seven projective planes. This visualizes one such surfaces orientation double cover in the 3-sphere.

Complex Analysis

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.

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.

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.

Below are some photos from an exihibition featuring in person 30+ person virtual reality lectures on spherical and hyperbolic geometry