PHOTO SERIES
PROJECTIVE TRIANGLES
TAYLOR SERIES
STILLS
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ANIMATIONS
SHORT FILMS
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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 THREE SPHERE & HYPERBOLIC SPACE
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.
In the 3-sphere we have worked to produce accurate visualizations of the six regular polytopes in $\mathbb{R}^4$ as well as 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.
Similar work in hyperbolic space allows the visualization of the interior of hyperbolic 3-manifolds. The first image above shows a Cayley graph of the figure-eight knot group embedded in $\mathbb{H}^3$, and the the second shows what one would see if there were a single rubber duck in the figure-eight knot complement (the multiple images arise from light traveling around nontrivial loops in the space). The final image is the Cayley graph of a group whose ideal boundary is the Appolonian gasket on $\mathbb{H}^2_\infty$.
FUTURE WORK
Ongoing work in the Allosphere is building the necessary foundation to produce perspectivally-correct visualizations of all eight Thurston geometries in dimension 3. This will allow us to produce fully immersive visualizations of the interior of the geometric components of 3-manifolds arising from geometrization.
I do most of my visual work in Mathematica. For people interested in playing around with some of this stuff themselves, below are some mathematica files together with short explanations of the mathematics and implementation.
HOPF & SEIFERT FIBRATIONS
The Hopf fibration $\mathbb{S}^3\to\mathbb{S}^2$ is a filling of the three sphere by pairwise-linked circles with parameter space s\$S^2$. The three sphere admits infinitely many twisted versions of this, called Seifert fibrations, where the complement of a Hopf link is filled with (p,q) torus knots. The mathematica notebooks below render fibers of the Hopf and Seifert fibrations, both in stereographic projection and in a perspectivally correct view (what you would actually see if you were inside of the round three sphere).
The 5 platonic solids are the five maximally symmetric tilings of $\mathbb{S}^2$. The maximally symmetric tilings of $\mathbb{S}^3$ are the 4-dimensional analogs of these familiar shapes - of which there are six. The mathematica notebooks below contain code to render the 5 cell, hypercube, 16 cell and 24 cell in stereographic projection.
Reflecting in the sides of certain hyperbolic polygons leads to beautiful tilings of the hyperbolic plane. The mathematica file here produces pictures of tilings based on the $(p,q,r)$ triangle tilings, in the Klein model, examples are shown below. The second notebook above is renders these pictures in the upper half plane and Poincare disk models (though as these models show more area of $\mathbb{H}^2$ away from the edges, it takes much longer to draw complete pictures).
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The hyperbolic plane can be realized as a subset of $\mathbb{R}\mathsf{P}^2$ (the Klein model) and in some cases tilings of $\mathbb{H}^2$ can be deformed into tilings of other properly convex subsets of projective space. The mathematica documents below contain code for producing visualizations of the deformations of $(p,q,r)$ triangle tilings. The calculation of this one parameter family of deformations for the reflection groups was completed using the description in the Master's thesis of Anton Valerievich Lukyanenko, available here.
The graph of a single variable complex function is a subset of $\mathbb{C}\times\mathbb{C}\cong\mathbb{R}^4$, making direct visualization impossible. Domain coloring is a useful technique that allows us to understand complex functions using color and saturation on the domain to represent the values of the output. The mathematica files below compute domain colorings of complex functions, using code inspired by an excellent post by Simon Wood on stack exchange (View Here). The images below are the example outputs given in the file.
