365 Days of 4D: January 8-14

This is a 3x3x3x3 grid of hypercubes, visualized with perspective projection. This one demonstrates camera movement in 4D.

This is the same hypercube but rotated in different ways. From left to right: corner-first intersection, edge-first intersection, cell-first intersection, and face-first intersection. Visualized with the cross-section method.

This is a hypersphere with hypercrystals protruding from it. The whole structure is rotating. Visualized with the cross-section method.

This is a 4x4x4x4 grid of hypercubes. Visualized with the cross-section method.

This is a 2x2x2x2 grid of hyperspheres with a hypercube in the center. The whole structure is rotating as it passes through the hyperplane. Visualized with the cross-section method.

This is a hypercrystal doing a tour of rotations. Visualized with the cross-section method.

This is a rapidly spinning hypercube. Visualized with the cross-section method.

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365 Days of 4D: January 15-21

A hypersphere doing a tour of rotations. The random spiky parts are due to a bug in Blender. Visualized with parallel projection. A hyperprism with a dodecahedral cross-section rotating in three planes. Visualized with the cross-section method. A pair of hyperspheres moving together, one in X and Y, one in X and W. Visualized with the cross-section method. A hypersphere rotating around a fixed point, moving steadily closer toward the center. Visualized with the cross-section method. A hypercube surrounded by eight hypercrystals. Visualized with the cross-section method. A tightly-packed hypergrid of hypercubes moving and rotating. Visualized with the cross-section method. A hyperprism with a dodecahedral cross-section passing through the hyperplane. Visualized with the cross-section method.

365 Days of 4D Animation - Week of January 1-7

Here's week 1 of my project! Sorry it's taken until June to post. This is a 4D hypercrystal, ticking like a metronome. A hypercrystal (not a mathematical term. I just made it up) is kind of like the 4D version of a 3D crystal structure. Visualized with the cross-section method.

365 Days of 4D: Project Introduction

In January 2018 I was a penniless, jobless PhD dropout who had recently graduated with an MS in Computer Science. During my Master's degree I researched 4D animation. What is 4D, you ask? That's a discussion for another day (but probably soon). I'll admit that, at first, I thought higher-dimensional geometry was an eccentric fringe topic in Mathematics. I didn't see a purpose and, frankly, I thought it was kind of boring. But that didn't stop my MS advisor from trying to recruit me. (Actually, he never knew this. I wonder if he would have tried to recruit me if he knew what I thought about 4D). Rewinding a bit... I was intrigued by the possibility of a Master's Degree, but I had my heart set on doing research in ray tracing. I had a chat with my soon-to-be advisor and we came up with an idea that fit both of our interests: path tracing in 4D. People had been doing Whitted-style ray tracing in 4D as early as 1992. To my knowledge, no one had ever attemp...

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