As a kid, I grew up exploring coastal British Columbia, sometimes bringing home treasure that I found. I would sometimes feel an intense need to find out, for example, what the view is like from a lighthouse in the distance. Similarly, as an adult I find myself intensely curious about, for example, "What would it look like to watch a black hole form in my living room?". In both cases, an answer exists - I just have to go out and find it (either by scrambling over boulders to get to the lighthouse, or by doing the math / writing the code).
In this spirit, I see mathematical art less as a creative process and more one of exploration and discovery. Just like when I was a kid, I rarely find anything interesting. However, I occasionally stumble up a treasure which I want to take home (with a 3D printer, or perhaps just a raytracer, depending on the context).
Films
3D Cross-Sections of a 4D Menger Sponge
00:05:00
Music: Composition by Han geul Lee ( used with permission)
Linktr.ee/xpkgg
Sometime in 2022 I became aware of the now famous central 2D diagonal cross-section of a 3D Menger sponge discovered in 2007 by Sébastien Pérez-Duarte. I became curious about what 3D diagonal cross-sections of a 4D Menger sponge would look like, and in summer 2022 I sat down, wrote the code, and got an answer. However, knowing the answer and understanding it are very different things.
To my astonishment, one of the 3D cross-sections of the 4D sponge contains four copies of the original central 2D cross-section found by Pérez-Duarte. I took me several months to understand why. This film aims to explain in an intuitive and visual manner this surprising result, and then walks through the full family of 3D cross-sections. All animations are done using a raytracer I wrote in C++ at the end of undergrad over ten years ago, which I managed to find recently on an old external hard-drive.
3D Cross-Sections of a second type of 4D Menger Sponge
00:05:00
Music: Composition by Han geul Lee ( used with permission)
Linktr.ee/xpkgg
Sometime in 2022 I became aware of the now famous central 2D diagonal cross-section of a 3D Menger sponge discovered in 2007 by Sébastien Pérez-Duarte. I became curious about what 3D diagonal cross-sections of a 4D Menger sponge would look like. Reasoning by analogy, I expected that the six-sided polygon with six-sided 2D star holes found in the 3D case would generalize to an eight-sided polyhedron with eight-pointed 3D star internal cavities. However, when I first did the calculation in summer 2022, I got something entirely different (which I go over in my other video), and I was at first quite confused.
I later came to realize that there are multiple ways to generalize the Menger sponge to 4D, and only one such generalization yields the cross-sections I expected. This video describes how this second type of 4D Menger sponge is constructed, explores why it yields the sections described above, and finally showcases the whole family of its 3D cross-sections.
