2023 Joint Mathematics Meetings
Hideki Tsuiki
Artists
Hideki Tsuiki
Professor of Mathematics and Computer Science
Kyoto University
Kyoto, Japan
Statement
When I found H-fractal (Hexagonal bipyramid imaginary cube fractal) and T-fractal (Triangular antiprismoid imaginary cube fractal), I was attracted by their beauty, in particular, the way their projected images change. Recently, I succeeded in characterizing the directions from which these objects are projected to positively-measured fractals, and I was also moved by the beauty of this result. I wondered how I can convey this beauty to a wider audience, and I came up with the idea of making precise models of these fractals with 3D-printers, make YouTube videos of them, and link a paper in the comment of the video.
Artworks
These three objects are fractal imaginary cubes, which are fractal objects with square projections along three orthogonal directions. Let \(F(k,D)\) be the fractal defined as the unique fixed-point of the operator \(F(X) = \frac{X + D}{k}\) on non-empty compact subsets of \(\mathbb R^3\) for + the Minkowski sum. The green one is known as Sierpinski Tetrahedron (4096 pcs), which is \(F(2,D_S)\) for \(D_S\) the four vertices of a regular tetrahedron. The other two, T-fractal (yellow, 729 pcs) and H-fractal (red, 729 pcs) are the only two fractal imaginary cubes of the form \(F(3,D)\) for \(D\) a nine-point set in \(\mathbb R^3\). Continuously changing fractals obtained as projected images of these fractals are beautiful and interesting.