Hideki Tsuiki
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.
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.