Fractal Imaginary Cubes
i.h.kyoto-u.ac.jpThis page may be a bit confusing, out of context.
By ‘imaginary cube’, Hideki Tsuiki means a three-dimensional object that is not a cube, but which nevertheless has square projections in three orthogonal directions, just like a cube does. Examples include the cuboctahedron and the regular tetrahedron.
His previous work on non-fractal imaginary cubes is written up at https://www.mdpi.com/1999-4893/5/2/273
I assume this means the four images at the top are the same structures at different angles?
Exactly.
I have a 3D-printed Sierpinski Tetrahedron. It's lovely.
Being able to see a shadow and learning that it has area zero made me think that there is something wrong with the definition of area.
None of the shadows you can physically see in the photos have 0 area because the construction of the fractal isn't perfect. What a perfect construction would look like and if it's ever theoretically possible to make physical are complicated questions. Beyond me for sure, how would an infinitely thin object even interact with light?
I would assume the problem with the idea is in the fractal physics rather than the definition of area, which has been solidly useful for me.
This has inspired me to imagine a fractal chandelier design, e.g made of cut glass
Such a cube, i.e., a set, is closed, right? Sooo, there exists a function on the space that is 0 on the cube, strictly positive otherwise, and infinitely differentiable.