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.
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.
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.
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