This is to set up the idea that extending the dimensions of objects higher and lower also makes sense with objects much different from cubes.
However, I have not checked
- whether the Cantor Set actually fits in with the Sierpiński triangle and tetrahedron
- what the four dimensional analog would be like
- what the fractal dimensions of those objects would be
The middle image of the following series shows the well-known Sierpiński Triangle: You start with a triangle and recursively remove the center triangle from a triangle, leaving three half-sized copies of the original triangle.
The image on the right extends the Sierpiński Triangle by one dimension to a tetrahedron: You start with a tetrahedron and recursively remove the center octahedron, leaving four half-sized copies of the original tetrahedron.
The image on the left is a depiction of the Cantor Set: You start with a line and recursively remove the center third of it, leaving two third-sized copies of the original line at the ends.
Note that drawing the Cantor Set with a computer program is quite
simple: We count
3**n - 1 and draw all points which
have no digit
1 in them in base three.
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