Fractals
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 wellknown Sierpiński Triangle: You start with a triangle and recursively remove the center triangle from a triangle, leaving three halfsized 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 halfsized 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 thirdsized copies of the original line at the ends.
Note that drawing the Cantor Set with a computer program is quite
simple: We count y
from 0
to 3**n  1
and draw all points which
have no digit 1
in them in base three.
1 2 3 4 5 
