I have a challenge for you today.
You have probably seen a fractal known as
the Menger Sponge. You start with a
cube (a square on all six faces). Then
you divide each face into 9 squares and cut the middle square out – creating a
square hole that goes all the way through.
Then repeat the process on a smaller scale – in the 8 other squares on
each face you divide each square into 9 smaller squares and cut out the
middle square creating another hole (smaller that the first holes) that goes
all the way through. After several
repetitions of this process you end up with a “Holey Cube” like the one
below.

A Sierpinski Pyramid is a fractal that
kind of does the same things with Tetrahedron, and cutting triangular holes.
What I need help with is this:
I would like to see what a Sierpinski
Octahedron would look like.
An octahedron is an eight faced Platonic
solid that has equilateral triangles as it’s eight faces.
If you divide each face into four
triangles and cut the middle triangle out, leaving a tetrahedral shaped hole,
I believe that you would end up with six smaller octahedrons that touch at
their vertices and edges. If you
continue the process on a smaller and smaller scale each time you would (I
think) end up with a fractal that would look similar to the Sierpinski
Pyramid. But I am having some problems
picturing it in my mind.
(Octahedron after one itteration.)
I don’t have the skills to draw one on
the computer. I hope there is someone
out there who could draw one for me, and show about three iterations.
If you are interested, and think you can
help me, please contact me at mbiom.edu@gmail.com.

Thanks,
David
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