## Wednesday, September 23, 2015

### I Hava a Challenge for You Today

9/23/2015

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