I'm thinking about the following problem.
First let me introduce the problem with a 2D example. The area of the triangle constructed by connecting the midpoints of a triangle is 1/4 of the the area of the full triangle.
Very straightforward and a picture is a nice clarification. I like how the center triangle can be found four times in the full triangle.
Onto 3 dimensions
This seems a lot less obvious then the 2D example. I printed the model using my 3D printer for better visualizing, but still it is not obvious.
I would like to stack the dual tetrahedron 27 times to visualize this quantity, but thus far I'm unable to do so. My first approach to divide each side in three and then connect corresponding points doesn't seem successful as there are pyramids with a square base to be found.
Is it possible to stack the tetrahedron? And if so, could someone provide me with some pointers on how to proceed.