What is the foundation under the great pyramid?  How is it
constructed?  Is there any chance it hasn't been fully researched? 
Could it be that the great pyramid is an octahedron?  Sounds crazy I
know, but I just have to ask.

I doubt that it is an octohedron.  This site mentions an internal
hillock (second paragraph next to the third photo), indicating that
Cheop's Pyramid was built on unleveled ground, well, on the stone
plateau.
http://en.wikipedia.org/wiki/Great_Pyramid_of_Giza

Well, it's not crazy to ask. But it is possible to use seismic
tomography to map out such a large, massive base, were it to exist.

The area under the Sphinx was not researched and it is claimed that it
contains either a sand or water filled cave with interesting material.