That's an interesting question, and difficult to visualize even in 3 dimensions. A closely related question is this: If you extend the slices to infinity, into how many regions do they divide the entire space? I.e., include the regions exterior to the square/cube/hypercube. This makes the boundary cuts more interesting. I don't know if it makes the problem harder or easier, but it does create more regions to count. In 2 dimensions, the number of exterior regions is 12. Adding the 4 interior regions gives a total of 16 regions for the entire plane. In 3 dimensions, well, not sure. By the way, even in 3 dimensions, I believe there are regions interior to the cube that do not intersect any vertices or edges of the cube: For each vertex, consider the plane defined by the three vertices that differ from that vertex in exactly one coordinate. For instance, for vertex (0, 1, 0) we get vertices (1, 1, 0), (0, 0, 0), and (0, 1, 1). These three vertices define a plane, and there are 8 such sets of them. Together, these 8 planes create an octahedron-shaped region in the center of the cube. You can think of it as a tiny internal dual of the cube. The vertices of the octahedron are the centers of the faces of the cube. If you then proceed to add the remaining planar slices, this interior octahedron gets sliced up into even more pieces. Do there exist interior regions that do not intersect any of the cube faces? How about in 4 dimensions? Tom Neil Sloane writes:
Take a unit square and cut along the lines joining any two vertices. This cuts the square into 4 pieces.
Now take a unit cube and make plane cuts though any three vertices: how many pieces are produced? (I don't know)
Same question for a unit d-dimensional cube, where the cuts are along hyperplanes through any d vertices.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun