For n = 3 there are 12 pieces: 3 regular octahedra centred in faces of the cube; 1 regular octahedron centred at vertices of the cube; 8 regular tetrahedra meeting octahedra in their faces. Note piece edge-length = sqrt(2) if cube edge-length = 2 . See the corresponding solid lattice of planes at https://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycomb WFL On 9/16/18, Dan Asimov <dasimov@earthlink.net> wrote:
Let the n-dimensional torus T^n be defined as the n-cube
Q_n = [-1, 1]^n
with its opposite faces identified. (That is, any point x of Q_n with coordinate x_k = ±1 for some k is identified with the point having the same coordinates except with x_k changed to -x_k.)
The vertices of Q_n are the 2^n points {-1, 1}^n.
Now we are going to cut T^n along each of the 2^(n-1) hyperplanes defined as the perpendicular bisectors of each pair of antipodal vertices of Q_n. This requires extending each hyperplane throughout the n-torus.
Puzzle: How many pieces do these planes cut the torus T^n into?
E.g., for n = 2 there are two pieces.
—Dan
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