Yes. Let L = { (a_1, a_2, a_3) | a_1 + 2a_2 + 3a_3 = 0 mod 7 } This also generalizes to d dimensions, using L = { (a_1, a_2, ..., a_d) | a_1 + 2a_2 + ... + d*a_d = 0 mod 2d+1 } I wonder what the Voronoi regions look like. On Tue, Sep 22, 2020 at 11:01 AM Dan Asimov <dasimov@earthlink.net> wrote:

Let L be any discrete subgroup of R^3 isomorphic to Z^3.

The the quotient R^3/L is a "flat 3-torus", and it carries a well-defined metric. Its metric can be be described as the result of identifying opposite faces of a parallelepiped (the bounded intersection of three slabs* in R^3) in the obvious way.

Of course the cubical 3-torus R^3/Z^3 is one example, but a flat 3-torus can have surprising properties. Note that the cubical 3-torus can be tessellated by 8 cubes each touching three of the other seven along two common square faces.

Puzzle: ------- Does there exist a flat 3-torus that can be tessellated by seven cubes each touching the other six along a common face?

—Dan

————— * A "slab" in R^3 is the closed region between two parallel planes.

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun