Welcome to MathOverflow

Question

8

3

I wonder if this has been studied:

What is the fewest number of unit cubes from which one can build an $n$-toroid?

The cubes must be glued face-to-face, and the boundary of the resulting object should be topologically equivalent to an $n$-torus, by which I mean a genus-$n$ handlebody in $\mathbb{R}^3$ (as per Kevin Walker's terminological correction). For example, 8 cubes are needed to form a 1-toroid:
CuboidTorus1
And it seems that 13 cubes are needed for a 2-toroid:
CuboidTorus2
I know how intricate is the analogous question for minimizing the number of triangles from which one can build a torus (cf. Császár's Torus), but I am hoping that my much easier question has an answer for arbitrary $n$. Thanks for ideas and/or pointers!

Addendum. Here is Steve Huntsman's 20-cube candidate for genus-5:
CuboidTorus5

edited Jun 6 at 16:46

asked Jun 6 at 2:05

Joseph O'Rourke
43.4k●1●50●215

4

Some terminological nitpicks: "n-torus" usually means "n-dimensional torus", not a genus n surface. The standard term for what you're talking about is "genus n handlebody" (assuming that the 3-dimensional context has already been established). Also, what exactly do you mean by "cuboid"? The Wikipedia article lists two different definitions, and the answer to your question will depend on which definition you intend. From your figures it appears that by "cuboid" you just mean "cube". – Kevin Walker Jun 6 at 2:58

2

@Steve: Your 20-cube "6-torus" (the symmetric one) is actually a "5-torus", or rather a genus 5 handlebody. – Kevin Walker Jun 6 at 3:01

7

The "very porous" cube of side $2L+1$ has $4L^3+9L^2+6L+1$ cubes and genus $2L^3+3L^2$. This gives an asymptotic bound of $2n$ to get a genus-$n$ handlebody. – Marco Golla Jun 6 at 10:45

3

Presumably an approximately spherical object made of the same porous material would give a slightly better asymptotic bound. – Tom Goodwillie Jun 6 at 14:00

2

A very porous approximation to a sphere is less efficient (fewer holes per cube) than a very porous cube. This may be easier to see in the $2$-dimensional version. – Douglas Zare Jun 7 at 7:50

show 14 more comments

Building an $n$-toroid from cubes

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.