Torus in $\mathcal{R}^3$

Question

Hi

I'm interested in packing the 3 space as dense as possible using equally sized tori whose major radius is much bigger than their minor radius in.

Do you have any idea how to attack this problem? I'm fairly new to this topic and I haven't found many papers for non-convex objects. (I think that the torus is somehow the easiest non convex object.)

I thought about writing some computer simulation to get a feeling for the problem. Also I think that the densest packing will be an irregular packing.

Any comments are appreciated.

Thank you

Andy

Answer 1

This is a remark about Villarceau circles

The torus completely decrivbed by two radii, say $(R,r)$. ($R>r$ and your torus is $r$-nbhd of a circle of radius $R$). Let us denote by $d(R,r)$ the density with which the space can be packed by your torii. Then $$\limsup_{r\to 0}\\ d(R,r) \ge C{\cdot}d(R,1)$$ For some constant $C\ge \tfrac{\pi}{4}{\cdot}\frac{R-1}{R+1}$.

To prove it you simply pack $(R,1)$-torus with $(R,r)$-tori with density near $C$; it is easy to do if you know what is Villarceau circles.

Answer 2

Not an answer, just a remark.

It is possible to fill space with disjoint hoops, i.e., congruent geometric circles. Informal sources here are two articles by Evelyn Sander: first, second. This was also the topic of an earlier MO question: "Is it possible to partition $\mathbb{R}^3$ into unit circles?". Perhaps you could explore these constructions for insights for your thickened hoops. Unfortunately, the constructions are not continuous (justifying your intuition re irregularity), and Dan Asimov showed in the 1990's that "it is not possible to fill a region of infinite volume" "using continuous families of disjoint hoops" (to quote Sander).