The Warwick Mathematics Society Website

User login

Upcoming events

  • No upcoming events available
Add to iCalendar
more

There are 497 members of the Warwick Mathematics Society, of which 0 are new today!
We're 99% of the way toward our target of 500 members.
You can join up on the UWSU website.

Are spheres the most inefficient shape for packing?

Post Icon Posted: Submitted by cj on 3 May 2008 - 11:20pm.

Joined: 2006-10-10
Posts: 519

I saw this claim in a respectable journal. Ok, I lie, it was in the letters page of Viz. The question still stands though!

Consider a region of 3-space, and a big bag of identical 3-dimensional shapes (with diameter 2?) Is it possible to find such a shape so that the optimal packing for the shape is more ineffective than the (recently discovered) optimal packing for the unit sphere? Hope that makes sense :/ I presume this question is either absolutely impossible, or has some very easy counterexample. Get to it! _

‹ Calculus of Variations Analysis II ›
Post Icon Posted: 4 May 2008 - 12:28am

Joined: 2007-02-14
Posts: 100

What do you mean more ineffective? Surely a hollow sphere is less effective?
Post Icon Posted: 4 May 2008 - 2:40am

Joined: 2006-10-10
Posts: 519

No you noob >.< I mean that, if you had like a 1000m by 1000m box, and poured in water until it reached the top, which shape, optimally packed, would let you pour in the most water?
Post Icon Posted: 4 May 2008 - 10:17am

Joined: 2007-10-03
Posts: 375

Well that question is just silly because any shape with zero volume would do. Like a (hollow) sphere with a point-sized hole somewhere on it would work quite nicely, or packing it with the standard 3D immersed Klein bottle...
And you can't really find a way to only change the problem slightly in order to get rid of that problem, except if you add some quite arbitrary condition like simple conectedness or something...

But if we think of only simply connected volumes, then I think the sphere might be the worst.
Post Icon Posted: 4 May 2008 - 1:24pm

Joined: 2006-11-02
Posts: 1005

A hollow sphere of non-zero volume with only one hole is still simply connected and it's worse than the normal sphere though. You could try considering only convex shapes but I expect that would make the problem much easier.
Post Icon Posted: 4 May 2008 - 3:47pm

Joined: 2006-10-10
Posts: 519

Solid shapes ._.
Post Icon Posted: 4 May 2008 - 4:54pm

Joined: 2006-11-02
Posts: 1005

"Solid" is quite ambiguous though. Convexity basically guarantees that but removes a lot of other shapes which we'd probably label as solid. I'm not sure if there's a possible rigorous definition of solid.
Post Icon Posted: 5 May 2008 - 3:06am

Joined: 2007-10-03
Posts: 375

Yes sorry I was totally in the fairies about the simple connectedness...
So yes, wonder what a good condition would be for the problem to become interesting.
Post Icon Posted: 5 May 2008 - 2:32pm

Joined: 2006-10-01
Posts: 427

The definition of 'solid' you're after is just a topological 3-manifold with boundary (every point has a neighbourhood homeomorphic to an open set in $ \{ (x,y,z) \in \mathbb{R}^3 : z\geq 0 \} $ with the relative topology), but this is way too easy: even if we ask for simple connectedness we can just 'drill' a little way into our spheres and make them even less efficient.

Convexity poses a more interesting question though, and I believe the answer may be in the affirmative...

Back to the Maths Forum