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Letters
The Ice Cube ProofSERGEI TABACHNIKOV, PIERRE DELIGNE,AND SINAI ROBINS
The Mathematical Intelligencer encourages comments
about the material in its pages. Letters to the editor
should be sent to the editor-in-chief.
EEditor-in-Chief’s Note: In a recent article in thesepages, ‘‘Proofs (Not) from the Book,’’
1Sergei
Tabachnikov gives a proof2
of Pick’s formula* thatbegins: ‘‘Place a unit cube of ice at each lattice point in theplane and let the ice melt. The water will evenly distributein the plane and, in particular, the amount of water insidethe polygon will equal its area.’’ We thank ProfessorsTabachnikov, Deligne, and Robins for permission topublish verbatim the flurry of correspondence that ensued.
Sept. 6, 2014
Dear Tabachnikov,
I liked the symmetry idea you explained in the Intelligencerfor Pick’s formula, but it makes no sense that you put aUNIT CUBE of ice at each lattice point. You should put acylinder with base a circle of radius r centered at eachlattice point, and choose r small enough (so that the cyl-inder for an interior/exterior point sits wholly inside/outside). At a vertex, a cube would not give the correctfraction inside.
Best, Pierre Deligne
***
Sept. 7, 2014
Dear Professor Deligne,
Thank you for your remark: I fully agree with it.
On a related note, Pick’s formula isn’t valid in higherdimensions. One wonders whether a kind of averaging(melting ice) could be applicable in multi-dimensionalsetting. For example, it seems that the argument works forlattice polytopes whose facets are centrally symmetric (thisimplies that the polytope is centrally symmetric as well)…
Once again, thank you for your interest.
Best regards, yours, Sergei
***
Sept. 7, 2014
Dear Sinai,
It seems to me that the ‘‘melting ice’’ argument could work inany dimension for polytopes whose facets are centrally sym-metric (which implies that the polytope is centrally symmetricaswell).Does itmake sense toyou? Is anything like this known?
Thanks, Sergei
***
Sept. 7, 2014
Hi Sergei,
Yeah, it sounds like it’s correct—do you need *all* of thefaces of the polytope to be centrally symmetric perhaps?
Inconclusive experimental demonstration of the ice-cube proofof Pick’s Theorem. The authors in this exchange discuss theshape and size of the chunks of ice, but seem unaware that icecubes slip and slide as they melt. Photograph by Stan Sherer.
*Pick’s formula computes the area of a plane polygon with vertices at points of the standard lattice Z2 as a function of the number lattice points in the polygon’s
interior and on its boundary.
� 2014 Springer Science+Business Media New York
DOI 10.1007/s00283-014-9517-6
Then the polytope is called a Zonotope. There is a theoremthat says that summing all the solid angles of an integerZonotope, at all lattice points, gives precisely its volume. Isthis what you’re after? I was going to present it at somepoint in our course, in fact.
The easiest way that I know to prove this is to use theFourier transform of (the indicator function of) the Zono-tope, plus the Poisson summation, and then use Stokes’sformula to retract the computation of the Fourier transformto the boundary components, which cancel out in pairs dueto the symmetry of the Zonotope.
Regards, Sinai
***
Sept. 7, 2014
I wrote notes in between your lines. Best, Pierre Deligne
‘‘For example, it seems that the argument works forlattice polytopes whose facets are centrally symmetric…’’
I agree.
‘‘…(this implies that the polytope is centrally symmetricas well)…’’
How do you see this? I expect you must assume here thepolytope to be convex, which is not needed for Pick forcentrally symmetric faces. Example: a cube with a smallercube sticking out of one of the faces (or carved into it).
***
Sept. 7, 2014
Dear Professor Deligne,
Yes, you are right: convexity should be assumed. In caseyou are interested, I am attaching two relevant papers, byG. C. Shephard3 and by P. McMullen.4
Concerning Pick’s formula in this more general setting, Iasked Sinai Robins (who coauthored a book on this sub-ject).5 He said that this fact is known for zonotopes, theconvex polytopes whose all faces (not only codimensionone) are centrally symmetric.
It seems that the ‘‘ice cylinders’’ proof is considerablysimpler.
However, I was after a more general statement (and youseem to agree): it suffices to have only facets (codim 1faces) to be symmetric. Perhaps it’s too good to be true…?What do you think?
Thank you, yours, Sergei
***
Sept. 7, 2014
Dear Sinai,
Thank you. Two comments:
• the ‘‘ice cube proof’’ seems to be simpler than Fourier &Poisson, doesn’t it?
• I was after a more ambitious conjecture, perhaps naively:it suffices to have only facets centrally symmetric. Do youhave a counterexample at hand? I guess, dim 4 is the firstcase to consider.
Best, Sergei
***
Sept. 7, 2014
Yes, indeed you are right—it suffices to have only thefacets centrally symmetric, and then the sum of the solidangles (the proportion of a small ice cube centered at eachinteger point which lies inside P) at all integer points,equals the volume of P.
Is this what you meant? (I checked it with Fourierarguments, just to make sure.);)
Regards, Sinai
***Sept. 8, 2014
Dear Professor Deligne,
I got a confirmation from an expert.It was stupid of me not to think about it earlier and not
to mention in the article. :(
Best regards, yours, Sergei
***
Sept. 8, 2014
Dear Tabachnikov,
Thank you for the papers. About your comment:‘‘It seems that the ‘ice cylinders’ proof is considerably
simpler.’’I do not know how really different the 2 proofs are. ‘‘Ice
melting’’ gives a convincing picture. To transform it into aproof, I do not want to get bogged down into hydrody-namics. I was thinking it would be technically simpler todiffuse a circle centered around each point by Brownianmotion, and one knows this is the same as diffusing by theheat equation evolution. For it, the change in heat inside isgiven by a flux across the boundary, to which the symmetryargument can be applied. On the other hand, a natural wayto look at the heat equation, and for instance to see theconvergence to a uniform distribution, is by Fouriertransform.
The volume is an affine invariant. Solid angles are not.In one of the papers you sent me, it is said that forzonotopes, some sums of interior angles are affine invari-ants, I have not yet looked at how the proof of thatinvariance relates to our story. By the way, instead of acircle around each point, one can in your argument useany centrally symmetric chunk of ice, and instead of aninterior angle use the fraction of the chunk inside. What-ever sum of interior angles is affine invariant for zonotopesshould remain the same when they are replaced by thosefractions.
THE MATHEMATICAL INTELLIGENCER
Best, Pierre Deligne
***
Sept. 9, 2014
Hi Sergei,
The reference I found for this theorem is Corollary 7.6,page 28, of the paper by Alexander Barvinok and JamesPommersheim, An Algorithmic Theory of Lattice Points inPolyhedra.6 But the initial feeling that I had, in my quickresponse email to yours, also gives a proof, which is slightlymore Fourier-based, but roughly the same ideas.
There’s also another related result to your question,whichI’ll go over tomorrow in the Geometry-Topology seminar,and which leads to further research questions. It came sort ofunexpectedly out of some work that I did on multiple-tilingsof Euclidean space by translations of a convex polytope. It’sTheorem 6.2 in http://arxiv.org/abs/1103.3163
(appeared in Combinatorica, 2012).7
But basically if you apply this theorem with the discreteset \Lambda = the integer lattice, then since we know thatthe k in that theorem must equal the volume of the poly-tope P, we have the corollary that:
The sum of the solid angles at all the integer points ofALL translates of a real polytope P = volume of P if and onlyif P is a multi-tiler.
In particular, by the main theorem of that paper, it implies(but is not necessarily implied by) that P is centrally sym-metric and its facets are centrally symmetric. Note that we areallowing any real polytope, but also we are assuming more,namely that the volume constraint holds for all translates.
But we still have the following open question: can weclassify all integer polytopes P such that vol P = the sum ofall solid angles (relative to P) at all integer points ?
One thing that I find fascinating is that the solid angle isnot an affine invariant, and yet these sorts of theoremsabout their sum being the volume persist for some classesof polytopes.
Ok, ciao for now, Sinai
***
Sept. 10, 2014
Dear Professor Deligne,
I am forwarding to you the latest message from Robins.The Corollary in the Barvinok-Pommersheim paper,
which I am attaching, concerns more general valuations (Iguess, your ‘‘centrally symmetric chunks of ice’’ is just that),but the next Corollary 7.7 concerns precisely the solid
angles. The authors call it a ‘‘101st generalization of Pick’sformula.’’
Thank you, yours, Sergei
REFERENCES1 Tabachnikov, S., ‘‘Proofs (Not) from the Book,’’ Math. Intelligencer
36(2), 9–14 (2014).2 This proof is from C. Blatter, Math. Mag. 70, 200 (1997).3 Shephard, G. C., Polytopes with Centrally Symmetric Faces. Canad.
J. Math. 19, 1206–1213 (1967).4 McMullen, P., Polytopes with Centrally Symmetric Facets. Israel J.
Math. 23(3–4), 337–338 (1976).5 Beck, M., and Robins, S., Computing the continuous discretely:
Integer-point enumeration in polyhedra, Undergraduate Texts in
Mathematics. Springer, New York, 2007.6 Barvinok, A., and Pommersheim, J. E., An algorithmic theory of
lattice points in polyhedra, New Perspectives in Algebraic Combi-
natorics. Berkeley, CA, 1996-1997, 91–147; Math. Sci. Res. Inst.
Publ. 38, Cambridge Univ. Press, Cambridge, 1999.7 Gravin, N., Robins, S., and Shiryaev, D., Translational tilings by a
polytope, with multiplicity. Combinatorica 32, 629–649 (2012).
Sergei Tabachnikov
Department of Mathematics
Pennsylvania Sate University
University Park, PA 16802
USA
e-mail: [email protected]
Pierre Deligne
School of Mathematics
Institute for Advanced Study
Einstein Drive, Princeton, NJ 08540
USA
e-mail: [email protected]
Sinai Robins
Department of Mathematics
Brown University
151 Thayer Street
Providence, RI 02912
USA
and
Division of Mathematical Sciences
Nanyang Technological University
21 Nanyang Link
Singapore 637371
e-mail: [email protected]
� 2014 Springer Science+Business Media New York
