Tomato Packing and Lettuce-Based Crypto

Home , E8 lattice

A. S. Mosunov and L. A. Ruiz-Lopez

University of Waterloo Joint PM & C&O Colloquium

March 30th, 2017 Spring in Waterloo Love is everywhere...

Picture from https://avatanplus.com/files/resources/mid/ 57f917fb6e738157a505ae3c.png. I see love among people...

Picture from http://www.cosmo.com.ua/upload/image/000_(6).jpg. ...and among spheres!

Figure:1 Spheres kissing in R3

1 Picture from http://www.artnet.com/WebServices/images/ll1015941llgE9VCfDrCWvaHBOAD/

tom-otterness-kissing-spheres-(large).jpg. What is a ”kissing number”? What is a ”kissing number”? It is not the number of studio albums by Kiss.

Picture from http: //jenixe.com/wp-content/uploads/2015/11/382884_kiss_band.jpg. What is a ”kissing number”?

I The Kissing Number Problem. Given a positive integer n, what is the largest number of n-dimensional unit spheres that n can “kiss” a unit sphere centred at the origin of R ? I This number is called the kissing number τn. I It is not important what the radius is, as long as all the spheres have the same positive radius. Examples for n = 1,2

2 Figure: τ1 = 2

3 Figure: τ2 = 6

2Figure 1 in [1]. 3Figure 1 in [3]. Example for n = 3

4 Figure: τ3 = 12

4Figure 3 in [3]. Why τ3 = 12? Intuition. How would you stack tomatoes or oranges so to occupy as much space as possible?

Pictures from https://thumbs.dreamstime.com/x/ heap-tomatoes-shared-pyramid-15156144.jpg and http: //blog.kleinproject.org/wp-content/uploads/2012/05/Oranges2.png. Why τ3 = 12?

I Such a packing is called the face-centred cubic packing (FCC). Every sphere is “kissed” by 12 other spheres.

Figure:5 Face-centred cubic packing

I Atoms in a crystal can arrange in a form of FCC packing (nickel, crystallization of palladium or krypton). π I The density of FCC packing is equal to √ ≈ 0.74. 18 I The centres of spheres constitute a lattice in a three dimensional space. 5Picture from http://web2.clarkson.edu/projects/nanomat/ Chapter1html/chapters/images/fccc.jpg. What is a lattice?

n I Let v1,v2,...,vn be linearly independent vectors in R . The set

n Λ = {a1v1 + a2v2 + ... + anvn :(a1,a2,...,an) ∈ Z }

is said to be a lattice with basis v1,v2,...,vn. 1 I Example. The integers Z form a lattice in R with a basis {(1)}. 2 I Example. The points of R which have integer coordinates constitute a lattice with a basis {(1,0),(0,1)}. More examples of lattices √ 2 n 1 3 o I The hexagonal lattice in R has a basis (1,0), 2 , 2 . I By matching centres of spheres with the points of a hexagonal lattice, one achieves the optimal sphere kissing conﬁguration. I In this example, spheres have radius 1/2.

Figure:6 Hexagonal lattice

6Figure 2 in [3]. More examples of lattices 3 I The face-centred cubic lattice in R has a basis {(0,1,1),(1,0,1),(1,1,0)}.

I By matching centres of spheres with the points of a hexagonal lattice, one achieves the optimal sphere√ kissing conﬁguration. I In this example, spheres have radius 1/ 2. I This conﬁguration is not unique!

Figure:7 Face-centred cubic lattice

7 Picture from

https://www.researchgate.net/profile/Mahmood_Rashid/publication/261764396/figure/fig1/AS:

216394273693697@1428603823308/

A-unit-3D-FCC-lattice-with-12-basis-vectors-on-the-Cartesian-coordinates.png. Why τ3 = 12?

I This construction gives us τ3 ≥ 12. How would we prove that τ3 = 12? I The proof is not obvious. In fact, Isaac Newton and David Gregory argued about this in 1694: Newton believed that the correct answer is 12, and Gregory believed that it is 13.

I The equality τ3 = 12 was proved by Sch¨uteand van der Waerden in 1953.

I Related problem. In 1611, Kepler conjectured that no packing of balls of the same radius in three dimensions has density greater than the FCC packing.

I Kepler’s conjecture was proved by Thomas C. Hales in 2000. 3 Other optimal configurations in R 3 I There are optimal lattice packings in R that are different from FCC packing. I There are optimal irregular packings. These packings do not arise from lattices. E.g. the icosahedron configuration.

Figure:8 Icosahedron conﬁguration / Icosahedron graph

8Figures 3 and 10 in [3]. n Other optimal conﬁgurations in R

I Perhaps, the kissing number τn can always be achieved by a lattice packing?

I No. For n = 9, the nonlattice packing known as “P9a” contains spheres that kiss 306 others. However, it is known 9 that kissing numbers of all lattice packings in R do not exceed 272. 10 I Same applies to the nonlattice packing “P10c” in R . 7 I The densities of (conjectured) optimal lattice packings in R , 8 9 R and R are 0.29530, 0.25367 and 0.14577, respectively. 2 3 Compare this to 0.90690 in R and 0.74048 in R . Known kissing numbers

I In the dimensions where kissing numbers are known, there exist highly symmetrical lattices.

n τn Lattice 1 2 Z 2 6 Hexagonal lattice 3 12 FCC lattice 4 24 D4 8 240 E8 24 196560 Leech lattice 4 D4 lattice in R

I The D4 lattice has a basis

{(1,0,0,1),(1,0,1,0),(1,0,0,−1),(0,1,1,0)}.

I Our optimal sphere kissing conﬁguration has 24 lattice points, namely

(±1,±1,0,0), (±1,0,±1,0), (±1,0,0,±1), (0,±1,0,±1), (0,±1,±1,0), (0,0,±1,±1). √ I In this example, spheres have radius 1/ 2, not 1. I Notice that√ any two lattice points diﬀer by a vector of length at least 2 = 2 · √1 . Thus the spheres may touch but they do 2 not overlap. 8 E8 lattice in R

I The E8 lattice is spanned by 8 basis vectors (see the Figure below). 8 I Our optimal sphere kissing conﬁguration includes 2 4 = 112 lattice points of type (06,±12) and 128 vectors of type ((±1/2)8) with an even number of negative components. √ I In this example, spheres have radius 1/ 2, not 1.

9 Figure: Dynkin diagram of E8 lattice with basis vectors at vertices

9Diagram from C. Stewart, Lectures Notes on the Geometry of Numbers, 2009. 24 The Leech lattice R I The Leech lattice is spanned by 24 basis vectors (see the Figure below). I Our optimal sphere kissing conﬁguration includes three types of lattice points, with 97152 + 1104 + 98302 = 196560 points in total.

Figure:10 Matrix whose columns are basis vectors of the Leech lattice

10Diagram from C. Stewart, Lectures Notes on the Geometry of Numbers, 2009. Delsarte’s method

I All of these constructions give a lower bound on τn. What about an upper bound?

I In 1972, the French mathematician Phillipe Delsarte n suggested the following approach. Let x1,...,xm ∈ R be vectors associated to kissing points. Then kxi k = 1 for all i,

(xi ,xj ) = kxi k · kxj k · cosθij = cosθij .

I Also, the matrix (cosθij )i=1,...,m is positive semideﬁnite: j=1,...,m

2 kt1x1 + ... + tmxmk = (t1x1 + ... + tmxm,t1x1 + ... + tmxm) m m = ∑ ∑ ti tj cosθij ≥ 0. i=1 j=1

I Further, for any i 6= j the angle θij between xi and xj satisﬁes π 1 θij ≥ 3 , so cosθij ≤ 2 . The Gagenbauer polynomials

I The Gagenbauer polynomials are deﬁned recursively as follows:

(n) (n) G0 (t) = 1, G1 (t) = t, (n) (n) (n) (2k+n−4)tGk−1(t)−(k−1)Gk−2(t) Gk (t) = k+n−3 .

I In 1943, Schoenberg proved that if (xij ) is a positive semideﬁnite matrix, then so is (f (xij )), where

d (n) f (t) = ∑ ck Gk (t) k=0 is a non-negative combination of Gagenbauer polynomials with c0 > 0. Main Theorem

I Theorem. (Delsarte, Goethals and Seidel, 1977) Let f (t) be a non-negative combination of Gagenbauer polynomials such 1 that c0 > 0 and f (t) ≤ 0 for all t ∈ −1, 2 . Then f (1) τn ≤ . c0

(n) I Proof. Since the sum of the entries of (Gk (cosθij )) is non-negative,

m d m (n) ∑ f (cosθij ) = ∑ ck ∑ Gk (cosθij ) i,j=1 k=0 i,j=1 m (n) 2 ≥ c0 ∑ G0 (cosθij ) = c0m . i,j=1 Main Theorem

I On the other hand,

m ∑ f (cosθij ) = mf (1) + ∑ f (cosθij ) ≤ mf (1). i,j=1 i6=j

I After combining the two estimates, we see that any sphere kissing conﬁguration with m points must satisfy

2 c0m ≤ mf (1).

I By setting m = τn, we see that f (1) τn ≤ . c0 Which polynomials give tight bounds?

I In 1973, Odlyzko and Sloane used the following polynomials to ﬁnd a tight upper bound on τ8 and τ24:

1 12 f (t) = t − t2 t + (t + 1), 8 2 2

1 12 12 12 f (t) = t − t + t2 t + t + (t + 1). 24 2 4 4 2

I For n = 4, one can prove that τ4 ≤ 25, but better estimate is impossible. Recall that D4 lattice yields τ4 ≥ 24.

I In 2003, Oleg Musin announced the proof of τ4 = 24. The paper was published in Annals of Mathematics. He developed a clever modiﬁcation of Delsarte’s method. References

[1] N. Elkies, Lattices, linear codes, and invariants, part I, Notices of the AMS 47 (10), pp. 1238 – 1245, 2000. [2] T. C. Hales, Cannonballs and honeycombs, Notices of the AMS 47 (4), pp. 440 – 449, 2000. [3] F. Pfender, G. M. Ziegler, Kissing numbers, sphere packings, and some unexpected proofs, Notices of the AMS, pp. 873 – 883, 2004. But what about real applications? (Don’t ask me, ask Luis)