Worst Packing Shapes Yoav Kallus Princeton Center for Theoretical - - PowerPoint PPT Presentation

▶

worst packing shapes

Worst Packing Shapes Yoav Kallus Princeton Center for Theoretical - - PowerPoint PPT Presentation

Jun 20, 2023 29 likes •411 views

Worst Packing Shapes Yoav Kallus Princeton Center for Theoretical Sciences Princeton University Physics of Glassy and Granular Materials, YITP July 17, 2013 Y. Kallus (Princeton) Worst Packing Shapes YITP 07/17/2013 1 / 16 From Hilberts

slide-1

SLIDE 1

Worst Packing Shapes

Yoav Kallus

Princeton Center for Theoretical Sciences Princeton University

Physics of Glassy and Granular Materials, YITP July 17, 2013

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 1 / 16

slide-2

SLIDE 2

From Hilbert’s 18th Problem

“How can one arrange most densely in space an infinite number of equal solids

f a given form, e.g., spheres with given

radii or regular tetrahedra with given edges, that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as large as possible?”

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 2 / 16

slide-3

SLIDE 3

Packing non-spherical shapes

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 3 / 16

Damasceno, Engel, and Glotzer, 2012.

slide-4

SLIDE 4

The Miser’s Problem

A miser is required by a contract to deliver a chest filled with gold bars, arranged as densely as possible. The bars must be identical, convex, and much smaller than the chest. What shape of gold bars should the miser cast so as to part with as little gold as possible?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 4 / 16

slide-5

SLIDE 5

Ulam’s Conjecture

“Stanislaw Ulam told me in 1972 that he suspected the sphere was the worst case of dense packing of identical convex solids, but that this would be difficult to prove.”

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 5 / 16

1995 postscript to the column “Packing Spheres”

slide-6

SLIDE 6

Ulam’s Last Conjecture

“Stanislaw Ulam told me in 1972 that he suspected the sphere was the worst case of dense packing of identical convex solids, but that this would be difficult to prove.”

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 5 / 16

1995 postscript to the column “Packing Spheres”

slide-7

SLIDE 7

Ulam’s Last Conjecture

“Stanislaw Ulam told me in 1972 that he suspected the sphere was the worst case of dense packing of identical convex solids, but that this would be difficult to prove.” Naive motivation: sphere is the least free solid (three degrees of freedom vs. six for most solids).

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 5 / 16

1995 postscript to the column “Packing Spheres”

slide-8

SLIDE 8

In 2D disks are not worst

0.9069 0.9024 0.8926(?)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 6 / 16

slide-9

SLIDE 9

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-10

SLIDE 10

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-11

SLIDE 11

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-12

SLIDE 12

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-13

SLIDE 13

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-14

SLIDE 14

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-15

SLIDE 15

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-16

SLIDE 16

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-17

SLIDE 17

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-18

SLIDE 18

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-19

SLIDE 19

Why can we improve over circles?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 7 / 16

slide-20

SLIDE 20

Why can we improve over circles?

f(θ)

ϕ ϕ

6

i=0 f(πi 3 + ϕ)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 8 / 16

slide-21

SLIDE 21

Why can we improve over circles?

f(θ)

ϕ ϕ

6

i=0 f(πi 3 + ϕ)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 8 / 16

slide-22

SLIDE 22

Why can we improve over circles?

f(θ)

ϕ ϕ

6

i=0 f(πi 3 + ϕ)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 8 / 16

slide-23

SLIDE 23

Why can we improve over circles?

f(θ)

ϕ ϕ

6

i=0 f(πi 3 + ϕ)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 8 / 16

slide-24

SLIDE 24

Why can we improve over circles?

f(θ)

ϕ ϕ

6

i=0 f(πi 3 + ϕ)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 8 / 16

slide-25

SLIDE 25

Why can we improve over circles?

f(θ)

ϕ ϕ

6

i=0 f(πi 3 + ϕ)

f (θ) = 1 + ǫcos(8θ)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 8 / 16

slide-26

SLIDE 26

Why can we not improve over spheres?

Lemma

Let f be an even function S2 → R. 12

i=1 f (Rxi) is independent of R if and only

if the expansion of f (x) in spherical harmonics terminates at l = 2.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 9 / 16

YK, arXiv:1212.2551

slide-27

SLIDE 27

Why can we not improve over spheres?

Lemma

Let f be an even function S2 → R. 12

i=1 f (Rxi) is independent of R if and only

if the expansion of f (x) in spherical harmonics terminates at l = 2.

Theorem (YK)

The sphere is a local minimum of the optimal packing fraction among convex, centrally symmetric bodies.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 9 / 16

YK, arXiv:1212.2551

slide-28

SLIDE 28

In 2D disks are not worst

0.9069 0.9024 0.8926(?)

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 10 / 16

slide-29

SLIDE 29

Reinhardt’s conjecture

0.9024

Conjecture (K. Reinhardt, 1934)

The smoothed octagon is an absolute minimum of the optimal packing fraction among convex, centrally symmetric bodies.

Theorem (F. Nazarov, 1986)

The smoothed octagon is a local minimum.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 11 / 16

K. Reinhardt, Abh. Math. Sem., Hamburg, Hansischer Universit¨

at, Hamburg 10 (1934), 216

F. Nazarov, J. Soviet Math. 43 (1988), 2687

slide-30

SLIDE 30

Regular heptagon is locally worst packing

0.8926(?)

Theorem (YK)

Any convex body sufficiently close to the regular heptagon can be packed at a filling fraction at least that of the “double lattice” packing of regular heptagons. Note: it is not proven, but highly likely, that the “double lattice” packing is the densest packing of regular heptagons.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 12 / 16

YK, arXiv:1305.0289

slide-31

SLIDE 31

Regular heptagon is locally worst packing

0.8926(?)

Theorem (YK)

Any convex body sufficiently close to the regular heptagon can be packed at a filling fraction at least that of the “double lattice” packing of regular heptagons.

Conjecture

The regular heptagon is an absolute minimum of the optimal packing fraction among convex bodies.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 12 / 16

YK, arXiv:1305.0289

slide-32

SLIDE 32

Higher dimensions

In 2D, the circle is not a local minimum of packing fraction among c. s. convex bodies. In 3D, the sphere is a local minimum of packing fraction among c. s. convex bodies. What can we say about spheres in higher dimensions?

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 13 / 16

slide-33

SLIDE 33

Higher dimensions

In 2D, the circle is not a local minimum of packing fraction among c. s. convex bodies. In 3D, the sphere is a local minimum of packing fraction among c. s. convex bodies. What can we say about spheres in higher dimensions? Note that in d > 3 we do not know the densest packing of spheres. But we do know the densest lattice packing in d = 4, 5, 6, 7, 8, and 24.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 13 / 16

slide-34

SLIDE 34

Extreme Lattices

A lattice Λ is extreme if and only if ||Tx|| ≥ ||x|| for all x ∈ S(Λ) = ⇒ det T > 1 for T ≈ 1. Contact points S(Λ) of the

ptimal lattice.
Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 14 / 16

YK, arXiv:1212.2551

slide-35

SLIDE 35

Extreme Lattices

A lattice Λ is extreme if and only if ||Tx|| ≥ ||x|| for all x ∈ S(Λ) = ⇒ det T > 1 for T ≈ 1. In d = 6, 7, 8, 24, the optimal lattice is redundantly extreme, and so the ball is reducible.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 14 / 16

YK, arXiv:1212.2551

slide-36

SLIDE 36

d = 4 and d = 5

In d = 4, 5, if ||Tx|| ≥ ||x|| for all x ∈ S(Λ) \ {x0}, and ||Tx0|| > (1 − ǫ)||x0||, then 1 − det T < Cǫ2 (compared with Cǫ for d = 2, 3).

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 15 / 16

YK, arXiv:1212.2551

slide-37

SLIDE 37

d = 4 and d = 5

In d = 4, 5, if ||Tx|| ≥ ||x|| for all x ∈ S(Λ) \ {x0}, and ||Tx0|| > (1 − ǫ)||x0||, then 1 − det T < Cǫ2 (compared with Cǫ for d = 2, 3).

1 − ǫ

(ρ(K) − ρ(B))/ρ(B) ∼ ǫ2 (V (B) − V (K))/V (B) ∼ ǫ The ball is not a local minimum of the

ptimal packing fraction.
Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 15 / 16

YK, arXiv:1212.2551

slide-38

SLIDE 38

Summary of new results

In d = 2, the heptagon is a local minimum of the

ptimal packing fraction, assuming the “double

lattice” packing of heptagons is their densest

packing. The disk is not a local minimum.

In d = 3, the ball is a local minimum among centrally symmetric bodies. In higher dimensions, at least with respect to Bravais lattice packing of centrally symmetric bodies, the ball is not a local minimum.

Y. Kallus (Princeton)

Worst Packing Shapes YITP 07/17/2013 16 / 16