CONSTRAINED SPHERICAL CIRCLE PACKINGS

Tibor Tarnai & Patrick W. Fowler Budapest Sheffield

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Late neolithic stone carving

Ashmolean Museum, Oxford Scotland, around 2500 BC

H. Bosch, Garden of delights

Prado, MadridAround 1600 AD

Pollen grain

Psilotrichum gnaphalobrium, Africa Electron micrograph, courtesy of Dr G. Riollet

The Tammes problem(the unconstrained problem)

How must n equal circles (spherical caps) be packed on a sphere without overlapping so that the angular diameter dn of the circles will be as large as possible?

The graph

• Vertex: centre of a spherical circle

• Edge: great circle arc segment joining the centres of two circles that are in contact

Solutions to the Tammes problem

3 4 5 6 7

8 9 10

11 12 24

d5 = d6

d11 = d12

Solution for n = 24: snub cube

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Spiral circle packing

Zs. Gáspár, 1990

n = 100

(apple peeling)

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Axially symmetric packing

LAGEOS, courtesy of Dr A. Paolozzi Golf ball

n = 426 n = 286

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

• Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Principle of the heating technique and symmetry

Magic numbers

2)]6/(2[ qqTn

)]6/(2)[1( qqTn

22 cbcbT 5,4,3q

1cb 2cb

where

,

(tetrahedron, octahedron, icosahedron)

(circles at the vertices)

(no circles at the vertices)

Subgraphs of multisymmetric packings

Octahedral packing

30 48 78

144 198 432

Icosahedral packing

60 120 180

360 480 750

Packing of 72 circles

tetrahedral octahedral icosahedral

d = 24.76706° d = 24.85375° d = 24.83975°

Packing of 192 circles

octahedral icosahedral

d =15.04103° d =15.17867°

Packing of 492 circles

both icosahedral

Icosahedral packings for large n

R.H. Hardin & N.J.A. Sloan, 1995

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Pentagon packing

Random packing Dandelion, Salgótarján Sculptor: István Kiss

Modified heating technique

Local optima for n = 24

Octahedral symmetry

Local optima for n = 72approximation of icosahedral papilloma virus

A map computed from electron cryo-micrographs, courtesy of Dr. R.A. Crowther

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Gamma Knife

Graphs of antipodal packings

Further results by J.H. Conway, R.H. Hardin & N.J.A. Sloane,1996

d5x2 = d6x2

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Problem of packing of triplets of circles

How must 3N non-overlapping equal circles forming N triplets be packed on a sphere so that the angular diameter of the circles will be as large as possible under the constraint that, within each triplet, the circle centres lie at the vertices of an equilateral triangle inscribed into a great circle of the sphere?

Method

AS surface area of the sphereAi area of the circles Aij area of double overlaps Aijk area of triple overlaps

ijkijiS AAAA Penalty

0,0 that sopenalty Minimize ijkij AA

Graphs of conjectural solutions

d2x3 = d3x3

d3x3 = d4x3

Graph of conjectural solution

Rattling triangle

The graphs as polyhedra

Compounds of triangles

2 3 4

5 6 7

The most symmetrical view

2 3 4

5 6 7

Solution for N = 2

Solution is not unique.

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Problem of packing of quartets of circles

How must 4N non-overlapping equal circles forming N quartets be packed on a sphere so that the angular diameter of the circles will be as large as possible under the constraint that, within each quartet, the circle centres lie at the vertices of a regular tetrahedron?

Linnett’s theory of valence

Valence model of diatomic molecules

Linnett’s valence configu-rations constructed from quartets of spin-up and spin-down electrons

Graphs of conjectural solutions

d4x4 = d5x4

Graphs of conjectural solutions

d7x4 = d8x4

Graphs as polyhedra

Compounds of tetrahedra

N = 1 N = 2

N = 3 N = 4

d4x4 = d5x4

Compounds of tetrahedra

N = 5 N = 6

N = 7 N = 8

d7x4 = d8x4

d4x4 = d5x4

Memorial to Thomas Bodley

Merton College Chapel, Oxford, 1615

Soccer ball

The graph of 8 quartets

Contents• Introduction

• Spiral packing

• Axially symmetric packing• Multisymmetric packing (TT & Zs. Gáspár, 1987)

– Pentagon packing (T.T. & Zs. Gáspár, 1995)

• Antipodal packing (T.T., 1998)

• Packing of triplets (P.W.F. & T.T., 2005)

• Packing of quartets (P.W.F. & T.T., 2003)

• Packing of twins (P.W.F. & T.T., 2005)

• Conclusions

Problem of packing of twin circles

How must 2N non-overlapping equal circles forming N twins be packed on a sphere so that the angular diameter of the circles will be as large as possible, where a twin is defined as two circles that are touching each other?

Expectation

• The diameter of circles in packing of N twins is equal to the diameter of circles in unconstrained packing of n = 2N circles.

• For given N, the number of different solutions of twin packings is equal to the number of perfect matchings in the graph of the unconstrained packing of n = 2N circles.The expectation is fulfilled in the case of the known solutions of the unconstrained packing problem: 2N = 4, 6, 8, 10, 12, 24

Number of solutions

4 6 8 10 12 14

16 18 20 22 24

1 (3) 1 (8) 3 (14) 6 (20) 5 (125) 8 (64)

11 (92) 76 (142) 54 (558) 120 (120) 385 (7744)

First number: reduced by symmetry

Number in parentheses: total number for labelled vertices

Perfect matchings of the icosahedron

Conclusions

• Different constrained packing problems were surveyed.

• A number of putative solutions were presented.

• Some applications in science, art and technology were shown.

• A bonus for the researcher: the beauty of solutions.

Acknowledgements

• We thank Günther Koller and Sándor Kabai for help with computer graphics.

• The work was supported by OTKA grant no. T046846.