constrained spherical circle packings
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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) - PowerPoint PPT PresentationTRANSCRIPT
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.