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CONSTRAINED SPHERICAL CIRCLE PACKINGS

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)

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CONSTRAINED SPHERICAL CIRCLE PACKINGS

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  1. CONSTRAINED SPHERICAL CIRCLE PACKINGS Tibor Tarnai & Patrick W. Fowler Budapest Sheffield

  2. 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

  3. Late neolithic stone carving Scotland, around 2500 BC Ashmolean Museum, Oxford

  4. H. Bosch, Garden of delights Around 1600 AD Prado, Madrid

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

  6. 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?

  7. The graph • Vertex: centre of a spherical circle • Edge: great circle arc segment joining the centres of two circles that are in contact

  8. Solutions to the Tammes problem 6 3 4 5 7 d5 = d6 10 8 9 11 12 24 d11 = d12

  9. Solution for n = 24: snub cube

  10. 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

  11. Spiral circle packing (apple peeling) n = 100 Zs. Gáspár, 1990

  12. 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

  13. Axially symmetric packing n = 426 n = 286 LAGEOS, courtesy of Dr A. Paolozzi Golf ball

  14. 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

  15. Principle of the heating technique and symmetry

  16. Magic numbers (circles at the vertices) (no circles at the vertices) where (tetrahedron, octahedron, icosahedron) ,

  17. Subgraphs of multisymmetric packings

  18. Octahedral packing 30 48 78 144 198 432

  19. Icosahedral packing 60 120 180 360 480 750

  20. Packing of 72 circles tetrahedral octahedral icosahedral d = 24.76706°d = 24.85375°d = 24.83975°

  21. Packing of 192 circles octahedral icosahedral d =15.04103°d =15.17867°

  22. Packing of 492 circles both icosahedral

  23. Icosahedral packings for large n R.H. Hardin & N.J.A. Sloan, 1995

  24. 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

  25. Pentagon packing Random packing Dandelion, Salgótarján Sculptor: István Kiss

  26. Modified heating technique

  27. Local optima for n = 24 Octahedral symmetry

  28. Local optima for n = 72approximation of icosahedral papilloma virus A map computed from electron cryo-micrographs, courtesy of Dr. R.A. Crowther

  29. 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

  30. Gamma Knife

  31. Graphs of antipodal packings d5x2 = d6x2 Further results by J.H. Conway, R.H. Hardin & N.J.A. Sloane,1996

  32. 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

  33. 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?

  34. Method AS surface area of the sphere Aiarea of the circles Aijarea of double overlaps Aijkarea of triple overlaps

  35. Graphs of conjectural solutions d2x3 = d3x3 d3x3 = d4x3

  36. Graph of conjectural solution Rattling triangle

  37. The graphs as polyhedra

  38. Compounds of triangles 2 3 4 5 6 7

  39. The most symmetrical view 2 3 4 5 6 7

  40. Solution for N = 2 Solution is not unique.

  41. 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

  42. 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?

  43. Linnett’s theory of valence

  44. Valence model of diatomic molecules Linnett’s valence configu- rations constructed from quartets of spin-up and spin-down electrons

  45. Graphs of conjectural solutions d4x4 = d5x4

  46. Graphs of conjectural solutions d7x4 = d8x4

  47. Graphs as polyhedra

  48. Compounds of tetrahedra N = 1 N = 2 N = 3 N = 4 d4x4 = d5x4

  49. Compounds of tetrahedra N = 6 N = 5 d4x4 = d5x4 N = 8 N = 7 d7x4 = d8x4

  50. Memorial to Thomas Bodley Merton College Chapel, Oxford, 1615

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