Extra ways to see: An Artist's Guide to Map Operations

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Map operations convert one subdivision of a surface into another, thereby suggesting alternate ways to decorate or build the surface. Over twenty map operations are described visually via their truchet tiles. Talk given at 2011 ISAMA Chicago.

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  • Extra ways to see: An Artist's Guide to Map Operations

    1. 1. Extra Ways to See:An Artist’s Guide to Map Operations James Mallos ISAMA 2011 Chicago, Illinois
    2. 2. In 1954, Buckminster Fuller disclosed the geodesic dome and ignited the topic of symmetrical designs on the sphere in America and Europe.
    3. 3. Elsewhere, symmetrical designs on the sphere had never gone out of style. Japanese temari balls. www.japan-cc.com
    4. 4. It’s now 57 years later, and we’re rich with designs on the57 years later, we are rich in inventions on the sphere. sphere.
    5. 5. We have: tensegrities...Diamond pattern Zig-zag pattern Circuit pattern Anthony Pugh, An Introduction to Tensegrity Star pattern Lawrence Pendred, www.pendred.net
    6. 6. ...tensegrity membranes... Shigematsu, Tanaka, and Noguchi, IASS-IACM 2008
    7. 7. ...unit-weavers...Iq’s, Holger Strom Twogs, James Mallos
    8. 8. ...unit origami... John Horigan, www.ozonehouse.com/john/origami
    9. 9. ...nexorades... TaffGoch sketchup.google.com/3dwarehouse
    10. 10. ...weave patterns... Akleman, Chen, Xing, and Gross, Siggraph 09
    11. 11. ...puzzles... George W. HartGoldberg Puzzle, George W. Hart, www.georgehart.com
    12. 12. ...and mechanisms. Juno’s Spinner, Junichi Yananose, www.polyhedra.jp
    13. 13. There are plenty of other surfaces...
    14. 14. Gyroid surface, mathworld.wolfram.com...will our designs work there?
    15. 15. Map operations• A map operation converts one subdivision of a surface (the base map) into another (the resultant map.) Ra() Base Map Resultant Map
    16. 16. What is a map?
    17. 17. A map is:A graph drawn on a closed surface in such a way that: • the vertices are represented as distinct points on the surface, • the edges are represented as curves on the surface intersecting only at the vertices, • if we cut the surface along the graph, what remains is a disjoint union of connected components, called faces, each topologically equivalent to an open disk. Abstracted from Lando and Zvonkin, Graphs on Surfaces and Their Applications
    18. 18. Which of these are maps?
    19. 19. Maps are embedded general graphs.They can have vertices and faces of valence {1, 2, 3, ...}
    20. 20. These maps all have one or more triangle faces:
    21. 21. Often the base map is a computer surfacemodel and the goal is to achieve certaincharacteristics in the resultant map. 3D models courtesy of INRIA via the Aim@Shape Shape Repository
    22. 22. bip artite ches s-co lorab le The most important map operations yield maps with guaranteed properties —no matter the original map.3-va lent -faced quadrangle
    23. 23. Operation Guaranteed Property Su() Bipartite Pa() Chess-colorable Ra() Bipartite and quadrilateral-faced Me() Chess-colorable and 4-valent Ki() Triangle-faced Tr() 3-valent Le() 3-valent Or() Bipartite and quadrilateral-faced Ex() Chess-colorable and 4-valent Gy() Pentagon-faced Sn() 5-valent Mt() Triangle-faced and chirally chess-colorable Be() 3-valent
    24. 24. Ki() Performed Algorithmically M Ki(M)
    25. 25. Ki(M) yields a map where each original edge is the diagonal of its own quadrilateral.
    26. 26. The same map that cuts the surface intodisks, also implicitly chops the surfaceinto quadrilaterals.Did I mention they are all quadrilaterals?This sounds like job for truchet tiles!
    27. 27. Truchet Tiles for Map OperationsId DuSu PaRa MeKi Tr
    28. 28. Id(M)
    29. 29. Du(M)
    30. 30. Su(M)
    31. 31. Pa(M)
    32. 32. Ra(M)
    33. 33. Me(M)
    34. 34. Ki(M)
    35. 35. Tr(M)
    36. 36. Truchet Tiles for some Chiral Map Operations Pr Ca Gy Sn
    37. 37. Gy(M)
    38. 38. Sn(M)
    39. 39. Some Practical Applications
    40. 40. Unit-weaver / Map Operation CorrespondencesIq’s TwogsRa() Tr()
    41. 41. Tensegrity / Map Operation Correspondences Diamond pattern Sn() Chiral edge = strut Circuit pattern Me() Zig-zag pattern Tr() Chiralized by strut
    42. 42. Zig-ZagTensegrity
    43. 43. CircuitTensegrity
    44. 44. DiamondTensegrity
    45. 45. StarTensegrity
    46. 46. Elementary Tensegrities From Sn()Sn( ) =Sn( ) =Sn( ) = Tensegrity simulations with Springie
    47. 47. Weaving / Map Operation Correspondence Chess coloring of Me() Rinus Roelofs, Bridges 2010
    48. 48. Truchet Tile for Plain WeavingChess Coloring (Me())
    49. 49. PlainWeaving
    50. 50. Will our spherical designs really work on other surfaces? • conformal distortions must be tolerated (i.e., in length and area, but not angle) • variations in both extrinsic and intrinsic curvature must tolerated: convex/concave/flat extrinsic curvature; and gaussian positive/ negative/zero intrinsic curvature.
    51. 51. Thanks!James Mallos

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