Tensegrities In Hyperbolic Space

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Tensegrities In Hyperbolic Space

  1. 1. Problems related to the Kneser-Poulsen conjecture Maria Belk
  2. 2. A Bunch of Discs
  3. 3. A Bunch of Discs <ul><li>Suppose we rearrange the discs so that the distances between centers increases. </li></ul>
  4. 4. A Bunch of Discs <ul><li>Suppose we rearrange the discs so that the distances between centers increases. </li></ul>
  5. 5. A Bunch of Discs <ul><li>What do you think happens to the area of the union of the discs? </li></ul>
  6. 6. A Bunch of Discs <ul><li>What do you think happens to the area of the union of the discs? </li></ul>Probably increases.
  7. 7. A Bunch of Discs <ul><li>What do you think happens to the area of the intersection? </li></ul>
  8. 8. A Bunch of Discs <ul><li>What do you think happens to the area of the intersection? </li></ul>
  9. 9. A Bunch of Discs <ul><li>What do you think happens to the area of the intersection? </li></ul>
  10. 10. A Bunch of Discs <ul><li>What do you think happens to the area of the intersection? </li></ul>Probably decreases.
  11. 11. Kneser-Poulsen Conjecture <ul><li>Conjecture (Kneser 1955, Poulsen 1954) If the distances between the centers of the discs have not decreased, then the area of the union has either increased or remained the same. </li></ul><ul><li>Also conjectured: The area of the intersection has either decreased or remained the same. </li></ul>
  12. 12. Kneser-Poulsen Theorem <ul><li>Theorem (Bezdek and Connelly, 2002) If the distances between the centers of the discs have not decreased, then: </li></ul><ul><li>The area of the union has either decreased or remained the same. </li></ul><ul><li>The area of the intersection has either increased or remained the same. </li></ul>
  13. 13. Kneser-Poulsen in other spaces? <ul><li>We can ask the analogous question in: </li></ul><ul><li>Higher dimensions   ,  </li></ul><ul><li>Spherical Space   </li></ul><ul><li>Hyperbolic Space   </li></ul>
  14. 14. Kneser-Poulsen in other spaces? <ul><li>We can ask the analogous question in: </li></ul><ul><li>Higher dimensions   ,  </li></ul><ul><li>Spherical Space   ,  </li></ul><ul><li>Hyperbolic Space   </li></ul>
  15. 15. Kneser-Poulsen in other spaces? <ul><li>We can ask the analogous question in: </li></ul><ul><li>Higher dimensions   ,  </li></ul><ul><li>Spherical Space   ,  </li></ul><ul><li>Hyperbolic Space   ,  </li></ul>
  16. 16. Notation <ul><li>p = (   ,   ,   ,  ,   ) is a configuration of points in   . </li></ul>         
  17. 17. Notation <ul><li>p = (   ,   ,   ,  ,   ) is a configuration of points in   . </li></ul><ul><li>q is also a configuration of  points in   . </li></ul>         
  18. 18. Notation <ul><li>We say that q is an expansion of p , if </li></ul><ul><li>   ,        ,    </li></ul>                   
  19. 19. Notation <ul><li>We say that p continuously expands to q if there is a continuous motion from p to q , in which the distances change monotonically. </li></ul>
  20. 20. Notation <ul><li>We say that p continuously expands to q if there is a continuous motion from p to q , in which the distances change monotonically. </li></ul>
  21. 21. Notation <ul><li>We say that p continuously expands to q if there is a continuous motion from p to q , in which the distances change monotonically. </li></ul>
  22. 22. Continuous Case <ul><li>In   ,   , and   , Csikós has shown: </li></ul><ul><li>Theorem (Csikós 1999, 2002) If there is a continuous expansion between the two configurations, then the volume behaves appropriately. </li></ul>Why? Because  =     , where   = size of Wall between discs  and    = change in distance between   and   Wall
  23. 23. Continuous Case Wall Why? Because  =     , where   = size of Wall between discs  and    = change in distance between   and      
  24. 24. Continuous Case Why? Because  =     , where   = size of Wall between discs  and    = change in distance between   and      
  25. 25. Continuous Case <ul><li>For more than 2 balls: </li></ul> =     The walls come from the Voronoi diagram. Walls
  26. 26. Kneser-Poulsen in other spaces? <ul><li>Theorem (Csikós, 2006) Euclidean, Hyperbolic, and Spherical spaces are the only reasonable spaces to ask the question in. </li></ul><ul><li>Counterexamples exist if the space is </li></ul><ul><li>Not homogeneous </li></ul><ul><li>Not isotropic </li></ul><ul><li>Not simply connected </li></ul>
  27. 27. On a Cylinder: <ul><li>An expansion, where the area of the union decreases: </li></ul>
  28. 28. On a Cylinder: <ul><li>An expansion, where the area of the union decreases: </li></ul>
  29. 29. What is known? <ul><li>In Euclidean space: </li></ul><ul><li>Gromov:  balls in    dimensions. </li></ul><ul><li>Bern and Sahai: Discs in two dimensions if there is a continuous expansion. </li></ul><ul><li>Csikós: Any dimension if there is a continuous expansion. </li></ul><ul><li>Bezdek and Connelly: Discs in two dimensions (no continuous expansion needed). </li></ul>
  30. 30. What is known? <ul><li>Spherical and Hyperbolic Space: </li></ul><ul><li>Csikós: Any dimension if there is a continuous expansion. </li></ul>
  31. 31. Outline <ul><li>Sketch of proof for dimension 2. </li></ul><ul><li>The problems in extending this proof to higher dimensions? </li></ul><ul><li>Hyperbolic and spherical spaces? </li></ul><ul><li>Tensegrities in Hyperbolic space </li></ul><ul><li>Remaining Questions </li></ul>
  32. 32. Outline <ul><li>Sketch of proof for dimension 2. </li></ul><ul><li>The problems in extending this proof to higher dimensions? </li></ul><ul><li>Hyperbolic and spherical spaces? </li></ul><ul><li>Tensegrities in Hyperbolic space </li></ul><ul><li>Remaining Questions </li></ul>
  33. 33. Outline <ul><li>Sketch of proof for dimension 2. </li></ul><ul><li>The problems in extending this proof to higher dimensions? </li></ul><ul><li>Hyperbolic and spherical spaces? </li></ul><ul><li>Tensegrities in Hyperbolic space </li></ul><ul><li>Remaining Questions </li></ul>
  34. 34. Outline <ul><li>Sketch of proof for dimension 2. </li></ul><ul><li>The problems in extending this proof to higher dimensions? </li></ul><ul><li>Hyperbolic and spherical spaces? </li></ul><ul><li>Tensegrities in Hyperbolic space </li></ul><ul><li>Remaining Questions </li></ul>
  35. 35. Outline <ul><li>Sketch of proof for dimension 2. </li></ul><ul><li>The problems in extending this proof to higher dimensions? </li></ul><ul><li>Hyperbolic and spherical spaces? </li></ul><ul><li>Tensegrities in Hyperbolic space </li></ul><ul><li>Remaining Questions </li></ul>
  36. 36. Proof of Kneser-Poulsen: <ul><li>The proof due to Bezdek and Connelly has two main components: </li></ul><ul><li>Lemma: If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul><ul><li>Lemma: If q is an expansion of p , in dim  , then there is a continuous expansion in  . </li></ul><ul><li>Since        , the conjecture holds. </li></ul>
  37. 37. Lemma 1 <ul><li>Lemma (Bezdek and Connelly) : If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul><ul><li>Idea of Proof: Use Cylindrical Shells to relate the volume in dimension  to the surface area in dimension  . </li></ul>
  38. 38. Lemma 1 <ul><li>Lemma (Bezdek and Connelly) : If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul>Sketch of Proof: Discs (in 1 dimension, equal radii, for simplicity)
  39. 39. <ul><li>Lemma (Bezdek and Connelly) : If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul><ul><li>Create Voronoi Diagram </li></ul>Discs (in 1 dimension, equal radii, for simplicity) Sketch of Proof:
  40. 40. <ul><li>Lemma (Bezdek and Connelly) : If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul><ul><li>Create Voronoi Diagram </li></ul>Discs (in 1 dimension, equal radii, for simplicity) Sketch of Proof:
  41. 41. <ul><li>Lemma (Bezdek and Connelly) : If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul><ul><li>Create Voronoi Diagram </li></ul><ul><li>Create balls in 3 dimensions </li></ul>Discs (in 1 dimension, equal radii, for simplicity) Sketch of Proof:
  42. 42. <ul><li>Lemma (Bezdek and Connelly) : If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul><ul><li>Create Voronoi Diagram </li></ul><ul><li>Create balls in 3 dimensions. </li></ul><ul><li>Consider cylindrical shells. </li></ul>Discs (in 1 dimension, for simplicity) Sketch of Proof:
  43. 43. Sketch of Proof <ul><li>Result of Cylindrical Shells for each Voronoi region separately: </li></ul>
  44. 44. Sketch of Proof <ul><li>Differentiate to get:    Vol. in dim.  = Surface area in dim  </li></ul><ul><li>If there is a continuous motion in dim  , then the surface area changes monotonically between the 2 configurations. </li></ul><ul><li>Thus, the volume in dim  does not change. </li></ul>
  45. 45. Lemma 2 <ul><li>Lemma (well-known): If q is an expansion of p , in dim  , then there is a continuous expansion in  . </li></ul><ul><li>Proof: Place p and q in orthogonal subspaces, then the following motion works: </li></ul>
  46. 46. The problems of extending this proof to higher dimensions.
  47. 47. Recall <ul><li>Lemma (Bezdek and Connelly) If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul>
  48. 48. Question <ul><li>Lemma (Bezdek and Connelly) If there is a continuous expansion from p to q in dim  , then volume in dim  does not decrease. </li></ul><ul><li>Question: If p is an expansion of q in   , in what dimension is there a continuous expansion? </li></ul>4 1 3 2 p q 1 4 3 2
  49. 49. Example <ul><li>This expansion in 2 dimensions requires 3 dimensions for a continuous expansion. </li></ul>4 1 3 2 p q 1 4 3 2
  50. 50. Another Example
  51. 51. Another Example <ul><li>Continuous contraction in dimension 4. </li></ul>
  52. 52. In  Dimensions <ul><li>An analogous construction in   : </li></ul><ul><li>Start with Tetrahedron: </li></ul>
  53. 53. In  Dimensions <ul><li>An analogous construction in   : </li></ul><ul><li>Start with Tetrahedron. </li></ul><ul><li>Attach “flaps” to each face. </li></ul>
  54. 54. In  Dimensions <ul><li>An analogous construction in   : </li></ul><ul><li>Start with Tetrahedron. </li></ul><ul><li>Attach “flaps” to each face. </li></ul><ul><li>The result requires 6 dimensions to continuously contract. </li></ul>
  55. 55. In  dimensions <ul><li>Analogous construction in   : </li></ul><ul><li>Start with a  -simplex. </li></ul><ul><li>Attach “flaps” to each facet. </li></ul><ul><li>Theorem (Belk and Connelly) Requires  dimensions to continuously contract, because the bar framework is rigid in   . </li></ul>
  56. 56. What does this mean? <ul><li>Question: If q is an expansion of p in   , in what dimension is there a continuous expansion? </li></ul><ul><li>Answer: There is a continuous expansion in   , and we cannot do better than that. </li></ul><ul><li>If we wanted to use a similar proof for higher dimensions, we would need to improve the other lemma. </li></ul>
  57. 57. Hyperbolic and Spherical Spaces
  58. 58. Hyperbolic and Spherical Space <ul><li>Theorem (Csikós, 2002) If there is a continuous expansion in   (or   ), then Kneser-Poulsen holds in   (or   ). </li></ul><ul><li>Question: If q is an expansion of p in   or   , in what dimension is there a continuous expansion? </li></ul>
  59. 59. Spherical Space <ul><li>Question: If q is an expansion of p in   , in what dimension is there a continuous expansion? </li></ul><ul><li>Since p and q sit in   , there is a continuous expansion in   , which remains on the sphere   . </li></ul><ul><li>Therefore: There is a continuous expansion in   . </li></ul>
  60. 60. Spherical Space <ul><li>This is not enough to prove Kneser-Poulsen in   , but we do get: </li></ul><ul><li>Theorem (Csikós 2002) Kneser-Poulsen holds for  balls in   . </li></ul><ul><li>Since  balls continuously expand in   . </li></ul>
  61. 61. Hyperbolic Space <ul><li>Question: If q is an expansion of p in   , is there a continuous expansion from p to q in   for any  ? </li></ul><ul><li>This is unknown. </li></ul>
  62. 62. Tensegrities
  63. 63. Tensegrity <ul><li>A tensegrity  p  is a configuration p and a graph  where each edge of the graph is labeled as a cable, strut, or bar. </li></ul>Must remain the same length Bar Can become longer Strut Can become shorter Cable
  64. 64. Tensegrity <ul><li>Example: </li></ul>This tensegrity can flex — the vertices can be moved while maintaining the cable/strut conditions.
  65. 65. Tensegrity <ul><li>Example: </li></ul>We can stretch the strut until the points are collinear.
  66. 66. Tensegrity <ul><li>Example: </li></ul>Now, the tensegrity is rigid — the vertices cannot be moved while maintaining the cable/strut conditions.
  67. 67. Tensegrity <ul><li>A more complicated example: </li></ul>This tensegrity is also rigid.
  68. 68. Tensegrity <ul><li>Another example: </li></ul><ul><li>This tensegrity is rigid in   , but it flexes in   . </li></ul>
  69. 69. What tensegrities are we interested in? <ul><li>Suppose q is an expansion of p . Create the tensegrity: </li></ul><ul><li>The configuration is p . </li></ul><ul><li>If the distance between two vertices increases from p to q , make the edge between the vertices a strut. </li></ul><ul><li>If the distance remains the same, make the edge a bar. </li></ul>
  70. 70. What tensegrities are we interested in? <ul><li>This creates tensegrities where: </li></ul><ul><li> is the complete graph (that is, there is an edge between any two vertices). </li></ul><ul><li>Every edge is either a bar or strut. </li></ul><ul><li>There exists another configuration q , which satisfies the bar and strut conditions. This means the tensegrity is not globally rigid . </li></ul>
  71. 71. Global Rigidity <ul><li>Global Rigidity: A tensegrity is globally rigid if there is no other configuration satisfying the cable, strut, and bar conditions. </li></ul>Globally Rigid Globally Rigid Not Globally Rigid
  72. 72. In Euclidean Space <ul><li>In   , for large enough  , every tensegrity is either: </li></ul><ul><li>Globally rigid, or </li></ul><ul><li>Flexible </li></ul><ul><li>Because: If it is not globally rigid, there is a motion connecting the two configurations. </li></ul>
  73. 73. In Hyperbolic Space <ul><li>Open Problem: In   (for large enough  ), is every tensegrity either globally rigid or flexible? </li></ul><ul><li>Lemma (Belk) The following tensegrities are either globally rigid or flexible in   . </li></ul><ul><li>Tensegrities with fewer than 4 points, and </li></ul><ul><li>Tensegrities with  points in general position that span   . </li></ul><ul><li>Why? </li></ul><ul><li>Case Checking </li></ul><ul><li>Minimal Tensegrities (tensegrities in general position in  dimensions) </li></ul><ul><li>Pogorelov Map (  :            , complicated function with some nice properties) </li></ul>
  74. 74. In Hyperbolic Space <ul><li>First tensegrity for which it is not known: </li></ul>Globally rigid in   ? (It is globally rigid in   .)
  75. 75. Minimal Tensegrities <ul><li>Start with 2 simplices with exactly one point in common. </li></ul>or
  76. 76. Minimal Tensegrities <ul><li>Start with 2 simplices with exactly one point in common. </li></ul><ul><li>Replace the edges of simplices with struts. </li></ul>or
  77. 77. Minimal Tensegrities <ul><li>Start with 2 simplices with exactly one point in common. </li></ul><ul><li>Replace the edges of simplices with struts. </li></ul><ul><li>Add cables between all remaining vertices. </li></ul>or
  78. 78. Minimal Tensegrities <ul><li>These are the only rigid tensegrities in dim    with  vertices in general position. </li></ul><ul><li>They are globally rigid in   and   . </li></ul>or
  79. 79. Pogorelov Map <ul><li> p , q   ( r , s ) </li></ul><ul><li>p , q = configurations in   </li></ul><ul><li>r , s = configurations in   </li></ul><ul><li>With the property that: </li></ul><ul><li>   ,        ,        ,        ,    </li></ul>
  80. 80. Pogorelov Map <ul><li>The problem is that the Pogorelov Map can significantly change the configuration. </li></ul>Hyperbolic Space Euclidean Space
  81. 81. Pogorelov Map <ul><li>We can fix this problem for minimal tensegrities, by specifying where the point of intersection of the simplicies goes in Euclidean space.. </li></ul>Hyperbolic Space Euclidean Space
  82. 82. Result <ul><li>Theorem (Belk): Kneser-Poulsen holds for 4 balls in   (for any  ). </li></ul>
  83. 83. Remaining Questions
  84. 84. Euclidean Space <ul><li>Euclidean Space: </li></ul><ul><li>Kneser-Poulsen in    ? </li></ul><ul><li>Could the simplex with “flaps” provide a counter-example? </li></ul>
  85. 85. Spherical <ul><li>Spherical Space: </li></ul><ul><li>Kneser-Poulsen in    ? </li></ul><ul><li>If q is an expansion of p in   , is there a continuous expansion in   ? </li></ul>
  86. 86. Hyperbolic Space <ul><li>Hyperbolic Space: </li></ul><ul><li>Kneser-Poulsen in    ? </li></ul><ul><li>If q is an expansion of p in   , is there a continuous expansion in   (for any  )? </li></ul><ul><li>Is global rigidity for tensegrities equivalent in Hyperbolic space and Euclidean space? </li></ul>

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