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Problems related to the Kneser-Poulsen conjecture Maria Belk
A Bunch of Discs
A Bunch of Discs ,[object Object]
A Bunch of Discs ,[object Object]
A Bunch of Discs ,[object Object]
A Bunch of Discs ,[object Object],Probably increases.
A Bunch of Discs ,[object Object]
A Bunch of Discs ,[object Object]
A Bunch of Discs ,[object Object]
A Bunch of Discs ,[object Object],Probably decreases.
Kneser-Poulsen Conjecture ,[object Object],[object Object]
Kneser-Poulsen Theorem ,[object Object],[object Object],[object Object]
Kneser-Poulsen in other spaces? ,[object Object],[object Object],[object Object],[object Object]
Kneser-Poulsen in other spaces? ,[object Object],[object Object],[object Object],[object Object]
Kneser-Poulsen in other spaces? ,[object Object],[object Object],[object Object],[object Object]
Notation ,[object Object],         
Notation ,[object Object],[object Object],         
Notation ,[object Object],[object Object],                   
Notation ,[object Object]
Notation ,[object Object]
Notation ,[object Object]
Continuous Case ,[object Object],[object Object],Why?  Because    =        , where     = size of Wall between discs    and       = change in distance between     and      Wall
Continuous Case Wall Why?  Because    =        , where     = size of Wall between discs    and       = change in distance between     and         
Continuous Case Why?  Because    =        , where     = size of Wall between discs    and       = change in distance between     and         
Continuous Case ,[object Object],   =          The walls come from the Voronoi diagram. Walls
Kneser-Poulsen in other spaces? ,[object Object],[object Object],[object Object],[object Object],[object Object]
On a Cylinder: ,[object Object]
On a Cylinder: ,[object Object]
What is known? ,[object Object],[object Object],[object Object],[object Object],[object Object]
What is known? ,[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Proof of Kneser-Poulsen: ,[object Object],[object Object],[object Object],[object Object]
Lemma 1 ,[object Object],[object Object]
Lemma 1 ,[object Object],Sketch of Proof: Discs (in 1 dimension, equal radii, for simplicity)
[object Object],[object Object],Discs (in 1 dimension, equal radii, for simplicity) Sketch of Proof:
[object Object],[object Object],Discs (in 1 dimension, equal radii, for simplicity) Sketch of Proof:
[object Object],[object Object],[object Object],Discs (in 1 dimension, equal radii, for simplicity) Sketch of Proof:
[object Object],[object Object],[object Object],[object Object],Discs (in 1 dimension, for simplicity) Sketch of Proof:
Sketch of Proof ,[object Object]
Sketch of Proof ,[object Object],[object Object],[object Object]
Lemma 2 ,[object Object],[object Object]
The problems of extending this proof to higher dimensions.
Recall ,[object Object]
Question ,[object Object],[object Object],4 1 3 2 p q 1 4 3 2
Example ,[object Object],4 1 3 2 p q 1 4 3 2
Another Example
Another Example ,[object Object]
In    Dimensions ,[object Object],[object Object]
In    Dimensions ,[object Object],[object Object],[object Object]
In    Dimensions ,[object Object],[object Object],[object Object],[object Object]
In    dimensions ,[object Object],[object Object],[object Object],[object Object]
What does this mean? ,[object Object],[object Object],[object Object]
Hyperbolic and Spherical Spaces
Hyperbolic and Spherical Space ,[object Object],[object Object]
Spherical Space ,[object Object],[object Object],[object Object]
Spherical Space ,[object Object],[object Object],[object Object]
Hyperbolic Space ,[object Object],[object Object]
Tensegrities
Tensegrity ,[object Object],Must remain the same length Bar Can become longer Strut Can become shorter Cable
Tensegrity ,[object Object],This tensegrity can  flex  — the vertices can be moved while maintaining the cable/strut conditions.
Tensegrity ,[object Object],We can stretch the strut until the points are collinear.
Tensegrity ,[object Object],Now, the tensegrity is  rigid  — the vertices cannot be moved while maintaining the cable/strut conditions.
Tensegrity ,[object Object],This tensegrity is also rigid.
Tensegrity ,[object Object],[object Object]
What tensegrities are we interested in? ,[object Object],[object Object],[object Object],[object Object]
What tensegrities are we interested in? ,[object Object],[object Object],[object Object],[object Object]
Global Rigidity ,[object Object],Globally Rigid Globally Rigid Not Globally Rigid
In Euclidean Space ,[object Object],[object Object],[object Object],[object Object]
In Hyperbolic Space ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
In Hyperbolic Space ,[object Object],Globally rigid in    ? (It is globally rigid in    .)
Minimal Tensegrities ,[object Object],or
Minimal Tensegrities ,[object Object],[object Object],or
Minimal Tensegrities ,[object Object],[object Object],[object Object],or
Minimal Tensegrities ,[object Object],[object Object],or
Pogorelov Map ,[object Object],[object Object],[object Object],[object Object],[object Object]
Pogorelov Map ,[object Object],Hyperbolic Space Euclidean Space
Pogorelov Map ,[object Object],Hyperbolic Space Euclidean Space
Result ,[object Object]
Remaining Questions
Euclidean Space ,[object Object],[object Object],[object Object]
Spherical ,[object Object],[object Object],[object Object]
Hyperbolic Space ,[object Object],[object Object],[object Object],[object Object]

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Tensegrities In Hyperbolic Space

  • 1. Problems related to the Kneser-Poulsen conjecture Maria Belk
  • 2. A Bunch of Discs
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  • 23. Continuous Case Wall Why? Because  =     , where   = size of Wall between discs  and    = change in distance between   and      
  • 24. Continuous Case Why? Because  =     , where   = size of Wall between discs  and    = change in distance between   and      
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