Tensegrities In Hyperbolic Space

Problems related to the Kneser-Poulsen conjecture Maria Belk

A Bunch of Discs ,[object Object]

A Bunch of Discs ,[object Object],Probably increases.

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]

Notation ,[object Object],         

Notation ,[object Object],[object Object],         

Notation ,[object Object],[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]

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]

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],[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.

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 ,[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]

Hyperbolic Space ,[object Object],[object Object]

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]

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

Pogorelov Map ,[object Object],[object Object],[object Object],[object Object],[object Object]

Pogorelov Map ,[object Object],Hyperbolic Space Euclidean Space

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]

Tensegrities In Hyperbolic Space

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