Tensegrities and differential forms. Oleg Karpenkov, University of - PowerPoint PPT Presentation

Tensegrities and differential forms. Oleg Karpenkov, University of Liverpool 11 June 2019 Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Given a graph G in the plane. Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Given a graph G in the plane. We have the following three equivalent statements: ◮ G is infinitesimally flexible; Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Given a graph G in the plane. We have the following three equivalent statements: ◮ G is infinitesimally flexible; ◮ G admits a non-zero tensegrity; Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Given a graph G in the plane. We have the following three equivalent statements: ◮ G is infinitesimally flexible; ◮ G admits a non-zero tensegrity; ◮ Rigidity matrix G is not of full rank. Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Given a graph G in the plane. We have the following three equivalent statements: ◮ G is infinitesimally flexible; ◮ G admits a non-zero tensegrity; ◮ Rigidity matrix G is not of full rank. Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Needle Tower of Kenneth Snelson (1969) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

“Tensegrity” = “Tension” + “Integrity” Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definitions Definition Let G = ( V , E ) be an arbitrary graph with n vertices (without loops and multiple edges). G Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definitions Definition Let G = ( V , E ) be an arbitrary graph with n vertices (without loops and multiple edges). ◮ A configuration P is a collection ( p 1 , p 2 , . . . , p n ) , p i ∈ R d . p 4 p 3 p 2 p 1 G P G ( P ) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definitions Definition Let G = ( V , E ) be an arbitrary graph with n vertices (without loops and multiple edges). ◮ A configuration P is a collection ( p 1 , p 2 , . . . , p n ) , p i ∈ R d . ◮ A tensegrity framework is the pair ( G , P ), denote G ( P ). p 4 p 3 p 2 p 1 G G ( P ) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definitions Definition Let G = ( V , E ) be an arbitrary graph with n vertices (without loops and multiple edges). ◮ A configuration P is a collection ( p 1 , p 2 , . . . , p n ) , p i ∈ R d . ◮ A tensegrity framework is the pair ( G , P ), denote G ( P ). ◮ A stress w acting on G ( P ). p 4 p 3 w 3 , 4 w 2 , 4 w 1 , 4 w 2 , 3 w 1 , 3 p 2 w 1 , 2 p 1 G G ( P ) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definitions Definition Let G = ( V , E ) be an arbitrary graph with n vertices (without loops and multiple edges). ◮ A configuration P is a collection ( p 1 , p 2 , . . . , p n ) , p i ∈ R d . ◮ A tensegrity framework is the pair ( G , P ), denote G ( P ). ◮ A stress w acting on G ( P ). ◮ A force-load F acting on G ( P ). p 4 p 3 f 3 , 4 f 2 , 4 f 1 , 4 f 2 , 3 f 1 , 3 p 2 f 1 , 2 p 1 G G ( P ) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definition ◮ w is called a self stress if at every vertex: � w i , j ( p j − p i ) = 0 . { j | j � = i } w 1 , 3 ( p 1 − p 3 ) w 3 , 4 ( p 4 − p 3 ) w 2 , 3 ( p 2 − p 3 ) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definition ◮ w is called a self stress if at every vertex: � w i , j ( p j − p i ) = 0 . { j | j � = i } ◮ F is called an equilibrium force-load if at every vertex: � f i , j = 0 . { j | j � = i } f 1 , 3 f 3 , 4 f 2 , 3 Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Definition ◮ w is called a self stress if at every vertex: � w i , j ( p j − p i ) = 0 . { j | j � = i } ◮ F is called an equilibrium force-load if at every vertex: � f i , j = 0 . { j | j � = i } ◮ ( G ( P ) , w ) is called a tensegrity if w is a self stress for G ( P ) w 1 , 3 ( p 1 − p 3 ) w 3 , 4 ( p 4 − p 3 ) w 2 , 3 ( p 2 − p 3 ) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Projective classical mechanics Projective classical mechanics Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: why to use projective static Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: why to use projective static 4 3 Let: 5 6 1 2 G Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: why to use projective static 4 3 Let: 5 6 1 2 G Generic cases: 1) the lines p 1 p 2 , p 3 p 4 , p 5 p 6 have a common point; p 4 p 3 p 5 p 6 p 2 p 1 Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: why to use projective static 4 3 Let: 5 6 1 2 G Generic cases: 1) the lines p 1 p 2 , p 3 p 4 , p 5 p 6 have a common point; p 4 p 3 p 5 p 6 p 2 p 1 2) the lines p 1 p 2 , p 3 p 4 , p 5 p 6 are parallel; p 4 p 3 p 5 p 6 p 1 p 2 G Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: why to use projective static 4 3 Let: 5 6 1 2 G Generic cases: 1) the lines p 1 p 2 , p 3 p 4 , p 5 p 6 have a common point; p 4 p 3 p 5 p 6 p 2 p 1 2) the lines p 1 p 2 , p 3 p 4 , p 5 p 6 are parallel; p 4 p 3 p 5 p 6 p 1 p 2 G Q: How to merge generic cases? Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: why to use projective static 4 3 Let: 5 6 1 2 G Generic cases: 1) the lines p 1 p 2 , p 3 p 4 , p 5 p 6 have a common point; p 4 p 3 p 5 p 6 p 2 p 1 2) the lines p 1 p 2 , p 3 p 4 , p 5 p 6 are parallel; p 4 p 3 p 5 p 6 p 1 p 2 G Q: How to merge generic cases? A: Projective static(I.Izmestiev) Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: projective static Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: projective static Definition. We say that a decomposable 2-form in Λ 2 ( R 3 ) is a force in R P 2 . Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: projective static Definition. We say that a decomposable 2-form in Λ 2 ( R 3 ) is a force in R P 2 . p = ( a , b ) → dp = adx + bdy + dz . Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: projective static Definition. We say that a decomposable 2-form in Λ 2 ( R 3 ) is a force in R P 2 . p = ( a , b ) → dp = adx + bdy + dz . Definition. Let F = dp 1 ∧ dp 2 be a nonzero force. Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Details: projective static Definition. We say that a decomposable 2-form in Λ 2 ( R 3 ) is a force in R P 2 . p = ( a , b ) → dp = adx + bdy + dz . Definition. Let F = dp 1 ∧ dp 2 be a nonzero force. ◮ The projective line ˜ p 1 ˜ p 2 is the line of force . Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Usage of projective statics ◮ To remove parallel/non-parallel cases. Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Usage of projective statics ◮ To remove parallel/non-parallel cases. ◮ To generalize to higher dimensional case. Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Quantum tensegrities What happens if we switch to differential forms with non-constant coefficients? Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Quantum tensegrities What happens if we switch to differential forms with non-constant coefficients? The answer was hinted by observing Point-hyperplane frameworks, slider joints, and rigidity preserving transformations by Y. Eftekhari, B. Jackson, A. Nixon, B. Schulze, Sh. Tanigawa, W. Whiteley Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Quantum tensegrities What happens if we switch to differential forms with non-constant coefficients? The answer was hinted by observing Point-hyperplane frameworks, slider joints, and rigidity preserving transformations by Y. Eftekhari, B. Jackson, A. Nixon, B. Schulze, Sh. Tanigawa, W. Whiteley We know the answer for df ( x , y ) + dz . Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Quantum tensegrities. Definitions. F = ( f 1 , . . . , f n ). Then dF is ( dF 1 = df 1 + dz , dF 2 = df 2 + dz , . . . , dF n = df n + dz ) . Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

Quantum tensegrities. Definitions. F = ( f 1 , . . . , f n ). Then dF is ( dF 1 = df 1 + dz , dF 2 = df 2 + dz , . . . , dF n = df n + dz ) . Definition Let F = ( f 1 , . . . , f n ), functions with finitely many critical points. A tensegrity ( G ( dF ) , w ) is a triple: a graph ( G , F , w ), where — f i are at vertices of G ; — w i , j are at edges of G . Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.