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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

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Tensegrities and differential forms.

Oleg Karpenkov, University of Liverpool 11 June 2019

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Given a graph G in the plane.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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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.

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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.

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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.

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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.

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Needle Tower of Kenneth Snelson (1969)

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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“Tensegrity” = “Tension” + “Integrity”

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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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.

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Definitions

Definition

Let G = (V , E) be an arbitrary graph with n vertices (without loops and multiple edges).

◮ A configuration P is a collection (p1, p2, . . . , pn), pi ∈ Rd.

G G(P) P p1 p2 p3 p4

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Definitions

Definition

Let G = (V , E) be an arbitrary graph with n vertices (without loops and multiple edges).

◮ A configuration P is a collection (p1, p2, . . . , pn), pi ∈ Rd. ◮ A tensegrity framework is the pair (G, P), denote G(P).

G G(P) p1 p2 p3 p4

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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SLIDE 12

Definitions

Definition

Let G = (V , E) be an arbitrary graph with n vertices (without loops and multiple edges).

◮ A configuration P is a collection (p1, p2, . . . , pn), pi ∈ Rd. ◮ A tensegrity framework is the pair (G, P), denote G(P). ◮ A stress w acting on G(P).

G G(P) p1 p2 p3 p4

w1,2 w1,3 w1,4 w2,3 w2,4 w3,4

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Definitions

Definition

Let G = (V , E) be an arbitrary graph with n vertices (without loops and multiple edges).

◮ A configuration P is a collection (p1, p2, . . . , pn), pi ∈ Rd. ◮ 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).

G G(P) p1 p2 p3 p4

f1,2 f1,3 f1,4 f2,3 f2,4 f3,4

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Definition

◮ w is called a self stress if at every vertex:

  • {j|j=i}

wi,j(pj − pi) = 0.

w1,3(p1 − p3) w2,3(p2 − p3) w3,4(p4 − p3)

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Definition

◮ w is called a self stress if at every vertex:

  • {j|j=i}

wi,j(pj − pi) = 0.

◮ F is called an equilibrium force-load if at every vertex:

  • {j|j=i}

fi,j = 0.

f1,3 f2,3 f3,4

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Definition

◮ w is called a self stress if at every vertex:

  • {j|j=i}

wi,j(pj − pi) = 0.

◮ F is called an equilibrium force-load if at every vertex:

  • {j|j=i}

fi,j = 0.

◮ (G(P), w) is called a tensegrity if w is a self stress for G(P) w1,3(p1 − p3) w2,3(p2 − p3) w3,4(p4 − p3)

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Projective classical mechanics

Projective classical mechanics

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: why to use projective static

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: why to use projective static

Let:

1 2 3 4 5 6 G

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: why to use projective static

Let:

1 2 3 4 5 6 G

Generic cases: 1) the lines p1p2, p3p4, p5p6 have a common point;

p1 p2 p3 p4 p5 p6

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: why to use projective static

Let:

1 2 3 4 5 6 G

Generic cases: 1) the lines p1p2, p3p4, p5p6 have a common point;

p1 p2 p3 p4 p5 p6

2) the lines p1p2, p3p4, p5p6 are parallel;

p1 p2 p3 p4 p5 p6 G

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: why to use projective static

Let:

1 2 3 4 5 6 G

Generic cases: 1) the lines p1p2, p3p4, p5p6 have a common point;

p1 p2 p3 p4 p5 p6

2) the lines p1p2, p3p4, p5p6 are parallel;

p1 p2 p3 p4 p5 p6 G

Q: How to merge generic cases?

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: why to use projective static

Let:

1 2 3 4 5 6 G

Generic cases: 1) the lines p1p2, p3p4, p5p6 have a common point;

p1 p2 p3 p4 p5 p6

2) the lines p1p2, p3p4, p5p6 are parallel;

p1 p2 p3 p4 p5 p6 G

Q: How to merge generic cases? A: Projective static(I.Izmestiev)

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: projective static

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: projective static

  • Definition. We say that a decomposable 2-form in Λ2(R3) is a

force in RP2.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: projective static

  • Definition. We say that a decomposable 2-form in Λ2(R3) is a

force in RP2. p = (a, b) → dp = adx + bdy + dz.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: projective static

  • Definition. We say that a decomposable 2-form in Λ2(R3) is a

force in RP2. p = (a, b) → dp = adx + bdy + dz.

  • Definition. Let F = dp1 ∧ dp2 be a nonzero force.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Details: projective static

  • Definition. We say that a decomposable 2-form in Λ2(R3) is a

force in RP2. p = (a, b) → dp = adx + bdy + dz.

  • Definition. Let F = dp1 ∧ dp2 be a nonzero force.

◮ The projective line ˜

p1 ˜ p2 is the line of force.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Usage of projective statics

◮ To remove parallel/non-parallel cases.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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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.

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Quantum tensegrities

What happens if we switch to differential forms with non-constant coefficients?

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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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.

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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.

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Quantum tensegrities. Definitions.

F = (f1, . . . , fn). Then dF is (dF1 = df1 + dz, dF2 = df2 + dz, . . . , dFn = dfn + dz).

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Definitions.

F = (f1, . . . , fn). Then dF is (dF1 = df1 + dz, dF2 = df2 + dz, . . . , dFn = dfn + dz).

Definition

Let F = (f1, . . . , fn), functions with finitely many critical points. A tensegrity (G(dF), w) is a triple: a graph (G, F, w), where — fi are at vertices of G; — wi,j are at edges of G.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Definitions.

F = (f1, . . . , fn). Then dF is (dF1 = df1 + dz, dF2 = df2 + dz, . . . , dFn = dfn + dz).

Definition

Let F = (f1, . . . , fn), functions with finitely many critical points. A tensegrity (G(dF), w) is a triple: a graph (G, F, w), where — fi are at vertices of G; — wi,j are at edges of G.

Definition

A self-stress condition on (G(dF), w) at a function Fi

  • Pi,k∈Crit(fi)

(−1)ind(Pi,k)

{j|j=i}

wi,jdFj(Pi,k) ∧ dFi(Pi,k)

  • = 0.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Definitions.

Definition

A self-stress condition on (G(dF), w) at a function Fi

  • Pi,k∈Crit(fi)

(−1)ind(Pi,k)

{j|j=i}

wi,jdFj(Pi,k) ∧ dFi(Pi,k)

  • = 0.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Definitions.

Definition

A self-stress condition on (G(dF), w) at a function Fi

  • Pi,k∈Crit(fi)

(−1)ind(Pi,k)

{j|j=i}

wi,jdFj(Pi,k) ∧ dFi(Pi,k)

  • = 0.

Remark

It might be also useful to consider critical points separately.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Definitions.

Definition

A self-stress condition on (G(dF), w) at a function Fi

  • Pi,k∈Crit(fi)

(−1)ind(Pi,k)

{j|j=i}

wi,jdFj(Pi,k) ∧ dFi(Pi,k)

  • = 0.

Remark

dFi ∧ dFj = dz ∧ dfj at critical points. So dFj ∧ dFi → dfj ∧ dz

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Definitions.

Definition

A self-stress condition on (G(dF), w) at a function Fi

  • Pi,k∈Crit(fi)

(−1)ind(Pi,k)

{j|j=i}

wi,jdFj(Pi,k) ∧ dFi(Pi,k)

  • = 0.

Compare:

  • Pi,k∈Crit(fi)

(−1)ind(Pi,k)

{j|j=i}

wi,jdfj(Pi,k)

  • = 0.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Geometry.

Lines of forces: dfi ∧ dfj = 0.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Geometry.

Lines of forces: dfi ∧ dfj = 0. Line of force is not necessarily parallel to grad fi.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Geometry.

Lines of forces: dfi ∧ dfj = 0. Splitting of the line of force.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example.

Example

For each point Pi = (xi, yi) consider fi = (x − xi)2 + (y − yi)2. After rescaling of stresses one has a classical tensegrity.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example.

Example

This works for point-hyperplane frameworks (by Y. Eftekhari,

  • B. Jackson, A. Nixon, B. Schulze, Sh. Tanigawa, W. Whiteley):

hyperplanes → linear functions.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example.

Remark

Note that the definition in higher dimensions is the same.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example.

Remark

Note that the definition in higher dimensions is the same.

Example

In R3 consider potentials: fa,b,c = ke (x − a)2 + (y − b)2 + (z − c)2 (ke — the Coulomb constant) and take the unit stresses.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example.

Remark

Note that the definition in higher dimensions is the same.

Example

In R3 consider potentials: fa,b,c = ke (x − a)2 + (y − b)2 + (z − c)2 (ke — the Coulomb constant) and take the unit stresses. Then we have a classical Coulomb situation for points with unit charges in three-space.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example.

Example

G =

a b c d

  • Oleg Karpenkov, University of Liverpool

Tensegrities and differential forms.

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Quantum tensegrities. Example.

Example

G =

a b c d

  • and

a : f1 = (x − 3)2 + (y − 3)2; b : f2 = (x + 3)2 + (y − 3)2; c : f3 = (x + 3)2 + (y + 3)2; d : f4 = (x − 3)2 + (y + 3)2;

  • :

f5 = x2 + 5y2.

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example.

Example

G =

a b c d

  • and

a : f1 = (x − 3)2 + (y − 3)2; b : f2 = (x + 3)2 + (y − 3)2; c : f3 = (x + 3)2 + (y + 3)2; d : f4 = (x − 3)2 + (y + 3)2;

  • :

f5 = x2 + 5y2. Then

A B C D O 5λ 5λ λ λ −2λ −2λ −2λ −2λ

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.

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Quantum tensegrities. Example. A B C D O

5λ 5λ λ λ −2λ −2λ −2λ −2λ

Oleg Karpenkov, University of Liverpool Tensegrities and differential forms.