# A Taste of Differential Geometry: Ribbons and White's Formula

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A Taste of Differential Geometry: Ribbons and White's FormulaA
Taste of Differential Geometry: Ribbons and White’s Formula

Richard G. Ligo

I Linking number I Writhe I Twist

I White’s Theorem

Tangent vector to the “Pringles” curve:

P(t) =

For example:

A calculation at t = π:

P(π) =

U =

A calculation at t = π:

P(π) =

U =

The fruit of our calculations:

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

Ribbon definition

A curve and normal vector field along it define a ribbon.

White’s Formula

Richard G. Ligo

Ribbon definition

A curve and normal vector field along it define a ribbon.

White’s Formula

Richard G. Ligo

A local crossing occurs when the ribbon appears “edge-on.”

Conversely, a nonlocal crossing occurs when one section passes over another.

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

The linking number

To calculate the linking number, assign crossing values to the crossings between the two edges, compute their sum, and divide by two.

Lk(R) = −1+1+1+1 2 = 2

2 = 1

Intuitively, Lk(R) describes the “tangledness” of the ribbon edges of R.

White’s Formula

Richard G. Ligo

The writhe

To calculate the writhe, choose a ribbon edge, assign crossing values to its self-crossings, and compute their sum.

Wr(R) = +1 = 1

Intuitively, Wr(R) describes the “nonplanarity” of the chosen ribbon edge.

White’s Formula

Richard G. Ligo

The twist

To calculate the twist, assign crossing values to the local crossings, compute their sum, and divide by two.

Tw(R) = −1+1 2 = 0

2 = 0

Intuitively, Tw(R) describes how much the ribbon edges rotate about each other.

White’s Formula

Richard G. Ligo

Lk(R) = 1 Wr(R) = 1 Tw(R) = 0

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

Lk(R) = 1 Wr(R) = 1 Tw(R) = 0

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

Lk(R) = 1 Wr(R) = 1 Tw(R) = 0

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

Another example:

Lk(R) = −1+1+1+1+1+1+1+1 2 = 6

2 = 3

Another example:

Lk(R) = −1+1+1+1+1+1+1+1 2 = 6

2 = 3

2 = 1

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

2 = 1

2 = 1

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

The Gauss-Bonnet Theorem

Like White’s Formula, the Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties.

Theorem: If M is a compact, two-dimensional, Riemannian manifold with boundary ∂M, then∫

M K dA +

kg ds = 2πχ(M),

Where K is the Gaussian curvature on M, kg is the geodesic curvature on ∂M, and χ(M) is the Euler characteristic of M.

White’s Formula

Richard G. Ligo

The Gauss-Bonnet Theorem

Like White’s Formula, the Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties.

Theorem: If M is a compact, two-dimensional, Riemannian manifold with boundary ∂M, then∫

M K dA +

kg ds = 2πχ(M),

Where K is the Gaussian curvature on M, kg is the geodesic curvature on ∂M, and χ(M) is the Euler characteristic of M.

White’s Formula

Richard G. Ligo

So, what can you do with this?

Direct applications:

White’s Formula

Richard G. Ligo

So, what can you do with this?

Direct applications:

White’s Formula

Richard G. Ligo

Energy stored within a ribbon is given by the integral

E = 1

κ is the curvature at a point.

τ is the torsion at a point.

low κ and τ high κ, low τ low κ, high τ high κ and τ

White’s Formula

Richard G. Ligo

Energy stored within a ribbon is given by the integral

E = 1

κ is the curvature at a point.

τ is the torsion at a point.

low κ and τ high κ, low τ low κ, high τ high κ and τ

White’s Formula

Richard G. Ligo

Energy stored within a ribbon is given by the integral

E = 1

κ is the curvature at a point.

τ is the torsion at a point.

low κ and τ high κ, low τ low κ, high τ high κ and τ

White’s Formula

Richard G. Ligo

I Space curves can be used to define ribbons.

I The behavior of ribbons is constrained by White’s formula.

I Ribbons can be used to model real-life situations.

I Real-life situations desire to minimize stored energy.

I Methods from differential geometry allow us to predict configurations.

White’s Formula

Richard G. Ligo

I Can we model curves with nonuniform density?

I Is it possible to obtain results from purely geometrical methods?

White’s Formula

Richard G. Ligo

I The University of Iowa

References:

I M. Dennis & J. Hannay. The geometry of Calugareanu’s Theorem.

I J. Hearst & Y. Shi. The Kirchoff elastic rod, the nonlinear Shrodinger equations, and DNA supercoiling.

I K. Hu. Writhe of DNA induced by a terminal twist.

I M. Podvratnik. Torsional instability of elastic rods.

Introduction

Definitions

Examples

Application

Conclusion

Richard G. Ligo

I Linking number I Writhe I Twist

I White’s Theorem

Tangent vector to the “Pringles” curve:

P(t) =

For example:

A calculation at t = π:

P(π) =

U =

A calculation at t = π:

P(π) =

U =

The fruit of our calculations:

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

Ribbon definition

A curve and normal vector field along it define a ribbon.

White’s Formula

Richard G. Ligo

Ribbon definition

A curve and normal vector field along it define a ribbon.

White’s Formula

Richard G. Ligo

A local crossing occurs when the ribbon appears “edge-on.”

Conversely, a nonlocal crossing occurs when one section passes over another.

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

The linking number

To calculate the linking number, assign crossing values to the crossings between the two edges, compute their sum, and divide by two.

Lk(R) = −1+1+1+1 2 = 2

2 = 1

Intuitively, Lk(R) describes the “tangledness” of the ribbon edges of R.

White’s Formula

Richard G. Ligo

The writhe

To calculate the writhe, choose a ribbon edge, assign crossing values to its self-crossings, and compute their sum.

Wr(R) = +1 = 1

Intuitively, Wr(R) describes the “nonplanarity” of the chosen ribbon edge.

White’s Formula

Richard G. Ligo

The twist

To calculate the twist, assign crossing values to the local crossings, compute their sum, and divide by two.

Tw(R) = −1+1 2 = 0

2 = 0

Intuitively, Tw(R) describes how much the ribbon edges rotate about each other.

White’s Formula

Richard G. Ligo

Lk(R) = 1 Wr(R) = 1 Tw(R) = 0

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

Lk(R) = 1 Wr(R) = 1 Tw(R) = 0

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

Lk(R) = 1 Wr(R) = 1 Tw(R) = 0

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

Another example:

Lk(R) = −1+1+1+1+1+1+1+1 2 = 6

2 = 3

Another example:

Lk(R) = −1+1+1+1+1+1+1+1 2 = 6

2 = 3

2 = 1

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

White’s Formula

Richard G. Ligo

2 = 1

2 = 1

Lk(R) = Wr(R) + Tw(R)

White’s Formula

Richard G. Ligo

The Gauss-Bonnet Theorem

Like White’s Formula, the Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties.

Theorem: If M is a compact, two-dimensional, Riemannian manifold with boundary ∂M, then∫

M K dA +

kg ds = 2πχ(M),

Where K is the Gaussian curvature on M, kg is the geodesic curvature on ∂M, and χ(M) is the Euler characteristic of M.

White’s Formula

Richard G. Ligo

The Gauss-Bonnet Theorem

Like White’s Formula, the Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties.

Theorem: If M is a compact, two-dimensional, Riemannian manifold with boundary ∂M, then∫

M K dA +

kg ds = 2πχ(M),

Where K is the Gaussian curvature on M, kg is the geodesic curvature on ∂M, and χ(M) is the Euler characteristic of M.

White’s Formula

Richard G. Ligo

So, what can you do with this?

Direct applications:

White’s Formula

Richard G. Ligo

So, what can you do with this?

Direct applications:

White’s Formula

Richard G. Ligo

Energy stored within a ribbon is given by the integral

E = 1

κ is the curvature at a point.

τ is the torsion at a point.

low κ and τ high κ, low τ low κ, high τ high κ and τ

White’s Formula

Richard G. Ligo

Energy stored within a ribbon is given by the integral

E = 1

κ is the curvature at a point.

τ is the torsion at a point.

low κ and τ high κ, low τ low κ, high τ high κ and τ

White’s Formula

Richard G. Ligo

Energy stored within a ribbon is given by the integral

E = 1

κ is the curvature at a point.

τ is the torsion at a point.

low κ and τ high κ, low τ low κ, high τ high κ and τ

White’s Formula

Richard G. Ligo

I Space curves can be used to define ribbons.

I The behavior of ribbons is constrained by White’s formula.

I Ribbons can be used to model real-life situations.

I Real-life situations desire to minimize stored energy.

I Methods from differential geometry allow us to predict configurations.

White’s Formula

Richard G. Ligo

I Can we model curves with nonuniform density?

I Is it possible to obtain results from purely geometrical methods?

White’s Formula

Richard G. Ligo

I The University of Iowa

References:

I M. Dennis & J. Hannay. The geometry of Calugareanu’s Theorem.

I J. Hearst & Y. Shi. The Kirchoff elastic rod, the nonlinear Shrodinger equations, and DNA supercoiling.

I K. Hu. Writhe of DNA induced by a terminal twist.

I M. Podvratnik. Torsional instability of elastic rods.

Introduction

Definitions

Examples

Application

Conclusion