A Foam Model for Bread Development

We find the sessile curve by finding its curvature.

Future Work

1. Create a large data base of fits and their residuals.

2. Use random volumes for the cells around a set distribution.

3. Repeat the word in a three-dimensional case.

Bread Models

&

Comparisons

Acknowledgments

The presenter would like to thank Rose-Hulman Institute of Technology for its generous support in funding the project, Professor David Finn for his inspiring direction and advice while investigating the topic, fellow group members: Kenneth Reed and Molly Stillman, and Matthew Obetz.

We compare two models for the final bread shape.

References

Ellipse

Sessile

D.L. Finn. The Shape of a Slice of French Bread, in preparation

Fitting an Ellipse

Fitting a Sessile

Example of a Sessile Fit

Ellipse Fitting Example

We find an ellipse to fit our data set by minimizing...

We plot the points from our curvature estimates and find a linear best fit line.

The curvature of a sessile is given by . Therefore, from our linear fit we obtain and .

Which we get by solving...

We can then find the sessile curve using a parametric equation in terms of an elliptical integral.

The Curvature is then given by...

This must be done numerically.

Curvature Comparisons

Base 20

Base 5

Curvature

Residual

Fit

Residuals

0.012743

0.053716

0.014032

0.06372

Sessile

Ellipse

0.012634

0.052981

0.014271

0.06490

(cc) image by anemoneprojectors on Flickr

Which is better?

Neither! They both provide equally good fits; however, the sessile fit comes with physical reasoning.

Curvature and the Convex Hull

Discrete Curvature

Convex Hull

What is a Convex Hull?

There are two types:

• Intrinsic
• Extrinsic

What is Curvature?

Signed Curvature

Here we look at the breads extrinsic curvature defined by...

Graham Scan

Convex Set: A set which contains all line segments connecting any two points in the set.

A Graham Scan is an algorithm for finding the convex hull for a convex set. Developed in 1972 by Ronald Graham. It runs with time complexity .

convex

concave (non-convex)

Method:

Convex Hull: The convex hull of a set of points P is the intersection of all convex sets containing P.

Continuing with the algorithm yields the expected square.

Now applying a Graham Scan to our configuration yields...

Calculate the angle formed with the start points, relative to the x-axis and sort the points accordingly (heap sort).

Proceed to the next two points in the set and calculate if the turn is a right or left turn. If the turn is left move on, if not, eliminate the next to last point and retry.

For a given set, find the point with the lowest y-coordinate.

A Foam Model for Bread Development

Discrete Curvature

But how do we find the convex hull for our configurations?

But how does one find the curvature of a set of points?

Moving Tangent

Circle

of

Curvature

sdf

Circle

of

Curvature

Discrete Tangents

Here we fit a secant to two sets of points sharing a common point, then average their slopes to find a 'tangent'.

Given 3 noncollinear points, one can find a circle which goes through all 3.

We take our set and fit circles to it

three points at a time.

Why can we use a Graham Scan?

Sample Curvatures

Circle

Curvature

Tangent

Curvature

Average

Base

Simply, we can't. Our figures are not convex. However, they fit our configurations tightly. Moreover, the fit is better with larger configurations and more refinements. We merely seek a reasonable approximation.

0.737563

0.723583

0.730573

3

0.570336

0.566566

0.574105

5

Note that there will be outliers resulting from the use of the convex hull.

10

0.409552

0.409102

0.408653

0.316613

0.315357

0.315985

20

Caleb McWhorter

B.A. Mathematics

B.S. Physics

Surface Evolver

What is Surface Evolver?

Gradient Descent

Example

What is Gradient Descent?

Take the function...

The algorithm then takes the following steps...

It is an algorithm for finding a minimum of a given function; it takes steps proportional to the gradient at a given point but in the opposite direction.

Surface Evolver is a program devolved by Ken Brakke which seeks a configuration with minimum energy using a (conjugate) gradient descent method given a set of forces and constraints.

We try a configuration of our own.

We choose a structure consisting of mostly hexagons because the minimization problem at hand closely resembles that of the Honeycomb Conjecture, which states that the best way to divide a surface into equal areas with least perimeter is with hexagons.

This was proposed by Pappus of Alexandria around c. 320 and proven by Thomas Hales in c. 1999.

After one minimization:

Here is our initial configuration:

...and after 10:

Finally, after 1,000 iterations:

This is not the only possible minimal configuration given the same structure as the gradient descent only seeks a local minimum, not a global. For example, here is another possible minimal configuration.

Here we see several more fully evolved configurations.

Next, we investigate whether looking at the curvature of the boundary allows us to find a accurate model for the structure.

How do we model the bread?

What model best represents the final bread shape?

Which types of shapes can bread foams take?

What forces act on the bread?

Bread

Baking

What forces act on the bread?

Graphs

What is a graph?

Example Structure

A graph is a collection of vertices...

...and edges.

So give the cell an area...

Now we use Green's Theorem

to transform the area constraint as such...

...more on Graphs

Our graphs will have faces...

Putting the equations and constraints gives the energy for one cell.

...and bodies.

And summing over all the edges yields the total energy with constraint for the whole figure.

...And what types of

shapes does bread form?

What processes

drive the formation

of bread...

Questions

Finished

Kneading:

This helps the production of gluten and creates air pockets in the dough, these can be seen after baking.

?

Baking:

How is bread baked?

Proofing:

Allow the yeast to ferment in warm water.

Bake the bread to allow it to solidify.