Refining Cellular Automatic Updating Rules to Model

Mean Curvature Flow

William B. Byrd, Walter L. Gerych, and Kateri A. SternbergMentor: Jeremy S. LeCrone

July 21, 2015

Abstract

In this paper we attempt to analyze and refine an algorithm for approximating meancurvature flow using cellular automata. It has been shown in a previous research programthat curvature can be accurately approximated from a cellular automatic representationof a curve. We use a combination of theoretical and experimental analysis to create andtest rules for evolving the curve based on these curvature calculations.

1 Introduction

Mean curvature flow is a well researched and well understood prin-ciple. Mean curvature flow is a geometric evolution equation. Weseek to approximate mean curvature flow in the case of simple closedcurves in R2

1.1 Continuous Mean Curvature Flow

Let Γ⊂ R2 be a curve that is twice differentiable, with a parametriza-tion γ : [−1, 1] → R2. We say that the curve Γ is simple if it doesnot intersect itself, and we say Γ is closed if γ(−1) = γ(1).

For 1 dimensional curves, the curvature can be defined in termsof circles. The curvature κ at a point p ∈ Γ is given by the inverseof the radius of the circle that best approximates Γ about p, calledthe osculating circle. The vector curvature ~κ has magnitude κ andpoints to the center of the circle. This allows us to express meancurvature flow with the equation dΓ

dt = ~κ.

1.2 Discrete Setting and Motivation

Our interest lies in studying the dynamic properties of a special caseof mean curvature flow, the case in which obstacles are present that

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stop the flow of the curve. With obstacles, simple closed curves willnot necessarily become more circular or collapse into a point unlikethe unobstructed flow. The curves will flow smoothly until theycome into contact with an obstacle.

Mean curvature flow with obstacles is a difficult subject to studyanalytically in a continuous setting. If that obstacle is in the interiorof the curve, the curve will no longer evolve at the points touchingthe obstacle. While before the flow was smooth and continuous,with obstacles the velocity experiences jumps and discontinuities.

In order to study the behavior of this flow with obstacles, wediscretized our curves. Our goal was to model mean curvature flowin a discrete cellular automatic setting.

The concept of cellular automata is widely known from Conway’sGame of Life. In it, cells in a grid can be either ”alive” or ”dead”.Every cell is updated based on a certain rule set on the local level;for the Game of Life, the number of living neighbors a cell has de-termines whether it will be alive or dead in the next generation.

Our project was to create updating rules that would evolve thediscrete cellular automatic system in a way that models mean cur-vature flow starting from a discrete representation of a curve. InSection 3, we discuss a way to ensure that the evolution is in thedirection normal to the curve. We discuss how to determine theamount of life to be added or removed as well as how the rulescould be implemented in an algorithm in Section 4. In addition,in Section 5, we discuss the relationship between our evolution andreal world time.

2 Background Information

We began our model using a method developed last year by LeCrone,Smith, and Zerger as a basis on which we could expand and im-prove [1]. The researchers developed a model that looks like meancurvature flow, but called for further research to create a more ac-

2

curate model for the discrete setting. In this section we will reviewthe general structure of the algorithm produced last year.

This evolution begins with a method for determining interfacepoints along the curve. Any cell with a center outside the pre-scribed curve is called dead and given a value of 0. Any cell whosecenter is on or inside the prescribed curve is considered alive andgiven a value of 1. The 8 cells immediately adjacent to a given cellare called its neighbors. The interface is defined as any live cellthat had at least one dead neighbor, which we take as the discreteapproximation of the evolving curve

The research continued by proving local area calculations canapproximate curvature. The idea is that if we draw a small circleabout a point on the curve, the area of the intersection of the inte-riors of the circle and curve can be used to calculate the curvature.This circle is known as a “template circle”.

For the discrete approximation of these calculations, they definedtheir discrete template circle to be the set of cells whose centers arewithin some radius of a chosen cell on the interface. In theory,smaller template circles are going to produce more accurate curva-ture calculations. Unfortunately, in the cellular setting, small radiicause problems since they don’t necessarily even look circular. Forexample, if the length of a cell was used as the radius for the tem-plate circle, the discrete approximation would be a 3 by 3 square.So last year the researchers decided that a discrete template circleof radius 7.5 times the length of a cell produced the best results fordiscrete calculations.

In order to prevent inaccurate curvature approximations in caseswhere multiple interfaces intersect the template circle, a burningalgorithm from Percolation Theory was integrated. With this al-gorithm, any cells on the live side of the interface in the templatecircle were set to 1 for the area calculation. This increased accuracyfor portions with high curvature and other degenerate situations.

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The amount of life in these “burned” areas is referred to as theburncount and is used as a rough measure of curvature. From this,updating rules are derived and implemented into an algorithm de-signed to model mean curvature flow.

The algorithm also allowed for obstacles to be introduced insidethe curve. These obstacles were made up of live cells that could notdie. The curve would flow in towards the obstacle, but the interfacecould not pass through the obstacle.

An improvement on this initial evolution was made this past yearby Dr. LeCrone so that cells can have varying amounts of life. Thecells are not simply alive or dead. Cells that are entirely containedin the curve are given a life concentration of 1 and those entirelyoutside are given a concentration of 0. The cells with the boundarypassing through them are assigned a life percentage by numericallyintegrating the area of the portion of the curve contained withinthe cell.

3 Determining the Change in Life

In order to improve the evolution, we have to determine how muchlife would be lost or gained due to any given cell in each generation.Developing new methods for determining how much life should beadded or lost is especially important because the cells can havefractional quantities of life which differs from the simple algorithmdeveloped previously wherein cells could only die or be born fully.

3.1 Use of Discrete Evolving Circles

We want the rules for our system’s evolution to model the evolu-tion of continuous curves evolving under mean curvature flow asclosely as possible. Before we start creating these rules, we need toknow how continuous curves should behave. Thankfully, we havean exact formula that predicts the flow of a continuous circle. Wethen use this information to create discrete approximations of thisflowing circle at different time steps. By examining the differences

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between the discrete approximations, we are able to recognize rela-tions between the burn count and the number of neighbors of cellsversus how much life those cells lose in each time step.

Our process starts by deriving the equation of a circle of radius runder mean curvature flow. We find that r′(t) =

√r2

0 − 2t, wherer0 is the initial radius. The derivation of this is as follows:

Since our circle is flowing under mean curvature flow, we knowthat for a circle of radius r as a function of time T , dr

dT = −~k where~k is the curvature of the circle and ~k is 1

r . Thus, drdT = −1

r . This im-plies that

∫r dr =

∫−1 dT . Integrating from elapsed time T = 0

to an arbitrary time T = t, we see

∫ r

r0

r(t) dr =

∫ t

0

−1 dT

1

2r(t)2 − 1

2r2

0 = −t

r(t)2 − r20 = −2t

r(t) =√r2

0 − 2t

Now that we have a justification that r(t) =√r2

0 − 2t, we cre-ate a series of discrete circles whose radii correspond to what theradius of a flowing circle should be at subsequent time steps. Sinceour goal is to create rules for evolution that mimic mean coverageflow, we can then study the changes between time steps in thesediscrete circles and analyze how the burncount, which is how weapproximate curvature, and number of a cell’s live neighbors in thediscrete approximations relate to the amount of life gained or lost.

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3.2 Separating by Neighbor Counts

(a) Template circle about a cell with 5 Neighbors (b) Template circle about a cell with 7 Neighbors

Figure 1: Differences Between Neighbor Counts

We attempted to find different rules for determining the amountof life to add or remove at a cell based on the number of immedi-ate living neighbors it had. The figure illustrates that cells with7 neighbors will allow for much larger intersections with the tem-plate circle than cells with 5 neighbors though the curvatures atthese two neighboring cells should be similar.

We initially decided to use regression techniques to find a preciserule mapping the burncount to the amount of life to add or removefor cells with 4 to 7 neighbors. After fitting our data with variouspolynomials, we found that a simple first order linear approxima-tion fit no worse than another approximation. We then created anindependent rule for each of these cell types based on the formulawe obtained.

However, linear approximation for each cell type had a root thatwas slightly too large. That is, the approximation was indicatingtoo large of a burncount that would correspond to zero percent oflife lost. To rectify this, we created a copy of our data that wasflipped and mirrored around what we intuitively believed the rootshould have been. We then concatenated these two matrices, andonce again fit this with a linear approximation. The root now cor-responded to what we thought zero life lost should have been.

Using the linear formula we obtained for each type of cell, wecreate an evolution for that cell type. As we mentioned previously,

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though, we created linear fits for only cell types with 4-7 neighbors.We could not create linear fits for cells with fewer neighbors be-cause our discrete circles had no cells with fewer than 4 neighbors.To determine rules for cells with 1-3 neighbors, different techniqueswould need to be implemented.

In theory, our evolution rule would calculate the burncount andneighbor count for each cell on the interface. Then, based on theneighbor count and burncount, our evolution rules would use theequation of the lines from our linear regressions to determine howmuch life should be added or removed. If the cell were to gainenough life such that it would have a life greater than 1, or loseenough life such that it would have a life of less than 0, the surplusor deficit would be passed on to a cell in the normal direction. SeeSection 4 for a discussion of how the normal direction is determinedin the discrete cellular automatic setting.

Unfortunately, this evolution quickly became asymmetric. For asymmetric curve, the curve should stay symmetric throughout theevolution, but ours were not. This was our first indication that ouralgorithm was not ideal.

In an attempt to improve our evolution, we decided to determinea burncount that should correspond to flatness, or no change inlife, for each cell type. To do this, we created a series of discretelines with slope ranging from 0 to π

4 . Since these lines should cor-respond to flatness, by calculating the burncount for different celltypes on these lines we know what burncounts should correspondto zero change in life.

Furthermore, we attempted to improve our evolution rules bycalculating how much life a cell wanted to lose, not just it’s changein life. Before, we merely calculated the life each cell lost, and cre-ated a linear regression from that. However, it is possible for a cellon the interface to want to lose more life than it contained, andthat surplus of life that the cell could not provide itself would be

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(a) 4 Neighbors (b) 7 Neighbors

Figure 2: Data for Burncount vs. Life Cell “Wants” to Remove

lost from a cell on the interior. We tried to capture this informationby finding all of the cells on the interior that lost life in our dis-crete circle approximations. If a cell of this type was touching onlyone interface cell that had been completely emptied in that timestep, the life the interior cell lost would be added to the life thatthe interface cell lost to determine how much life that interface cellwanted to lose. If an interior cell that lost life was touching multi-ple interface cells that had emptied themselves fully, the loss in lifeof the interior cell would be divided evenly between those interfacecells.

We then ran a linear regression on the life each cell wanted tolose based on that cell’s burncount for cells with 4 to 7 neighbors.Once again, we used the equation from a linear fit to determineevolution rules for cells of each neighbor type. We then added acondition that said that the burncount for each cell type had to bebelow what we had previously determined to be flat by analyzingthe discrete lines for each cell type.

This evolution did not work any better. The evolution still lostsymmetry, and, worse still, the interface became “crinkly” as shownin Figure 3.

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(a) Evolution (b) Discrete Circle

Figure 3: Difference Between Evolution and Discrete Circle

Even though the burncounts for cells with 6 or 7 neighbors hadthe most well defined pattern of the neighbor types, during our ac-tual evolution our rules worked the least well on those cell types.During the evolution this cellular automatic system, our rules wereelecting to add life to the 6- and 7-neighbor cells frequently. Con-versely, even though the spread of the burncounts for cells with 4or 5 neighbors was the least linear they behaved the best with ourrules.

Apparently differentiating cells based on neighbor counts andcreating a unique rule set for each of these types of cells is not theway to go.

3.3 Determining Regression for Life vs. Burncount

We decided to create a single rule for evolution that is based solelyon burncounts, and does not take the neighbor count into consid-eration. In an attempt to find a better linear regression to createour new evolution rule, we once again created a series of discretecircles evolving by mean curvature flow just as before, but with afiner cellular size.

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(a) Original (b) Zeros Removed (c) Final

Figure 4: Regression for Circles

Figure 4(a) shows the results of graphing each burncount for ev-ery generation versus the life each of those cells lost. At first glance,it does not look like we’ll be able to fit this data with a line. How-ever, there is a lot of misleading data in the graph. First, we candisregard the cells that did not lose any life, as shown in Figure 4(b).

Now it appears as if our data branches off into two differentlines. We determined that the left branch contains cells that hadcompletely emptied themselves between generations. In reality,those cells wanted to lose more life than they held, but the graphloses that information and only shows how much they emptied fromthemselves. We already talked about this issue in section 3.2 whenwe discussed trying to determine the life cells wanted to lose in or-der to determine more accurate evolution rules for cells of differentneighbor types. As we said before, the problem is that we are notcollecting information on the life that each cell wanted to lose, justhow much life they lost themselves. If we were graphing the lifeeach cell ”wanted” to lose, the data points on the left would bemuch lower, and the spread of the data would be much more linear.Following this justification, we disregard cells that completely emp-tied themselves. By doing this, we get a much more linear spread,as shown in Figure 4 (c). The data can now be fitted with a simplelinear regression.

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Figure 5: Regular Interface for Inverted Circles

In order to create rules for adding life, as opposed to losing lifeas before, we needed to analyze what growth should look like. Todo this, we created a series of discrete circles just like before, exceptthat we had the life inverted. The cells outside the circle were alive,and the interior of the circle was dead. The dead circle shrank eachgeneration.

While testing the inverted circles, it became apparent that re-peating the same calculations would not work for the purpose ofderiving updating rules based on burncount data. When a livingcircle is shrinking, there are many fully living cells on the interfacethat lose a portion of life. The analogue of this in the inverted caseis a fully dead cell gaining life. This poses a problem since our cur-rent algorithm only identifies the living cells that neighbor a deadcell as interface. Figure 5 shows our first analysis of the invertedcircle using the same paradigm with live interface cells. This isobviously terrible. The slope is negative, but we know it should bepositive. The issue is that we were only analyzing cells that werealready alive, and not considering cells that were dead that gainedlife. The reason that a higher burncount seemed to correspond toa lower amount of life added is because cells with high burncountswould most likely already be mostly full.

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(a) With Zeros (b) Zeros Removed (c) Final

Figure 6: Regression for Inverted Circles

We redefined the interface for inverted circles such that the in-terface was any cell that had both living and dead neighbors. Thismeans that interface cells no longer have to be alive. Analyzingthis data, we get Figure 6. This is similar to the data we saw in4(a) in that it branches. This time, the right branch is caused bycells that completely filled and wanted to add more than they couldhold. By disregarding these cells because they are giving us incom-plete and misleading information, akin to the data manipulationwe performed on the non-inverted circles, we get 6(c).

Figure 7: Combined Regression Graph

Figure 7 is a graph of both the life lost for the shrinking circlesand the life gained by the inverted circles versus the burncount ofthose cells, after we manipulated each of those data sets. In an idealworld, the portion of the graph that corresponds to the life gainedby the inverted circles would be to the right of the portion thatcorresponds to the loss of life. If this were the case, we could fit the

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bottom portion with one linear regression to create a rule for losinglife. Then we would have a region that corresponded to no changein life, and then the top portion could be fit with a regression thatcould be used to create a rule for adding life. However, it can beseen in Figure 7 that the data gathered doesn’t line up. This isdue to the fact that the burn counts from the top half of the dataare taken from dead cells just outside the interface and the bottomhalf is taken from living cells on the interface. The bottom half ofthe data can be used to create a rule for losing life, but to use thetop part of the data we would have to reclassify what we mean byinterface.

One solution to this problem is to isolate the areas where lifeshould be growing, and rather, shrink the dead region in thoseareas. This could be done using the same information gatheredfrom the shrinking circle. It would also accomplish one of our goalsof ensuring that a curves evolution is unaffected by whether or notthe life is on the inside or outside.

4 Evolving in the Normal Direction

Mean curvature flow in the continuous setting evolves in the direc-tion normal to the surface of the curve. In order to more accuratelymodel mean curvature flow, the curve must also evolve in the nor-mal direction in the discrete setting. To better control the directionof interface evolution, we need a reliable way to determine normalvectors. The following analytic result motivates how one can ac-complish this goal.

4.1 Theoretical Justification For Normal Vector Approximation

Let Γ ⊂ R2 be a simple closed curve that is twice differentiable,with a parametrization γ : [−1, 1] → R2. Any point p ∈ Γ, canbe rotated and translated such that there exists a function f forwhich f(0) = 0, f ′(0) = 0, and f is twice differentiable. That is,Γ = graph(f) locally. Let Rδ be the area that is in the interior ofΓ and enclosed by a δ-radius template circle centered at 0 as illus-

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Figure 8: A Rotated and Translated Curve Inside a Template Circle

trated in Figure 8. We will show that the normal vector ~0, which

is equal to

[01

]after the curve has been shifted and translated, can

be calculated from the area contained in Rδ.

Lemma 1. Let R′δ be the region bounded by a δ-radius templatecircle and y=0, centered at (0, 0). Then,

∫∫R′

δ(−[x

y

])dA

23δ

3=

[01

]Proof. The above double integral can be represented by∫ δ

−δ∫ 0

−√δ2−x2

[x

y

]dydx

23δ

3

The result follows from simple calculus.

∫ δ−δ∫ 0

−√δ2−x2

[xy

]dydx

23δ

3= −

∫ δ−δ

[x√δ2 − x2

−12(δ2 − x2)

]dx

23δ

3

= −

[−1

2(δ2 − δ2)3/2 + 12(δ2 − δ2)3/2

(−δ3

2 + δ3

6 )− (δ3

2 −δ3

6 )

]23δ

3

=

[0

23δ

3

]23δ

3=

[01

]14

Let R be the region bounded below f(x) and a template circleabout the point p as in Figure 8.

~N =

∫∫R(~p−

[xy

])dA

23δ

3

Theorem 2. limδ→0+~N is the outward unit normal vector at the

point p.

Proof. Let R′δ be the region bounded by the template circle andbelow the linear approximation of f about 0. As we have calculated,∫∫

R′δ(−[x

y

])dA

23δ

3=

[01

]Now we bound the error ~E in this approximation as a function ofδ and will show that ~E → 0 as δ → 0.To begin, let

~E0 = ~N −[01

]=

[E0x

E0y

]In order to ensure that the x and y components of our error vectorare both positive, let

~E =

[|E0x||E0y|

]=

[Ex

Ey

]We will look at each component Ex,Ey of the error individually.

Ex ≤∫ δ−δ∫ |f(x)|

0 |x|dydx23δ

3=

∫ δ−δ |xf(x)|dx

23δ

3

Now, since f(0) = 0 and f ′(0) = 0, by Taylor’s Remainder The-orem there exists an M ∈ R such that |f(x)| ≤ Mx2 in someneighborhood of zero.

Ex ≤∫ δ−δM |x

3|dx23δ

3=

2∫ δ

0 Mx3dx23δ

3=

2M 14δ

4

23δ

3=

3M

4δ

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Since limδ→03M4 δ = 0 then limδ→0Ex = 0.

Now, in the same manner for the y component,

Ey ≤∫ δ−δ∫ |f(x)|

0 |y|dydx23δ

3=

∫ δ−δ∫ |f(x)|

0 ydydx23δ

3=

∫ δ−δ

12 |f(x)|2dx

23δ

3

Substituting in the bound |f(x)| ≤Mx2 from above

Ey ≤∫ δ−δ

12M

2|x4|dx23δ

3=

∫ δ0 M

2x4dx23δ

3=

15M

2δ5

23δ

3=

3M 2

10δ2

Since limδ→03M2

10 δ2 = 0 then limδ→0Ey = 0. Since the limit as δ

goes to 0 of each component Ex, Ey is 0, limδ→0~E = 0.

So, limδ→0N =

[01

]as claimed.

4.2 Implementing Normal Vector in Discrete Setting

Figure 9: Error of Calculated Normal Vector for Discrete Circles at Interface Points Histogram

To find the normal vector in the discrete setting, we approximatethe integral from Theorem 2 by performing quadrature over theliving cells in the template circle. This discrete approximation tothe normal vector is fairly accurate as shown in Figure 9. For 100circles of radius varying from 1 to 5 with a resolution of 0.1 and us-ing a template circle of radius 0.75, all of the approximated normal

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vectors are within 1.5◦ of the actual normal vectors.

If a cell fully empties or fills itself in a generation of the evolu-tion, it may still need to remove or add more life from elsewhere inorder to maintain mean curvature flow. In this case, the cell mustelect another cell to alter.

Once a unit normal vector is determined, it is then used to findthe cell in the normal direction. To implement this, the algorithmtakes the dot product of the unit normal vector and a unit vectorin the direction of each of the eight neighboring cells.The algorithmthen takes the maximum dot product and returns the coordinatesof the cell associated with that dot product. This cell is deemedthe normal cell and is elected to lose or gain life after its normalneighbor. If the normal cell would have a surplus or deficit of lifeitself, a situation that hopefully rarely occurs, the normal cell is setto either 1 or 0 and the surplus/deficit is not passed further on.

5 Determining Actual Time

One of the goals for our project was to relate the generations in ourcellular automatic system to the passage of time in the correspond-ing continuous setting. In the continuous setting curves undergoingmean curvature flow evolve over time, but in the discrete cellularautomatic setting the system is updated in generation snapshots,with no immediately obvious way to determine how much timeelapses between each generation.

5.1 Theoretical Justification for Constant Area Loss

The rate of change of area for a simple closed curve undergoingmean curvature flow is constant, so the only information we needto determine the time between generations is the amount of arealost between those generations.

Theorem 3. [2] The rate of change of enclosed area of a simpleclosed curve under mean curvature flow is constant and equal to

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−2π.

For the reader’s convenience, we present a proof for this theorem.

Proof. Let Γ be a simple closed curve that is the boundary of thecompact two-dimensional manifold M . In reference to our cellularautomatic setting, Γ is the curve we are approximating, and M isthe surface composed of all of our living cells. The rate of changeof the enclosed area A can be expressed as

dA

dt=

∫Γ(t)

~v · ~n ds

Where Γ(t) is Γ after elapsed time t, ~v is the velocity vector foreach point on Γ, and ~n is the normal outward pointing vector foreach point on Γ. Since the curve is evolving by mean curvatureflow, ~v = −~κ, where ~κ is the curvature vector at each point on Γ.Thus, our expression becomes

dA

dt=

∫Γ(t)

~κ · ~n ds

We will use the Gauss-Bonnet theorem to evaluate this integral.The Gauss-Bonnet theorem states that

∫M

K dA+

∫Γ

κg ds = 2πχ(M)

Where K is the Gaussian curvature of M , κg is the geodesic cur-vature of Γ, and χ(M) is the Euler characteristic of M . Since ourcurve is embedded in R2, the Gaussian curvature of our surface is 0.

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Additionally, the geodesic curvature of Γ is equivalent to the meancurvature of Γ, so

∫Γ κg ds = −

∫Γ(t) ~κ · ~n ds. Since our surface is

topologically equivalent to a disk, and the Euler characteristic of adisk is 1, 2πχ(M) = 2π. Thus,

0−∫

Γ(t)

~κ · ~n ds = 2π

∫Γ(t)

~κ · ~n ds = −2π

dA

dt=

∫Γ(t)

~κ · ~n ds = −2π

Thus, the rate of change of the enclosed area of a simple closedcurve is constant.

Note that this does not imply that the time between one gener-ation will be the same as the time between the next, because theloss of area can be different for successive generations. Rather, itsimply gives us a way to determine the time between individualgenerations.

6 Possible Future Research

There are many directions in which this research program couldcontinue. One hope would be to actually implement our theoret-ical approach to real time into the algorithm. This would enablemore accurate comparisons because each generation is not a con-sistent amount of time in the current evolution.

Furthermore, the rules we derived in Section 3 for the amountof life to be lost or gained can be implemented into the currentalgorithm. These results can then be compared to known solutions

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of mean curvature flow to determine their accuracy as a discreteapproximation.

A different approach to the problem could be to define fully deadcells to be -1 and living cells to be +1. Then we could allow nega-tive and positive clumps to have the same rules for spreading. Thiswould be useful since we are only actually trying to model the flowof a curve and the flow should not depend on whether we choosethe life to be on the inside or outside of the curve.

7 Selfies #thestruggle #thejourney

Acknowledgments

This research was carried out as a part of the Summer Undergradu-ate Mathematics Research REU at Kansas State University fundedby NSF award #1262877. The authors would like to thank KansasState University and Starbucks for their hospitality.

References

[1] J. LeCrone, B. Smith, S. Zerger Using Cellular Automata to Model Mean Curvature Flow.2014.

[2] B. White Evolution of Curves and Surfaces by Mean Curvature. 2002.

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