Spare Easter Eggs? Here’s a little experiment in complexity…

 

Consumed too much chocolate? Sugar overdose? Unwanted Easter eggs stacking up? Before you start baking chocolate cakes, please do this little experiment for me: smash the eggs on the floor and measure the size of the fragments. They might just reveal some amazing universal laws of complexity, self-organisation and fractals.  I’ve banged on about power laws and city size elsewhere on this blog – basically city sizes (and lots of other things in the universe) follow eerily consistent power law size distributions. In Paul Krugman’s take on this subject – the excellent and very readable Self Organising Economy –  he mentions that if you throw a Grecian urn at a wall the resulting fragments will follow a power law and that the exact power shape (the slope of graph) will be unique the shape and materials of that urn.   It appears that fragmentation processes follow the same distributional patterns as Earthquakes, meteorites, tree branches, companies and cities.  Krugman doesn’t say much more on the Grecian urns and the internet hasn’t got much either so i wanted to test this idea myself. Unfortunately I have few Grecian urns at my disposal, and none that i can smash for the purposes of a poorly subscribed amateur blog. Conversely, chocolate eggs are in abundance this time of year, and I figured a refrigerated chocolate egg would be sufficiently brittle to use in lieu of priceless antique urn.
Grecian urn, probably expensive
So here are the steps (with details below):
1) Refrigerate chocolate Easter egg
2) Drop egg from height onto clean surface
3) Weigh or measure fragment sizes
4) Sort and plot on chart
5) Eat chocolate egg fragments
6) Send me your results…

This is what you are aiming for.
A smashed Easter Egg
Egg size fragments – size versus rank of size
Alas I only had time to smash one egg.  The results were ok, but I’d like to smash some more.

Now for the experimental details:

For the remainder of the article i will use the term “egg” to mean large chocolate eggs the kind you see at Easter. Do not drop bird eggs.

1) Refrigerate Easter egg

This is simple enough. The egg must be hard and brittle to smash. A soft egg might not fragment.
It must be a large thin shelled egg like this one:
 These kind of eggs are ubiquitous in British supermarkets around Easter.

2) Drop from height

For the experiment to be fair the eggs need to hit with an equal force. The easiest way for me to achieve this is to drop the eggs. Dropping from a constant height should result in a reasonably similar force upon hitting the ground (although depends on the mass of the egg). Dusting down my school textbooks it appears that the force on the egg is related to its mass and acceleration (Newtons 2nd law).  In this case the force is actually the deceleration as the egg becomes stationary after falling under gravity with an acceleration of 9.81m/s/s. This means that the higher the height from which the egg is dropped the greater speed at the split second before collision with the floor and therefore the more deceleration required to bring it to a stop and therefore the more force acting on the egg. Got that?  This is my understanding – hopefully a physicist will correct me if I’m wrong (I have an engineering degree but it was a while ago).  The force can also be explained as a change of momentum. Anyway, all I’m saying is that an egg is more likely to fragment at a great height (as Humpty Dumpty or anyone who has fallen off a tall building can attest). So I’m dropping my eggs from 1.2m height, after 1 hour refrigeration.  They smash into lots and lots of fragments.
Clearly its necessary to drop onto a hard and very clean surface. Most floors have had dirty feet all over them at some point so it might be best to drop onto cleaner hard surface. The surface must be hard and consistent.
I have used the floor – I have an exceptionally clean floor.
Further, the fragments will fly so it might be worth constructing some barriers. Its important to collect all the debris. The smaller fragments will travel further and missing them will introduce bias.

3) Measure fragment sizes

Once you have the fragments their individual size or mass must be measured. This is the boring part. Ideally we would weigh each fragment, but i did not have a sensitive scale.  Other methods might be to estimate the area or length of the pieces.  You could try measuring the water displacement of the fragments (cold water mind).  As I was short on time I crudely measured the area of the fragments using Powerpoint.  I basically made squares around the fragments and measured the areas using the square dimensions (see screenshot).
Using Powerpoint to very crudely measure areas

4) Plot chart

Once measured the fragments should be plotted on a chart. Use can use Excel or google sheets (free).
You then need to sort the fragment sizes and then rank. When you do an xy scatter chart of the resulting table it should look a bit like this:
Egg size fragments plotted with a power trend line fitted by Excel. Its not a great fit at the large fragment end.
Note the rank is along the x-axis and the fragment size is on the y-axis.
My original hypothesis is that the egg fragments will follow a power law. This is partially evident in the chart, but the data may exhibit a logarithmic distribution.  More experiments are required to confirm. Essentially we seem to be short of one big fragment from the Mini Eggs Easter Egg.
The log fit is slightly better. More experiments required.
The second hypothesis is that the power law steepness depends on the shape of the egg.
Here we can just break more eggs and plot in the same way.
Or final advanced step is to make the power law curve into a straight line.  You can do this by taking the log of both the rank and the data.  This might sound scary but is very easy in excel/google sheets. You simply write: LOG () in the formula bar.
Taking Logs is easy. You can also ask Excel to scale the axis using logs if you prefer
This will give the data straight lines which are easier to assess.
Power converted to logs
5) eat egg
The only final problem is what to do with all those chocolate egg pieces. I suggest you eat them. If you are on a diet and theyre destined for the bin, you can smash antique vases next year.
6) Submit conclusions: send me your results.

 

Conclusions/Discussion

 

Power laws are ubiquitous throughout nature and the fragmentation of rocks, and brittle objects seems to be no exception.  The idea that the slope of the power law might indicate the properties of the object or the nature of the fragmentation event seems integrating to me (is it used in forensics?).

 

In studying cities we find that city size distributions follow a power distribution.  This is known as Zipf’s Law.  My post here shows how I applied Zipf’s Law to settlements in Suffolk, England and China.  In the same way as for your broken egg fragments, you will observe one big fragment, a few medium sized fragment, and a lot of smaller fragments mirrors city sizes in countries.   A remarkable feature of Zipf’s law is that city sizes tend to distribute with a log slope of 1.  However, the slope can vary between systems of cities, and over time, which is where the science gets very interesting indeed.
Chinese cities by metropolitan population
The point that Krugman and others were making is that apparent order can arise out of apparent chaos and complexity.  In fact complex systems tend towards order.  Thomas Schelling identified such order from unstable disorder this in his studies of segregation.  It appears there is some kind of order in the internal structure of an Easter Egg that causes it to break into this pattern of sizes.

 

I don’t know if the Easter Egg tests will prove to all follow power laws, and if the log slopes will say anything about the shapes.  I will have to wait until next year to find out, because if I eat anymore chocolate now I’ll be sick.  If you have some spare eggs (and time!) please get smashing and send me some results.
To be continued erm… next Easter!

 

PS for those of you wondering how i have time to write this frivolous article this weekend consider the other mainstay of Easter in the UK: the rail replacement bus. I’ve got an article about this too.  Let’s just say it takes a long time to move about without a car, but the replacement bus is too distracting to read.

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