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Making soufflé tonight, I wondered if the six yolks took on the optimal circle packing configuration. They do not. It is only with seven congruent circles that the optimal packing places one in the center.

Q. Why don't the yolks in a bowl follow the optimal packing of congruent circles in a circle?


         
          Six yolks in a bowl.


          Circs567
          Image from Wikipedia. Optimal packings for $5,6,7$ circles.


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closed as off-topic by Wojowu, Peter LeFanu Lumsdaine, Alexey Ustinov, Ben McKay, Anton Fetisov Jan 23 at 0:54

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    $\begingroup$ They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique. $\endgroup$ – Wojowu Jan 20 at 23:01
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    $\begingroup$ Is not it clearly visible on your photograph that they are NOT circles? $\endgroup$ – Alexandre Eremenko Jan 20 at 23:06
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    $\begingroup$ I would think since gravity is pulling them down, a configuration with the lowest energy would include an egg at the center for most quantities. The optimal circle packing problem doesn't address the 3rd dimension. $\endgroup$ – Greg Schmit Jan 20 at 23:43
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    $\begingroup$ Consider a flat-bottomed stainless steel potential well of diameter slightly less than $3$ times the diameter of the yolks. Experimental evidence (namely, i.stack.imgur.com/QE8iT.jpg and i.stack.imgur.com/mmT4b.jpg ) suggests that both configurations are stable. At supercritical diameters (not pictured), the eggs seem to prefer configurations that minimize the total surface area and therefore avoid the configuration with $6$-fold symmetry. Thermal annealing proved uninsightful. $\endgroup$ – MTyson Jan 21 at 0:14
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    $\begingroup$ This belongs on math.stackexchange, not here $\endgroup$ – Ross Millikan Jan 21 at 3:57
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The system doesn't try to minimise the radius of the enclosing circle, but its potential energy. We can idealise this as non-overlapping disks in a convex rotationally symmetric potential $V$ with $V(0) = 0$. The configuration that was physically realised then has potential energy $5 V(d)$ (with $d$ the diameter of the yolks) while the configuration from Wikipedia would have potential energy $6 V(d)$.

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    $\begingroup$ However, the $d$ in $6 V(d)$ is smaller than the one in $5 V(d)$. You need to show that the inequality still holds. $\endgroup$ – Matthias Urlichs Jan 21 at 8:08
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    $\begingroup$ @MatthiasUrlichs Assuming perfect, rigid disks, the $d$ is the same. $\endgroup$ – Wojowu Jan 21 at 9:36
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    $\begingroup$ @MatthiasUrlichs If you look at the images in the question, the size of the circle of optimal 6- and 7-packings are the same, so removing any circle from the optimal 7-packing creates another optimal 6-packing. And if you remove a circle not on the center, you can move the other circles around a bit. So the $d$ is equal for both cases. $\endgroup$ – Janne Kokkala Jan 21 at 14:49
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Those packing rules only apply for rigid circles. Anyone who's ever cracked an egg knows that yolks are not rigid. As a result of that, you can clearly see that the sides of yolks are flattened as they touch another yolk.

So those packing rules simply don't apply.

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    $\begingroup$ There is energy to be saved by compressing the yolks out of circles. Presumably there is more to be saved in the $5+1$ configuration than in the circle of $6$. $\endgroup$ – Ross Millikan Jan 21 at 3:58
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What do you mean, "the yolks don't follow optimal packing"? Sure they do. The configuration with one yolk in the center has the exact same radius as the one with six yolks distributed along the edge.

It also has lower potential energy, thus the 6-circle solution you cited is a non-global optimum at best. In fact it's probably metastable, given egg yolks' general tendency to be squishy blobs instead of perfect circles.

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In addition to the excellent answers already added, it is important to note that in addition to Martin Hairer's description, this problem is distinct from circle packing in another way: the system tries to minimize the potential function at every point in time, subject to the physical laws governing the movement of egg yolks in a bowl. This is not generally equivalent to minimizing the potential function, and might lead to a local but not global minimum.

One example of this is a pencil balancing on its tip. This is an equilibrium state, but clearly the potential energy is suboptimal. This example is admittedly rather far from the egg yolk problem though, so an edit from someone with a more similar example would probably be appropriate.

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    $\begingroup$ Welcome to MO dreamconspiracy. An egg yolk balanced on top of a bowl instead of inside is perhaps a closer example of unstable equilibrium, but the configuration they've found above is stable and (essentially) the global minimum PE. $\endgroup$ – Alec Rhea Jan 21 at 6:45
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    $\begingroup$ @alecrhea a global minimum is essentially always stable in a system that is governed by physical laws where the gradient of potential is proportional to acceleration. That doesn't mean there are no other stable points or that these are never reached $\endgroup$ – DreamConspiracy Jan 21 at 6:48
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    $\begingroup$ Probably it's just a matter of different definitions, but I would call a pencil balancing on its tip an unstable equilibrium point. I guess that when you write "stable" you mean what I call "equilibrium point", not what I call "stable". $\endgroup$ – Federico Poloni Jan 22 at 12:52
  • $\begingroup$ @FedericoPoloni ah yes. I will edit the answer to fix the terminology $\endgroup$ – DreamConspiracy Jan 22 at 12:55
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In addition to the non-rigid quality of egg yolks and the noted third dimension, it would take precision to align six yolks in a circle. By random placement, they most often find a closer packing, i. e. one without a big honking space in the middle.

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  • $\begingroup$ @DreamConspiracy: did you mean to suggest that configurations with a space in the middle are unstable equilibria? I didn't really get that from your answer. $\endgroup$ – Nik Weaver Jan 22 at 16:37

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