Six yolks in a bowl: Why not optimal circle packing?












6












$begingroup$


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.





discrete-geometry classical-mechanics circle-packing







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
    $endgroup$
    – Wojowu
    2 hours ago








  • 3




    $begingroup$
    Is not it clearly visible on your photograph that they are NOT circles?
    $endgroup$
    – Alexandre Eremenko
    2 hours ago






  • 1




    $begingroup$
    I think the yolk in the middle is getting smushed, so the ones on the outside have larger radius effectively. There is an optimal packing with 6 circles where the 1 inside is smaller than 5 congruent circles outside.
    $endgroup$
    – Douglas Sirk
    2 hours ago






  • 3




    $begingroup$
    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
    2 hours ago






  • 1




    $begingroup$
    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
    1 hour ago
















6












$begingroup$


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.





discrete-geometry classical-mechanics circle-packing







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
    $endgroup$
    – Wojowu
    2 hours ago








  • 3




    $begingroup$
    Is not it clearly visible on your photograph that they are NOT circles?
    $endgroup$
    – Alexandre Eremenko
    2 hours ago






  • 1




    $begingroup$
    I think the yolk in the middle is getting smushed, so the ones on the outside have larger radius effectively. There is an optimal packing with 6 circles where the 1 inside is smaller than 5 congruent circles outside.
    $endgroup$
    – Douglas Sirk
    2 hours ago






  • 3




    $begingroup$
    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
    2 hours ago






  • 1




    $begingroup$
    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
    1 hour ago














6












6








6





$begingroup$


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.





discrete-geometry classical-mechanics circle-packing







share|cite|improve this question











$endgroup$




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.





discrete-geometry classical-mechanics circle-packing




discrete-geometry classical-mechanics circle-packing






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 hours ago







Joseph O'Rourke

















asked 3 hours ago









Joseph O'RourkeJoseph O'Rourke

84.7k16225692




84.7k16225692








  • 3




    $begingroup$
    They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
    $endgroup$
    – Wojowu
    2 hours ago








  • 3




    $begingroup$
    Is not it clearly visible on your photograph that they are NOT circles?
    $endgroup$
    – Alexandre Eremenko
    2 hours ago






  • 1




    $begingroup$
    I think the yolk in the middle is getting smushed, so the ones on the outside have larger radius effectively. There is an optimal packing with 6 circles where the 1 inside is smaller than 5 congruent circles outside.
    $endgroup$
    – Douglas Sirk
    2 hours ago






  • 3




    $begingroup$
    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
    2 hours ago






  • 1




    $begingroup$
    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
    1 hour ago














  • 3




    $begingroup$
    They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
    $endgroup$
    – Wojowu
    2 hours ago








  • 3




    $begingroup$
    Is not it clearly visible on your photograph that they are NOT circles?
    $endgroup$
    – Alexandre Eremenko
    2 hours ago






  • 1




    $begingroup$
    I think the yolk in the middle is getting smushed, so the ones on the outside have larger radius effectively. There is an optimal packing with 6 circles where the 1 inside is smaller than 5 congruent circles outside.
    $endgroup$
    – Douglas Sirk
    2 hours ago






  • 3




    $begingroup$
    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
    2 hours ago






  • 1




    $begingroup$
    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
    1 hour ago








3




3




$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
2 hours ago






$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
2 hours ago






3




3




$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
2 hours ago




$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
2 hours ago




1




1




$begingroup$
I think the yolk in the middle is getting smushed, so the ones on the outside have larger radius effectively. There is an optimal packing with 6 circles where the 1 inside is smaller than 5 congruent circles outside.
$endgroup$
– Douglas Sirk
2 hours ago




$begingroup$
I think the yolk in the middle is getting smushed, so the ones on the outside have larger radius effectively. There is an optimal packing with 6 circles where the 1 inside is smaller than 5 congruent circles outside.
$endgroup$
– Douglas Sirk
2 hours ago




3




3




$begingroup$
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
2 hours ago




$begingroup$
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
2 hours ago




1




1




$begingroup$
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
1 hour ago




$begingroup$
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
1 hour ago










2 Answers
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22












$begingroup$

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 $6 V(d)$.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$













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      2 Answers
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      2 Answers
      2






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      22












      $begingroup$

      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 $6 V(d)$.






      share|cite|improve this answer









      $endgroup$


















        22












        $begingroup$

        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 $6 V(d)$.






        share|cite|improve this answer









        $endgroup$
















          22












          22








          22





          $begingroup$

          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 $6 V(d)$.






          share|cite|improve this answer









          $endgroup$



          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 $6 V(d)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Martin HairerMartin Hairer

          3,9611628




          3,9611628























              0












              $begingroup$

              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.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                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.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  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.






                  share|cite|improve this answer









                  $endgroup$



                  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.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 1 hour ago









                  GrahamGraham

                  1091




                  1091






























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