Six yolks in a bowl: Why not optimal circle packing?
$begingroup$
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.
Image from Wikipedia.
Optimal packings for $5,6,7$ circles.
discrete-geometry classical-mechanics circle-packing
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
$endgroup$
add a comment |
$begingroup$
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.
Image from Wikipedia.
Optimal packings for $5,6,7$ circles.
discrete-geometry classical-mechanics circle-packing
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
$endgroup$
1
$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
1 hour ago
2
$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
1 hour 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
57 mins ago
$begingroup$
@Wojowu: Could you verify that, or offer a citation? I am finding only that six equal circles around the perimeter is the uniquely optimal $6$-circle configuration.
$endgroup$
– Joseph O'Rourke
44 mins 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
29 mins ago
add a comment |
$begingroup$
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.
Image from Wikipedia.
Optimal packings for $5,6,7$ circles.
discrete-geometry classical-mechanics circle-packing
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
$endgroup$
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.
Image from Wikipedia.
Optimal packings for $5,6,7$ circles.
discrete-geometry classical-mechanics circle-packing
discrete-geometry classical-mechanics circle-packing
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
edited 1 hour ago
Joseph O'Rourke
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
84.7k16225692
84.7k16225692
1
$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
1 hour ago
2
$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
1 hour 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
57 mins ago
$begingroup$
@Wojowu: Could you verify that, or offer a citation? I am finding only that six equal circles around the perimeter is the uniquely optimal $6$-circle configuration.
$endgroup$
– Joseph O'Rourke
44 mins 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
29 mins ago
add a comment |
1
$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
1 hour ago
2
$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
1 hour 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
57 mins ago
$begingroup$
@Wojowu: Could you verify that, or offer a citation? I am finding only that six equal circles around the perimeter is the uniquely optimal $6$-circle configuration.
$endgroup$
– Joseph O'Rourke
44 mins 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
29 mins ago
1
1
$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
1 hour ago
$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
1 hour ago
2
2
$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
1 hour ago
$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
1 hour 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
57 mins 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
57 mins ago
$begingroup$
@Wojowu: Could you verify that, or offer a citation? I am finding only that six equal circles around the perimeter is the uniquely optimal $6$-circle configuration.
$endgroup$
– Joseph O'Rourke
44 mins ago
$begingroup$
@Wojowu: Could you verify that, or offer a citation? I am finding only that six equal circles around the perimeter is the uniquely optimal $6$-circle configuration.
$endgroup$
– Joseph O'Rourke
44 mins 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
29 mins 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
29 mins ago
add a comment |
1 Answer
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$begingroup$
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 $6 V(d)$.
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
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add a comment |
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$begingroup$
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 $6 V(d)$.
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
$endgroup$
add a comment |
$begingroup$
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 $6 V(d)$.
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
$endgroup$
add a comment |
$begingroup$
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 $6 V(d)$.
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
$endgroup$
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 $6 V(d)$.
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
answered 1 hour ago
Martin HairerMartin Hairer
3,9011628
3,9011628
add a comment |
add a comment |
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1
$begingroup$
They do. Five circles touching a center circle is an optimal configuration too. Optimal configurations aren't, in general, unique.
$endgroup$
– Wojowu
1 hour ago
2
$begingroup$
Is not it clearly visible on your photograph that they are NOT circles?
$endgroup$
– Alexandre Eremenko
1 hour 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
57 mins ago
$begingroup$
@Wojowu: Could you verify that, or offer a citation? I am finding only that six equal circles around the perimeter is the uniquely optimal $6$-circle configuration.
$endgroup$
– Joseph O'Rourke
44 mins 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
29 mins ago