Ch. 9: Symmetry and Optimization!

What is the longest stick that fits in this cubical box?

Is this the only solution?

What is the longest stick that fits in this cubical box? Is this the only solution? NO! Performing any symmetry of the cube gives you a picture of another solution.

What is the longest stick that fits in this cubical box? Is this the only solution? NO! Performing any symmetry of the cube gives you a picture of another solution. THUS, there are 4 solutions which are permuted By the symmetries of the cube!

What is the longest stick that fits in this cubical box? Is this the only solution? NO! Performing any symmetry of the cube gives you a picture of another solution. THUS, there are 4 solutions which are permuted By the symmetries of the cube! What is the largest cube that fits in a dodecahedron?

What is the longest stick that fits in this cubical box? Is this the only solution? NO! Performing any symmetry of the cube gives you a picture of another solution. THUS, there are 4 solutions which are permuted By the symmetries of the cube! What is the largest cube that fits in a dodecahedron? Is this the only solution?

What is the longest stick that fits in this cubical box? Is this the only solution? NO! Performing any symmetry of the cube gives you a picture of another solution. THUS, there are 4 solutions which are permuted By the symmetries of the cube! What is the largest cube that fits in a dodecahedron? Is this the only solution? NO! Performing any symmetry of the dodecahedron gives you a picture of another solution. THUS, there are 5 solutions which are permuted By the symmetries of the dodecahedron!

The solution to an optimization problem often has the same symmetries as the problem. When it does not, there must be multiple “tied” solutions that are permuted by the symmetries of the problem.

Why are bubbles spherical?

Why are honeycombs hexagonal?

Why are bubbles spherical? Why are honeycombs hexagonal? An icosahedral HIV virus Why are many viruses icosahedral?

Why are bubbles spherical? Why are honeycombs hexagonal? An icosahedral HIV virus Why are many viruses icosahedral? Nature’s solutions to optimization problems are often highly symmetric! Nature’s solutions to optimization problems are often highly symmetric!

Why are bubbles spherical? The Bubble Theorem: The sphere is the least-surface- area way to enclose a given volume. Nature’s solutions to optimization problems are often highly symmetric! Nature’s solutions to optimization problems are often highly symmetric!

What optimization problem is nature solving here? Does each bubble surface have the same symmetries as its frame?

What optimization problem is nature solving here? Does the bubble surface have the same symmetries as its frame?

To enclose 25 acres of grassland using the least possible length of fencing, in what shape should you build your fence?

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane.

PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). We wish to prove that Farmer Don’s winning fence is a circle, but for all we know now, it could have a crazy shape like this.

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). The horizontal line which is chosen to divide its area in half will automatically also divide its perimeter in half. Why?

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). The horizontal line which is chosen to divide its area in half will automatically also divide its perimeter in half. Why? (If say the top had more perimeter, then replacing the top with the reflection of the bottom would enclose 25 acres using less fence, contradicting the assumption that the original shape was the best.)

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). Replacing the top with the reflection of the bottom produces a TIED WINNER (with at least 2 symmetries). Replacing the top with the reflection of the bottom produces a TIED WINNER (with at least 2 symmetries). (same area and same perimeter)

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). The vertical line which is chosen to divide its area in half will automatically also divide its perimeter in half.

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). Replacing the right with the reflection of the left produces a TIED WINNER (with at least 4 symmetries). Replacing the right with the reflection of the left produces a TIED WINNER (with at least 4 symmetries). (same area and same perimeter)

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). Summary: Starting with a winner, you can build a tied winner with at least four symmetries: I, H, V, R 180. Why is this a symmetry?

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). Claim 1: Every line through the center divides the area and perimeter in half. Because R 180 is a symmetry that exchanges the two sides of the line.

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). Claim 2: Every line through the center meets the fence at right angles. Why is a non-right angle like this impossible?

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). Claim 2: Every line through the center meets the fence at right angles. Why is a non-right angle like this impossible? Because replacing one side with the reflection of the other would produce something impossible: a winner with an “innie-point”.

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). Summary: Starting with a winner, you can build a tied winner which (unlike in this picture) meets every line through the origin at right angles.

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). BRAINSTORM: Think of fence shapes that meet every radial line at right angles.

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). BRAINSTORM: Think of fence shapes that Meet every radial line at right angles. THE CIRCLE IS THE ONLY POSSIBILITY! so the tied winner must be a circle

THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. PROOF: Farmer Don found the least-perimeter possible way to enclose the given area (say 25 acres). BRAINSTORM: Think of fence shapes that Meet every radial line at right angles. THE CIRCLE IS THE ONLY POSSIBILITY! so the tied winner must be a circle So this must be a quarter-circle (and similarly for the other 3 quadrants)

Related question: why are bubbles spherical? THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane.

THE BUBBLE THEOREM: The sphere is the least-surface-area way to enclose a given volume. THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. Related question: why are bubbles spherical?

THE BUBBLE THEOREM: The sphere is the least-surface-area way to enclose a given volume. THE CIRCLE THEOREM: The circle is the least-perimeter way to enclose a given area in the plane. THE DOUBLE BUBBLE PROBLEM: What is the least-surface-area way to enclose and separate two (possibly different) volumes? What bubbles do when they colide Another mathematical possibility Which is best? Related question: why are bubbles spherical?