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Bubbles for Bubble’s Sake Lalu Simcik, PhD Cabrillo College SVCCM March 2013 www.cabrillo.edu/~lsimcik

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The corral problem Rectangular corral with constrained length of fence (say 1000 feet) Perimeter equation Area Equation transformed to area function with variable substitution

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The corral problem Vertex of a parabola –Midpoint of the quadratic formula roots –completing the square –Uniqueness For one animal Leads to proof that the ideal rectangle is a square (single corral case)

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Area function Parabola

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Got more animals?

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Variations Two animals Three animals Two animals by the river Three animals by the river Is there a pattern in all these examples?

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More variations What is the pattern in all of these examples?

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Presentation in Precalculus More autonomous style Double Jeopardy More animals

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Presentation in Calculus I n-Animals Using related rates in lieu of variable substitution Norman Window Corral

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Corrals of Infinite Internal Complexity Infinite number of internal walls, Zeno’s Paradox, for example

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Corrals of Infinite Internal Complexity Substituting out ‘w’ ……leading to: ….leading to

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Regular Polyhedra as Optimal Enclosures Well known that as the number of sides approaches infinity, the limit shape will be a circle.

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Proof that regular polygons are optimal n-sided area enclosures Less known: why is a regular n-sided polygon optimal over all other n-gons? Convex / concave? Flip out concave portion to prove by contradiction that convex is necessary to be optimal. Consider two neighboring sides of the optimal convex n-gon:

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Maximize outer triangle area Using Heron’s formula:

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Each adjacent central triangle is optimal Continuing And the drawing becomes: Conclusions: the outer triangle is necessarily isosceles. This is true for all adjacent sides (i.e. adjacent sides are always equal in length) Convex n-gon with equal sides is a regular n-sided polygon

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Did you know…… Sides Name n regular n-gon 3 equilateral triangle 4 square 5 regular pentagon 6 regular hexagon 7 regular heptagon 8 regular octagon 9 regular nonagon 10 regular decagon

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But maybe you didn’t know…. SidesName 11 regular hendecagon 12 regular dodecagon 13 regular triskaidecagon 14 regular tetradecagon 15 regular pentadecagon 16 regular hexadecagon 17 regular heptadecagon 18 regular octadecagon 19 regular enneadecagon 20 regular icosagon

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Perhaps don’t want to know SidesName 100 regular hectagon 1000 regular chiliagon 10000 regular myriagon 1,000,000 regular megagon

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Presentation in Calculus III The multivariable original corral problem continues without variable substitution (one animal, 1000 ft of fence) Maximize enclosed area using “Big D” does not work Confirm limitation with a surface plot of the

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Introduce the Method of Lagrange Maximize subject to the constraint: What rectangle has all four sides equal to one-fourth of the perimeter?

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The Aviary Maximize the volume subject to the constraint of a fixed amount of surface area Lagrange Multipliers method or substitution and the use of ‘Big D’ Proof of the cube as a minimal enclosure

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Approximations 3-D mesh software (Octave, Matlab) can offer visualizations of maximization

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Aviary with n-chambers Method of Lagrange n chambers

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Aviary continued Aviary with n compartments Aviary in the corner of the room What do all these problems have in common? Conjecture: Any optimal n-dimensional rectangular enclosure with finite or infinite rectangular internal or external additions, utilizes equal boundary material in all n dimensions.

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2-D or 3-D What do all the rectangular corrals have in common with the aviaries? “Equal boundary material used in xy or xyz directions” Sphere has equal material used in all possible directions Consider the regular polyhedra in the Isepiphan Problem (Toth,1948)

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Naturally

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Double bubble Side view is ~1.01 times the area of the top (looking down the longitudinal axis) Engineer 10% error – gets promotion Physicist 1% error – gets Nobel prize Mathematician 1% error – gets back to work

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A Little Bubble Lingo Spherical Bubble that are joined share walls. Edges are where walls and bubbles meet other walls and bubbles Three walls/bubbles make an edge Edges meet in groups of four (see the end of the straw)

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Bubble Lingo - Angles –Inter wall angle is 120° –Inter edge angle is arc cos(−1/3) ≈ 109.4712° (ref: Plateau, 1873)

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Question: What is a ‘Cube’ Bubble?

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Cube bubble Boundary conditions are 6 sides in 3-D Bubbles construct minimal aviary with the constraint of –Inter wall angle is 120° –Inter edge angle is arc cos(−1/3) ≈ 109.4712° (ref: Plateau, 1873) Cube angles are nearly 20°or 30° off from Plateau angles

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Platonic Solids Continued: Dodecahedron

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Dodecahedron Bubble Regular polyhedra (Platonic Solids) are minimal surfaces for a fixed volume (not fully proven) Boundary conditions cause bubbles to create the near- Platonic Solids Inter wall and inter edge angles defined by Plateau Dodecahedron edge angles are only 7° off from Plateau angles

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Tom Noddy on Letterman

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On the BBC: The Code

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Basalt Crystal Columns

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Honeycomb

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Dodec-bubble

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Icosahedron Bubble Question – can this exist?

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Icosahedron Bubble Requires 5 edges to meet (impossible!)

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Conclusion Have fun with optimization Have a robust example with seemingly endless possibilities Ask students “What is the overall pattern here?” Create new problems easily www.cabrillo.edu/~lsimcik

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