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Bubbles for Pre-Calculus and Beyond

Explore various optimization problems involving rectangular corrals and aviaries, including maximizing area, volume, and surface area. Learn about the concept of Lagrange multipliers and the use of boundary materials in different dimensions. Discover the connection between these problems and the regular polyhedra.

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Bubbles for Pre-Calculus and Beyond

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  1. Bubbles for Pre-Calculus and Beyond Lalu Simcik, PhD Cabrillo College CMC³ Fall 2011 www.cabrillo.edu/~lsimcik

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

  3. 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)

  4. Area function Parabola

  5. Got more animals?

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

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

  8. Presentation in Precalculus • More autonomous style • Double Jeopardy • More animals

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

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

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

  12. 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

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

  14. 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

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

  16. Aviary with n-chambers • Method of Lagrange n chambers

  17. 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 aviary with finite or infinite rectangular internal or external additions that exist, utilizes equal boundary material in all n dimensions.

  18. 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)

  19. 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

  20. 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)

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

  22. Question: What is a ‘Cube’ Bubble?

  23. 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

  24. Platonic Solids Continued: Dodecahedron

  25. 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

  26. Tom Noddy on Letterman

  27. Icosahedron Bubble • Question – can this exist?

  28. Icosahedron Bubble • Requires 5 edges to meet (impossible!)

  29. 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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