Bubble or Nothing

1. Bubble or Nothing Lalu Simcik, PhD Cabrillo College CMC³ Spring 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. Simplified, step by step presentation Offer A(x) to use even if student is blocked Prioritize use of vertex Avoid using ‘y’ Algebra II

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

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

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

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

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

14. 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:

15. Maximize outer triangle area • Using Heron’s formula:

16. 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

17. Did you know……

18. But maybe you didn’t know….

19. Perhaps don’t want to know

20. Presentation in Calculus III – returning to the rectangular corral problem • Rectangular corral with constrained length of fence (say 1000 feet) • Perimeter equation • Area Equation is a multi-variable function

21. Presentation in Calculus III • The multivariable corral problem continues without variable substitution • Maximize enclosed area using “Big D” does not work • Confirm limitation with a surface plot of the

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

24. Moving to 3-D: the Aviary Assuming fabric on all sides, including the floor 600 square feet of netting, find Maximal volume aviary

25. 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

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

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

30. 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 exists !!!), utilizes equal boundary material in all three dimensions.

31. 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)

32. 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

33. 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

34. 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)

35. 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

36. Tom Noddy on Letterman

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

38. 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