Bubble or Nothing

Lalu Simcik, PhD

Cabrillo College

CMC³ Spring 2011

www.cabrillo.edu/~lsimcik

The corral problem

• Rectangular corral with constrained length of fence (say 1000 feet)

• Perimeter equation

• Area Equation transformed to area function with variable substitution

xwwx 500100022

xxxA

xxxA

xwA

500

)500(2

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)

500

12

b

aa

bxv

feetwx 250

Area function Parabola

500

12

b

aa

bxv

feetwx 500

xxxA 5002

feetxw

feetx

250500

250

Got more animals?

Variations

• Two animals• Three animals

• Two animals by the river• Three animals by the

river

• Is there a pattern in all these examples?

More variations

What is the pattern in all of these examples?

Algebra II

• Simplified, step by step presentation

• Offer A(x) to use even if student is blocked

• Prioritize use of vertex

• Avoid using ‘y’

7. Llama Corral by the River: A farmer has 1000 ft of fence to make a rectangular Llama corral by a river (see picture). w x x River a) Create an equation that adds up fence lengths to equal the 1000 ft. (3 pts) b) Solve your equation for ‘w’ in part a) (2 pts) c) Use part b) to show that the Area Function (which is a function of x) is: (7 pts) xxxA 10002)( 2 d) Find the length in feet for x that will maximize the area of the corral. After you find this x, use it to find the length of w. (Hint: vertex!)

(5 pts)

Presentation in Precalculus

• More autonomous style

• Double Jeopardy

• More animals

3. With 1000 ft of fence, find the dimensions for a rectangular corral that maximize the area for 2 animals by the river. (see drawing on board) (8 pts) ↓↓↓ iii) Find the best x and w (7 pts) ↓↓↓

i) Perimeter equation:

ii) Find )(xA

)(xA = _________________

Presentation in Calculus I

• n-Animals

• Using related rates in lieu of variable substitution

• Norman Window Corral

Corrals of Infinite Internal Complexity

• Infinite number of internal walls, Zeno’s Paradox, for example

perimeterconstPwx

ii

ii

.

11

PwxtoSubject

xwwxAMaximize

:

),(

Corrals of Infinite Internal Complexity

• Substituting out ‘w’ ……leading to:

22

....

1)(

2

P

P

x

and

xP

x

xxPxA

v

….leading to

.

,

2

2

direction

orthogonaleachinused

fencetheofhalfagainonce

Pw

Px

Regular Polyhedra as Optimal Enclosures

• Well known that as the number of sides approaches infinity, the limit shape will be a circle.

2'

221 2

sinlim rn

nrHL

n

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:

Maximize outer triangle area

• Using Heron’s formula:

xccbbcb

xccb

xcbcb

dx

dA

thatshowneasilybecanitand

bcb

xccb

xcbcb

xA

cbbxcxs

24

1

22222

1

,...

2222)(

22

)(

22

1

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

2

,,02,024

10 2 c

xsoandxcsoxccbthatimpliesdx

dA

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

But maybe you didn’t know….

Sides Name

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

Perhaps don’t want to know

Sides Name

100 regular hectagon

1000 regular chiliagon

10000 regular myriagon

1,000,000 regular megagon

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

xwwxfA ,

100022 wx

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

20 xwwwxxwx fffDandff

wxfA ,

Introduce the Method of Lagrange

Maximize subject to the constraint:

What rectangle has all four sides equal to one-fourth of the perimeter?

ftp 1000xwwxf ),(

022),( pwxwxg impliesgfsogxwf 2,2,

802222),,(,22

pyieldspwxginngsubstitutiwandx

44,

82,2

pwand

pxboth

pandwxSince

Moving to 3-D: the Aviary

Assuming fabric on all sides, including the floor

600 square feet of netting, findMaximal volume aviary

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

methodDBigforyx

xyxyyxV

methodLagrangeforxyzzyxV

fixedisSxzyzxyS

s

)(),(

),,(

)(2

2

ftzyx

ftSLet

10

600 2

Approximations

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

Aviary with n-chambers

• Method of Lagrange n chambers

600)1(22),,(

,,)(

yznnxynxzzyxg

gVnxynxznyzVyznxV

))1(2(

)2)1((

)22(

yznnxznxyz

nxyyznnxyz

nxynxznxyz

zyyieldspairnd

xn

nyyieldspairst

21

21

planeyzinAreaplanexyinAreaplanexzinArean

nx

n

nx

n

nx

xn

nnx

n

nnxx

n

nnxxg

1

4

1

4

1

4

1

2)1(

1

22

1

22)(

222222

2

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.

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)

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

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

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)

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

Tom Noddy on Letterman

Icosahedron Bubble

• Requires 5 edges to meet (impossible!)

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