1. The Wisconsin Menger Sponge Project WMC Green Lake May 2012 Presenters: Roxanne Back and Aaron Bieniek

2. Today What is a Menger Sponge and how did this project get started? What is this project? How can I use this in my class? How do I begin?

3. "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.“ (Mandelbrot, 1983). Definition: A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. - Wolfram MathWorld

4.  The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions (another is Cantor dust)fractal Wacław SierpińskiCantor setCantor dust

5. The Menger Sponge Level 1 Level 2Level 3

6. Inspiriation Menger Mania

7. Nicholas Rougeux  2007 Level 2 Post-its

8. Jeannine Mosely – MIT Origami Club  1995- Level 3  150 pounds  66,000 + Business Cards  5 feet tall

9. University of Florida 2011  Kevin Knudson and Honor Students level 3

10. Nicholas Rougeux  Mengermania Website  2008 Attempt at a Level 4 (only 2.6 % complete) “The sponge is soaked.”

11. Fractals Reference App by Wolfram “Not all yellow sponges are named Bob”

12. Wolfram Alpha  http://www.wolframalpha.com/input/?i=menger+sponge http://www.wolframalpha.com/input/?i=menger+sponge

13. Who will help me build a level 3?  (And be more impressive than those previously built)

14. Who will help me build a level 3?  (And be more impressive than those previously built)  Why not just build a Level 4????

15. Who will help me build a level 3?  High School Students! Timeline  Pilot at Whitnall High School  Launch at WMC Green Lake Conference in May 2012  Collect Level 1’s and 2’s Sept. 2012-April 2013  Display and Celebrate completed Level 3 at WMC Green Lake Conference May 2013

16. Math, Menger, and Modeling  Volume  Surface Area  Fractal Dimension  Combinatorics  Limits  Closed form formulas  Scale (Ratio)

17. Dimension Figure Dimension No. of Copies Line segment12 = 2 1 Square24 = 2 2 Cube38 = 2 3 Any Self-Similar Figure dn = 2 d

18. The fractal dimension of a Menger Sponge  N = 3^d  20 self-similar pieces, magnification factor =3  Fractal dimension = log 20/log 3 ~2.73 Level 1 Level 2Level 3

19. Wisconsin Menger Sponge Project  http://wisconsinmengerspongeproject.wikispaces.com/ http://wisconsinmengerspongeproject.wikispaces.com/

20. The Modular Menger Sponge  Made with business cards  Level 0 made from 6 business cards to make a cube  20 cubes will make a Level 1; 20 Level 1 frames will make a Level 2  Scaling down the model after each iteration so it remains the same Level 0 size throughout, in an infinite way, would give one the Menger Sponge

21. Number of Cards to Build Each Level Unpaneled  A business card is considered a “unit,” U  U 0 =6, U 1 =6x20=120, U 2 =120x20=2400, U 3 =48000  U n =6x20 n

22. Number of Cards to Build Each Level (Paneled)  P 0 =12  Where two Level n-1 cubes are locked together, those sides won’t need paneling and those panels must be subtracted  P 1 =(8 corner P 0 cubes)+(12 edge P 0 cubes) = 8(P 0 -3 panels not needed) + 12(P 0 -2 panels not needed) = 8(P 0 -3)+12(P 0 -2) =8x9+12x10=192 units  P 2 =(8 corner P 1 cubes)+(12 edge P 1 cubes) = 8(P 1 -3x8 panels not needed) + 12(P 1 -2x8 panels not needed) = 8(P 1 -24)+12(P 1 -16) =8x168+12x176=3456 units  P 3 =(8 corner P 2 cubes) +(12 edge P 2 cubes) = 8 (P 2 -3x8 2 )+12(P 2 -2x8 2 ) = 66, 048 units  Suggests a general recursive formula  P n =8(P n-1 -3x8 n-1 )+12(P n-1 -2x8 n-1 )= 20P n- 1 -6x8 n

23. Number of Cards to Build Each Level (Paneled)  The recurrence can be solved to get a closed formula using generating functions: multiply the eqn by x n and sum over all n ≥ 1 to get  The generating function and use

24. Number of Cards to Build Each Level (Paneled)  Using Partial fractions  Which gives 6=A(1-20x)+B(1-8x), let x=1/8 to give A=-4 and then x=1/20 to give B=10  The generating function is thus:  P n =8x20 n +4x8 n

25. Volume of Each Level  V 0 =1 unit 3  V 1 =1- (1/3) 3 x 7  V 2 =1- (1/3) 3 x 7 – (1/3 x 1/3) 3 x 7 x 20  V 3 =1- (1/3) 3 x 7 – (1/3 x 1/3) 3 x 7 x 20 – (1/3 3 ) 3 x 7 x20 2  We recognize this contains a geometric series.

26. Volume of Each Level  A closed form of a geometric series:  The volume of the nth iteration:*  To find the volume of the Menger Sponge:

27. Surface Area of Each Level  A 0 = 6  A 1 = (6x8 + 6x4)/9 = 72/9 = 8  A 2 = ((6x8 + 6x4)x8 + 6x4x20)/(9x9)  A 3 = ((6x8 + 6x4)x8x8 + 6x4x20x(8) + 6x4x(20x20)))/(9x9x9)  A 4 = ((6x8 + 6x4)x8x8x8 + 6x4x20x(8x8) + 6x4x(20x20)x8 + 6x4x(20x20x20))/(9x9x9x9) *

28. Surface Area of the Menger Sponge

29. Let the Project Begin!