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The Wisconsin Menger Sponge Project

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

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The Wisconsin Menger Sponge Project

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  1. WMC Green Lake May 2012 The Wisconsin Menger Sponge Project 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. Fractals "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. Sierpinski Carpet 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)

  5. The Menger Sponge Level 2 Level 3 Level 1

  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

  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

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

  19. Wisconsin Menger Sponge Project • 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 LevelUnpaneled • A business card is considered a “unit,” U • U0=6, U1=6x20=120, U2=120x20=2400, U3=48000 • Un=6x20n

  22. Number of Cards to Build Each Level(Paneled) • P0=12 • Where two Level n-1 cubes are locked together, those sides won’t need paneling and those panels must be subtracted • P1=(8 corner P0cubes)+(12 edge P0 cubes) = 8(P0-3 panels not needed) + 12(P0-2 panels not needed) = 8(P0-3)+12(P0-2) =8x9+12x10=192 units • P2=(8 corner P1cubes)+(12 edge P1 cubes) = 8(P1-3x8 panels not needed) + 12(P1-2x8 panels not needed) = 8(P1-24)+12(P1-16) =8x168+12x176=3456 units • P3=(8 corner P2cubes) +(12 edge P2 cubes) = 8 (P2-3x82)+12(P2-2x82) = 66, 048 units • Suggests a general recursive formula Pn=8(Pn-1-3x8n-1)+12(Pn-1-2x8n-1)= 20Pn-1-6x8n

  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 xnand 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: • Pn=8x20n+4x8n

  25. Volume of Each Level • V0=1 unit3 • V1=1- (1/3)3 x 7 • V2=1- (1/3)3 x 7 – (1/3 x 1/3)3 x 7 x 20 • V3=1- (1/3)3 x 7 – (1/3 x 1/3)3 x 7 x 20 – (1/33)3 x 7 x202 • 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 * A0 = 6 A1 = (6x8 + 6x4)/9 = 72/9 = 8 A2 = ((6x8 + 6x4)x8 + 6x4x20)/(9x9) A3 = ((6x8 + 6x4)x8x8 + 6x4x20x(8) + 6x4x(20x20)))/(9x9x9) A4= ((6x8 + 6x4)x8x8x8 + 6x4x20x(8x8) + 6x4x(20x20)x8 + 6x4x(20x20x20))/(9x9x9x9)

  28. Surface Area of the Menger Sponge

  29. Let the Project Begin!

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