The Wisconsin Menger Sponge Project WMC Green Lake May 2012 Presenters: Roxanne Back and Aaron Bieniek.
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The Wisconsin Menger Sponge
Project
WM
C Green Lake May
2012
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?
"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
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)
The Menger Sponge
Level 1 Level 2 Level 3
InspiriationMenger Mania
Nicholas Rougeux
2007 Level 2 Post-its
Jeannine Mosely – MIT Origami Club
1995- Level 3
150 pounds
66,000 + Business Cards
5 feet tall
University of Florida 2011
Kevin Knudson and Honor Students level 3
Nicholas Rougeux
Mengermania Website
2008 Attempt at a Level 4
(only 2.6 % complete) “The sponge is soaked.”
Fractals Reference App by Wolfram
“Not all yellow sponges are named Bob”
Wolfram Alpha
http://www.wolframalpha.com/input/?i=menger+sponge
Who will help me build a level 3?
(And be more impressive than those previously built)
Who will help me build a level 3?
(And be more impressive than those previously built)
Why not just build a Level 4????
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
Math, Menger, and Modeling
Volume
Surface Area
Fractal Dimension
Combinatorics
Limits
Closed form formulas
Scale (Ratio)
Dimension
Figure Dimension No. of Copies
Line segment 1 2 = 2 1
Square 2 4 = 2 2
Cube 3 8 = 2 3
Any Self-Similar Figure
d n = 2 d
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 2 Level 3
Wisconsin Menger Sponge Project
http://wisconsinmengerspongeproject.wikispaces.com/
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
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
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
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 xn and sum over all n ≥ 1 to get
The generating function and use
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
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.
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:
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)
*
Surface Area of the Menger Sponge
Let the Project Begin!
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