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!