Doing and Assessing a Math Project
Designing a Math Project
Topic: Fractal Geometry
Group members: Ralph Baker, Laura Cang, Mina Nozar
November 23, 2009
1a) Doing a Math Project:
Topic: Fractal geometry
Description: From the position of Math 11 students, research Fractals
Format: Presentation
Description:
Fractal: pattern that repeats itself, with detailed structure at different magnification scales
Infinitely complex but bases on incredibly simple principles (additions & subtraction)
Mandelbrot and Julia Sets:
Most studied family of functions parameterized by a variable, c: z = z2 + c
Z and c are complex numbers
Julia set will vary on the complex plane. Some of these Julia sets will be connected, and some will be disconnected, and so this character. Mandlebrot set is the set for values of c where the Julia set is connected.
When z→0 the point is coloured black
When z -> infinity, the point is coloured. The colours are chosen arbitrarily but each colour represents a different rate of change
Iterative/recursive process: Millions and millions of operations to produce a complete set
In each iteration, the the output in as the input for the next iteration
Simplest: Self-similar - smallest pieces identical to the whole, except for a dialation
Ex: Sierpinski triangle/gasket/seive, Sierpinski carpet, Koch curve
Fractal dimension: Not a whole number!
Instances/examples of Fractals and scaling patterns
Nature: Clouds: puffs of puffs
Mountains: Peaks within peaks
Coastlines: Little bits looks like larger bits
Trees: Branches of Branches or Branches
Fern leaf: Series of fronds extending from the spine to the left and right,
each frond made of sub-fornds,
each sub-frond made of sub-fronds
Broccoli romanesco, a cauliflower-like plant: Head consisting of a spiral swirl of florets,
each floret a spiral swirl of a sub-florets,
each sub-floret a spiral swirl of sub-sub florets
Art & Architecture: African textile, sculpture, architecture, and hairstyle designs
Applications: Modeling, Image compression
References:
1. Ron Eglash, African fractals, in buildings and braids http://www.ted.com/index.php/talks/ron_eglash_on_african_fractals.html
2. Ron Eglash, African Fractals
3. Ian Stewart, The Magical Maze – Seeing the World Through Mathematical Eyes, John Wiley & Sons, Inc., 1997
4. Ian Stewart and Martin Golubitsky, Fearful Symmetry – Is God A Geometer?, Penguin Group, 1992
5. Hans Lauwerier, Fractals – Endlessly Repeaterd Geometrical Figures, Princeton University Press, 1991
6. http://en.wikipedia.org/wiki/Sierpinski_triangle
7. http://en.wikipedia.org/wiki/Sierpinski_carpet
1b) Assessing a Math Project: Individual, 500 word reflections.
2. Designing a Project
A project we, as teachers, could assign to our Math 11 students:
Students should be able to identify the basic building block of such a complex pattern.
a) Find an instance of fractalization in your life
b) Discern what the single building block (seed shape) would be
c) Draw or model the building block
d) Describe the instructions (transformations/translations) required to build the full fractal form
ie. what does the iterative process entail?
e) Determine the dimensionality of the fractal chosen
Assessing the project based on the following Rubric
| Description | 0 | 1 | 2 | 3 |
| Choice of fractal with description and where you found it with justification of your choice. | Inappropriate choice, lacking description | Incomplete justification for fractal choice | Well thought out. Description demonstrates understanding. | Description includes complete justification of choice and process for locating your fractal. Demonstrates full understanding of concept. |
| Identify the single building block | No block identified. | Indicated choice of building block. Description may be unclear or incomplete. | Description of unit is understandable. Indicates how you found the building block. | Clear description of block. Process for identification obvious and correct. |
| Drawing or model of single block | Drawing or model is incomplete, demonstrates little effort. | Drawing or model is adequate but unlabeled and incomplete. | Adequate drawing or model. Diagrams are labeled and demonstrates thought. | Aesthetically pleasing drawing or model. Diagrams are correct, clear and mathematically relevant. |
| Description of the iteration | Description uses no mathematically relevant language and is lacking clarity or is incorrect. | Description uses little to no mathematically relevant language and may be incomplete | Description is adequate and uses some mathematically relevant language | Description is clear, mathematically relevant, and demonstrates a high level of mathematical understanding |
| Model/Drawing with minimum 2 interations | Model/Drawing is incomplete and demonstrates little to no understanding of fractalization. | Iterations follow the description to some degree. | Model is clear and follows iterative description with few errors. | Model is aesthetically pleasing and demonstrates a high level of understanding and follows the iterative description closely |
Presentation
Instances/examples
Fern
Romanesco broccoli
Tree
Coastline
Sierpinski triangle/gasket/seive:
Plane fractal, one of the basic examples of similar-sets, named after the Polish mathematician Wacław Sierpiński who described it in 1915
Generation/Iteration algorithm:
Start with an equilateral triangle
Make a triangle with size ½ height and ½ width of the original triangle
Seed shape: triangle Scaling transformation size: 1/2
Make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner. A central triangle hole merges in the central hole
Repeat with each of the smaller triangles. Iterate infinite number of times!
Fractal dimention: Side can be doubled in size by taking three identical copies:
Sierpinski carpet: A plane fractal - generalization of the Cantor set to 2D - first described Wacław Sierpińskiin 1916
Generation/Iteration algorithm:
Seed shape: square Scaling transformation size: 1/3
Start with a square
Divide it into 9 squares, each one- third the size of the orignial square
Remove the central square
Repeat with each of the smaller squares. Iterate infinite number of times!
Fractal dimention: Takes 8 copies to make the size of the size increase by a factor of 3
Koch Curve: Another example of similar-sets, named for Swedish mathematician Helge von Koch
Seed shape: spike
Scaling transformation size: 1/3
Start with astraight line segment (stage 0)
Replace the middle third of this segment by two pieces, each as long as the middle third (joined like two sides of an equilateral triangle - spike)
At each succeeding stage, replace each line segment at its middle third by a spike.
Repeat the above process infinitely. The limit of this infinite process results in the Koch curve
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