Matrices Fractals.pdf - ncssm

“A fractal is an object or quantity that displaysself-similarity, in a somewhat technical sense,on all scales. The object need not exhibitexactly the same structure at all scales, but thesame "type" of structures must appear on allscales.“An object is said to be self-similar if it looks"roughly" the same on any scale.”- Wolfram MathWorld

Consider an initial image We want to produce a newimage that looks like

( x , y ) Consider the point 0 0 represented as a column vectoror a 2 x 1 matrix.xy00We want to create matrices that will transform the pointthrough matrix operations.acbdxy00orx0 e y0 f

About the x-axis1 0 x0 x0 0 1y y 0 0About the y-axis1 0x0 x0 0 1y y 0 0

Consider a rotation through an angle of B in the counterclockwisedirection. We will write the point using polarnotation as follows:x r cos( A)y r sin( A)

We want to find the l, m, n and p so thatx l m x y n p y

x r cos( A B)y r sin( A B)Using the angle-addition formulas:x r cos( A B) r cos( A)cos( B) sin( A)sin( B)y r sin( A B) r sin( A)cos( B) cos( A)sin( B)x r cos( A)cos( B) r sin( A)sin( B)y r sin( A)cos( B) r cos( A)sin( B)

x r cos( A)cos( B) r sin( A)sin( B)y r sin( A)cos( B) r cos( A)sin( B)x x cos( B) y sin( B)y y cos( B) xsin( B)Or in written as a matrix equationx cos( B) sin( B) x sin( ) cos( ) y B B y

cos( B) sin( B)sin( ) cos( ) B B So if we want to rotate the point (x,y) throughan angle of 45 degrees, we have a rotationmatrix equal tocos( B) sin( B)sin( ) cos( ) B B

S = Stretch matrixR = Rotational matrixMatrix for CBoN code = RS or SR?

One line of code for each “new” figure:a b c d e fWherex a b x e y c d y f Note: a, b, c, and d will contain both dilationsand rotations.

For each sub-figure in your fractal image, you will need a lineof code: a b c d e fChoose Multiple Reduction Copy Machine from the drop downmenu.Type your lines of code in the dialog box – You can copy andpaste these from a more user-friendly input doc.Change the depth to 1, then 2, then n where n is where youwant to “stop”. Click Restart after each depth change.Sit back and watch the beautiful fractal image unfold!

Create your very own fractal using Multiple ReductionsCopy Machines. Explain how you use matrices – rotation, translation anddilations to create step 1 of your fractal image. For each line of code, a,b,c,d,e,f, give R, S and T,R S x0 +Ty0Include a screen shot of step 1, 2 and nExplain why you think this fractal image is particularlyinteresting to you – beyond “It’s really pretty.”

Quiz or test question. Given steps 1, and 2,write the codes to produce the fractal image. Given the matrices, draw steps 1 and 2. Others?Extend to Iterated Function Systems

“ I was highly interested in this lab from the beginning, because it mademath into something creative and artistic, and it also incorporatedtechnology in an innovative way…. After I create my own design, I had tofigure out how to make it mathematically possible, which made me thinkabout how math is all around us, even in ways we usually don’tacknowledge…, I was able to put it within the program and see howcomplex it could become, which was fascinating considering how simple itwas to enter the data.It was amazing to see how something so mathematically simple as a fewsquares rotated, translated, compressed, stretched, and entered into amatrix could become so complex, with multiple layers and much visualinterest. Furthermore, it made me think of the reiterations we did earlierin the year, and see a new, more creative use for such mathematics, asthis software simply reiterated the original design within each of thesquares it saw.”

“Our unit in fractals was very interesting to me. I lovedseeing the patterns appear and observing how every tinychange in the transformation matrices significantlychanged the outcome of the fractal…We didn’t just learn about the formation of fractals; wealso learned a great deal about writing matrices totransform points. Rotation matrices in particular werevery interesting to me…However, since we derived this rotation matrix in class,there was no need for me to memorize it; it just madesense. Using this on my fractal project was also veryimportant to the design I wanted to produce.This was, by far, my favorite project of the year.”

Astronomy:GalaxiesRings of SaturnBio / Chem:Bacterial CulturesChemical ReactionsHuman AnatomyMoleculesPlantsPopulation GrowthOther:Clouds coastlines andBorderlinesData CompressionDiffusionEconomyFractal ArtFractal MusicLandscapesNewton’s MethodSpecial Effects (Star Trek)Weather

Text: The Computational Beauty of Nature:Computer Explorations of Fractals, Chaos, Complex Systems,and Adaptation, Gary Flake MIT Press, 1998.CBoN Website for Java applets:https://mitpress2.mit.edu/books/FLAOH/cbnhtml/home.html NCTM Mathematics Teacher Article:Hands-On Fractals and the Unexpected Mathematics, AlanGluchoff, April 2006, Vol 99.Fractals, Hunting the Hidden Dimension NOVA home video

NCSSM Post-AP Calculus Project:Newton’s Method in the Complex Planehttp://www.ncssm.edu/courses/math/apcalcprojects/newtonsmethod/ Winfeed – Free program that creates images for Mandelbrot,Iterated Functions Systems, Bifurcation Diagrams, and muchmore, Rick Parris at Phillips Exeter Academy.http://math.exeter.edu/rparris/winfeed.html NetLogo – Models Library Fractalshttp://ccl.northwestern.edu/netlogo/

Anja S. Greer ConferencePhillips Exeter AcademyJune 23 – June 28, 2013 Tactile Fractalshttp://www.nytimes.com/2013/01/22/science/uscexhibit-shows-fractals-built-from-paper.html?hpw

Complex Systems Project Intro to IFS – Iterated Functions System