Archive for June, 2009

Final Reflection

It’s hard for me to say whether or not this class is what I expected because I didn’t know exactly what to expect.  You never know what a class will turn out like because it depends so much on the attitudes of the students. Mr. Miller and I  developed a strong idea for what we wanted the class to be like:  lots of exploration and questioning.  Mostly, this was very successful.  Many students took to the freedom and used the time and opportunities we gave them to take their learning to a new level.  A few wasted their time and didn’t take things seriously.  To really judge things, though, I am going to have to focus on the first group because they are the ones we did this for.  I think that they discovered things about themselves and how they learn that will aid them in their futures.  I am so proud of what they accomplished!

I am going to take SO MUCH away from this class.  First of all, the hikes that Mr. Miller showed us were unbelievable and I can’t wait to visit some of these spots again.  Of course, it won’t be the same without the whole crew!  I am also taking away a deeper undersanding of Geometer’s Sketchpad and Cellular Automata.  These are two things that I’d never really been able to play around with and explore until now.  I’ve also made my first blog, which was a great experience!  The biggest things I will take have to do with the students.  First, I will take away some of the relationships I’ve built with the young adults in the class.  Some students I already knew well (such as my advisees), and our bonds were strengthened.  Others I was able to work with for the first time (Marquinika, Brenda, Karla, Dennis, Jessica, Lucretia).  These relationships and what I’ve learned from each student have changed my iterative path ever so slightly–but who knows what the Butterfly Effect has in store.  Another thing about students that I will take away is how they have an innate curiosity that is sometimes buried by the piles of “muck” they must learn in their regular classes.   These are important no doubt, but if a student has never been given the freedom to explore on their own, it is much less meaningful.  I was amazed at how giving the students cameras and a simple assignment to “find patterns” shifted their focus from socializing to searching for what makes the world around them move as it does.

The hardest thing was organizing the countless ideas and experiences that I wanted to share with my students.  There is so much I know they would benefit from seeing, but we had to try and draw a fine line between doing the right amount and DTM.  In the end, the organic nature of our planning put us right at that magical edge between order and chaos.  Another hard thing was for me to stop teaching and learn.  There is so much I can teach my students, but there is just as much that they can teach me.  I feel that I did accomplish this, however, and was able to help my students develop their own ideas and learn from what they had to share.

During this post-session, I learned how to use my camera to take much better close-up macro shots.  I usually take mostly landscape shots when I’m exploring nature because of the difficultites I’ve had with focusing on small objects.  Nevertheless, I was able to take some incredible close-up shots during our journeys and I now have an additional item in my photographer’s bag of tricks.

To close out my reflection I want to thank every single one of my students.  You all inspire me to keep working hard for you even though I won’t be teaching next year.  I hope you will keep working hard for yourselves!  I also want to thank Mr. Miller for mentoring and teaching me these last couple of years.  Aside from the new things you have shown me, you have also been a great sounding board for me to develop my own ideas and theories.  I hope we get a chance to run this postsession again!

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Sierpinski Triangle

I created this Sierpinski triangle in Geometer's Sketchpad by iterating the following algorithm hundreds of times:  Find the midpoint of each side of the triangle.  Connect the midpoints to form a new triangle.  Remove the inner triangle.  I ended up with this self-similar design.  This means that if you zoom in on any of the smaller triangles, it will look exactly like the entire figure!

Sierpinski 1: I created this Sierpinski triangle in Geometer’s Sketchpad by iterating the following algorithm hundreds of times:

  1. Find the midpoint of each side of the triangle.
  2. Connect the midpoints to form a new triangle.
  3. Remove the inner triangle.

I ended up with this self-similar design. This means that if you zoom in on any of the smaller triangles, it will look exactly like the entire figure!

Relaxed Sierpinski:  Some students found that if you relaxed rule #1 so that the inner triangles did not have to be on midpoints, you got some interesting new fractals.  Here is mine.  It's interesting how simply removing regularity makes the fracal somewhat sinister.  Notice that it is still self-similar however!

Relaxed Sierpinski: Some students found that if you relaxed rule #1 so that the inner triangles did not have to be on midpoints, you got some interesting new fractals. Here is mine. It’s interesting how simply removing regularity makes the fracal somewhat sinister. Notice that it is still self-similar however!

Cellular Sierpenski:  This is the same design as the shape above made in a completely different way.  I made this using a 1D cellular automata program with rule 90.  The program builds each successive line using a very simple set of rules on the line above. This is after about 100 iterations.

Cellular Sierpenski: This is the same design as the shape above made in a completely different way. I made this using a 1D cellular automata program with rule 90. The program builds each successive line using a very simple set of rules on the line above. This is after about 100 iterations.

Cellular Sierpenski 2:  This is the same rule after about 40,000 iterations.  Even though I've zoomed out, the shape still looks the same.  It now has over 17,000,000 squares!

Cellular Sierpenski 2: This is the same rule after about 40,000 iterations. Even though I’ve zoomed out, the shape still looks the same. It now has over 17,000,000 squares!

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Exploratorium

Fractal Cube:  This cube hanging above the entrance to the Exploratorium is not quite a three dimensions, but it is certainly larger than 2.

Fractal Cube: This cube hanging above the entrance to the Exploratorium is not quite a three dimensions, but it is certainly larger than 2.

Ice Crystals:  These fractal crystals form when water freezes.  You can actually watch as the branching pattern iterates accross the surface of the glass to form these shapes.

Ice Crystals: These fractal crystals form when water freezes. You can actually watch as the branching pattern iterates accross the surface of the glass to form these shapes.

Chaotic Diffusion:  Just like in the cellular automata, the simple rules of thermodynamics lead to the chaotic mixing of the two fluids.

Chaotic Diffusion: Just like in the cellular automata, the simple rules of thermodynamics lead to the chaotic mixing of the two fluids.

Chaotic Diffusion, Part Deux:  Again the laws of thermodynamics lead to chaos.  This time, the fluids are the same--air--but the fluid heated by the filament has different properties which appear in the shadows as it mixes with the cooler air.

Chaotic Diffusion, Part Deux: Again the laws of thermodynamics lead to chaos. This time, the fluids are the same–air–but the fluid heated by the filament has different properties which appear in the shadows as it mixes with the cooler air.

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Cellular Automata: The Game of Life

The Game of Life is a 2D cellular automata where cells either “live” (turn black) or “die” (turn white) based on how many other cells around them are alive or dead.  When you iterate, the cells interact to produce potentially complex patterns.  Below is an interesting pattern that I found that is not chaotic but reproduces the initial set up..

Generation1:  Starting with this pattern, it remains stable for infinite generations unless something is changed from the outside.  When a living cell is added to any corner, the shape begins growing.

Initial Condition: Starting with this pattern, it remains stable for infinite generations unless something is changed from the outside. When a living cell is added to any corner, the shape begins growing.

Generation 5:  After 5 generations, a very similar but much larger shape appears.

Generation 5: After 5 iterations, a very similar but much larger shape appears.

Generation 14: The initial shape has now blown up into this symmetrical "snow flake".

Generation 14: The initial shape has now blown up into this symmetrical “snow flake”

Generation 17:  After 17 iterations, the intial shape has been reproduced into 4 exact versions of itself.  If we were to add another living cell to a corner, the 4 would interact in a seemingly chaotic manner.  However, this current pattern would stay stable for inifinite iterations.

Generation 17: After 17 iterations, the initial shape has been reproduced into 4 exact versions of itself. If we were to add another living cell to a corner, the 4 would interact in a seemingly chaotic manner. However, this current pattern would stay stable for infinite iterations.

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Day 7: Chapel of the Chimes & Mountain View Cemetary

This is one of my favorite places in Oakland and I go there to take pictures often.  Here are some of the shots I took looking through the lens of “Patterns in Nature”.

The leaky faucet in this pool makes perfectly concentric circles in the water.

Ring Pool: The leaky faucet in this pool makes perfectly concentric circles in the water on top of a chaotic tessellation.

This plant caught my eye as a perfect example of iterative, self-similar design.  The wirey stem makes the better very clear.

Wired Foliage: This plant caught my eye as a perfect example of iterative, self-similar design. The wiry stem makes the pattern very clear.

The lighting in the Chapel of the Chimes is beautiful and sets the scene for this lone flower.

Flower in Mourning: The lighting in the Chapel of the Chimes is beautiful and sets the scene for this lone flower.

I LOVE this shot and I think it is one of my favorites so far.  The bend in the leaf around the major veins creates intersting color and focal changes.  Within the center section, the veins also create a pattern very reminiscent to Hilbert's Space-Filling Curve (below).  Nature uses these types of fractal shapes to fill space as efficiently as possible.

Space-Filling Leaf: I LOVE this shot and I think it is one of my favorites so far. The bend in the leaf around the major veins creates intersting color and focal changes. Within the center section, the veins also create a pattern very reminiscent to Hilbert’s Space-Filling Curve (above in red). Nature uses these types of fractal shapes to fill space as efficiently as possible.

Tree

Tree: Sometimes it is nice to remind ourselves that the most visible and perhaps definitive example of a natural fractal is a tree.  Looking up into this one, I can see the overlapping layers that form a beautiful and regular, yet fractal pattern.

5 is the magic number for this plant.  How many places can you find 5?

Five Fingers: 5 is the magic number for this plant. How many places can you find 5?  This reminded me of the hand fractal below it.

Summit:  These are the brave souls that made it to the summit of Mountain View Cemetary.  They were rewarded with this incredible panoramic view of the Bay Area.

Summit: These are the brave souls that made it to the summit of Mountain View Cemetary. They were rewarded with this incredible panoramic view of the Bay Area.

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Day 6: Leona Gorge

This was a very challenging uphill hike with lots of amazing fractals and a great view of the city.  Enjoy!

This one is pretty self-explanatory, but I like how the branching continues even under the waterline.

Branching Roots: This one is pretty self-explanatory, but I like how the branching continues even under the waterline.

I mostly like the colors in this one.  The bright green and blue contrast nicely and the puffy clouds mimmic the gaps between the leaves.

Sky Blue Sky: I mostly like the colors in this one. The bright green and blue contrast nicely and the puffy clouds mimmic the flowers that are mixed in with the leaves.

There are 3 different patterns in the water in this photo.  The chaotic waves become very regular as the water goes through the chute.  Once it drops into the lower pool, the water becomes even more chaotic than before.

Waves of 3, Wave to Me: There are 3 different patterns in the water in this photo. The chaotic waves become very regular as the water goes through the chute. Once it drops into the lower pool, the water becomes even more chaotic than before.

I wish the focus was a little better, but I like how the web connects to the spikey spheres.

Webbed Spheres: I wish the focus was a little better, but I like how the web connects to the spikey spheres.

I think Fractal Fennel could replace the Oak Tree as the symbol of our city.  Then, they could pay me a bunch of money to use this as their publicity photo!

Fennelland: I think Fractal Fennel could replace the Oak Tree as the symbol of our city. Then, they could pay me a bunch of money to use this as their publicity photo!

Cool focus.  Can you see the hidden fennel?
Cool focus. Can you see the hidden fennel?
A forest at the base of a tree.  Self-similarity, ho ho, he he.

Forest of Moss: A forest at the base of a tree.  Self-similarity, ho ho, he he.

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Mandelbrot Set

The Mandelbrot set is the most famous fractal in the world.  It’s created with a very simple equation: z = z^2 + c.  From simplicity comes infinite complexity.  No matter how much you explore and zoom, you will never find perfect repetition or regularity.  Here are a couple of interesting images I captured from the Mandelbrot Set.

This image looks like it's the full Mandelbrot set because it has the very iconic image in the center.  Nevertheless, this has actually been zoomed in about 10 times on the edge of the original set.  Talk about self-similarity!  Also notice the great-great-great-great-grandbaby Mandelbrot under the large great-great-great-grandbaby.

Mean Green: This image looks like it’s the full Mandelbrot set because it has the very iconic image in the center. Nevertheless, this has actually been zoomed in about 10 times on the edge of the original set. Talk about self-similarity! Also notice the great-great-great-great-grandbaby Mandelbrot under the much larger great-great-great-grandbaby.

Mandelbranches:  This is at the edge between two mini-Mandelbrots.  The branches coming off of the edge of each begin to overlap here.

Mandelbranhes: This is at the edge between two mini-Mandelbrots. The branches coming off of the edge of each begin to overlap here.

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