Fractal Tree -- Geometry Class Project

 

One of my goals this year is to bring fractals into my teaching, especially in geometry class. Today I am going to share the fractal tree. It can easily be done on a computer but is also great for practicing measuring with a ruler and protractor. Now a fractal tree is probably one of the most basic fractals and well-known or at least its cousin, the fern is certainly seen as a fractal. 

Christmas Fractal Lesson

 

Over the years I have shared my love for fractals with you. I began with my introduction to frozen fractals after Elsa sings about them in "Let It Go" in Disney's Frozen. A fractal is an object that has self-similarity, or each part looks like the whole. It introduces new ideas of symmetry, dimensionality, and more. Fractal geometry often explains some of the irregularities of our world. It can be a very complex topic to understand, but it is an important one. Fractals are making advances in our medical world, entertainment (movies, computer games and more) as well as science. There are some topics that kids even young kids can understand. Since they were introduced in Frozen, many kids have now heard of them. I feel it is important to teach kids about them and give them a true idea of what they are. 

Sierpinski Gasket -- Math Art -- Fractals

 

Today I am going to share some fun math art looking at a famous fractal--the Sierpinski Gasket or the Sierpinski Triangle. It is a perfect fractal to have kids create and goes well with geometry lessons. A fun way to introduce it and create it is the Chaos Game. Here is a video showing the Chaos Game with a triangle, square, and pentagon. With the triangle the Sierpinski Triangle will appear with enough iterations of the game. The rule as explained in the video is to begin with a random point. Then randomly choose a vertex. Connect your point to the vertex and find the midpoint. (Erase the line.) The midpoint is your new starting point. Repeat. This is a game you could easily play in a class as well. Don't watch the video first though. 

Exploring Pi with a Fractal & Pi Activity Round-Up

 

I really wanted to do something with pi for artwork in my classroom plus would love a good pi activity. I struggled with this one. I have seen the pi skyline like this one over at What Do We Do All Day? It is fun but not quite what I am looking for. There are different pi artworks if you google "pi art" but most is based on the digits of pi. To be honest I do not believe in having kids memorize the digits of pi, so much of the artwork is not my thing. As I was searching for ideas, I came across this YouTube video that intrigued me. I decided to make the "fractal" that has an area of pi! It is a spin from the Sierpinski Carpet, Menger Sponge, and the Wallis Sieve. Now fractals are supposed to be infinite, but I cannot draw them this way. I am working on taking this fractal to the third level. If you were able to go on infinitely the area of this picture would be pi. 

Frozen Fractals Lesson 2 and Olaf Craft

Last week I gave you an introduction to fractals in response to all the interpretations of frozen fractals from Disney's Frozen movie's song, "Let It Go." Today I thought I would continue my lesson on fractals. But first I wanted to share an easy craft to make an Olaf from the movie. Hazel got her Elsa and Anna dolls this week and she now wants the males so she can re-enact the entire movie. I told her we could make an Olaf. Looking at our supplies I came up with some styrofoam balls, a little bit of white and black Model Magic and some pipe cleaners (black and brown), googly eyes and an orange tear drop shaped foam piece. I cut the biggest and medium styrofoam balls in half and toothpicked them together. Then we used a small styrofoam ball and some of the white clay to make his head. We covered the other two balls as best we could with what we had left of the white clay. We did not have enough, but she didn't care. Then we used the black to form the buttons and mouth. I took a small piece of white for his tooth from the back. We added the brown pipe cleaners for arms and hair and then put the eyes on with tiny black pipe cleaner eye brows. The eyes and eye brows as well as the nose all needed to be glued on to get them to stick well.

Hazel is happy with how he turned out, so we will go with it. I will eventually get us some more white Magic Model to finish his bottom part and add the legs.

Now onto fractals. Last week we looked at the Koch Snowflake since we are talking frozen fractals. But I was thinking I should explain why study fractals besides to know what they are from a line in a song. Fractals are a very new thing in the math world. However they are being used in so many places and have been around forever. Fractals occur in nature and always have. Mathematicians and scientists somewhat ignored them for a long time due to how complex the shapes looked. For years we have simplified our shapes to fit into our cookie cutter basic shapes: circle, square, triangle, rectangle, etc. However things like a fern are not a basic shape and when you simplify it to such you lose some of its elements.

"Barnsley fern plotted with VisSim" by DSP-user - Own work, using model written by Mike Borrello This chart was created with VisSim.. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

As we learn more about fractals we are finding more uses for them. For those who remember Encyclopedia Encarta, the pictures on this CD were developed by programs written to make fractals similar to the wanted picture. Fractals are used in making movie backgrounds, video games as well as being explored in medicine. The lungs are now realized to be fractals as well as our blood vessels (veins and arteries). The more we learn about fractals in the human body the better our medical science will be.

"Thorax Lung 3d (2)" by AndreasHeinemann at Zeppelinzentrum Karlsruhe, Germany http://www.rad-zep.de - http://www.rad-zep.de. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

We will talk today about one of the more basic fractals and one that is easy to create. I often had my geometry students create this fractal in different ways. It is a wonderful way to teach about measuring, as well as midsegments of triangles. We are going to make a Sierpinski Triangle or sometimes called the Sierpinski Gasket. To start you need a triangle. You can use any triangle. Most commonly used are equilateral triangles, but any will work.

Now the rule is to draw the three midsegments of the triangle. A midsegment is a line segment that connects the midpoints of two sides of triangle. Its properties are that it is parallel to the third side and its length is equal to half the length of the third side.

The final part of the rule is to remove the triangle formed by the midsegments. To remove it, we will color it in.

Now we continue to the next stage by doing the same thing for all the non-colored in (nonremoved) triangles.

Sorry for some reason I did not take a picture of Stage 2 with the triangles removed. We continue our rule for Stage 3.

Finally we have Stage 3 which is where I stopped since I was not feeling well and I was getting a bit of a headache. However you can see how the rule can be applied infinitely as with all fractals.

The Sierpinski Triangle is also a great fractal to explore self-similarity. In the one below you can see how each color represents a shape that looks like the whole.

"Sierpinski-rgb". Licensed under CC BY-SA 3.0 via Wikimedia Commons.


The one below is zooming into magnify the smaller parts of the Sierpinski Triangle to show its infinite range. 

"Sierpinski zoom" by Mariko GODA - Own work.
Licensed under CC BY-SA 3.0 via Wikimedia Commons.

Finally I want to share with you a Sierpinski Pyramid that we made in one of the summer classes I took at Yale. We made it by forming regular tetrahedrons from envelopes. I do not totally remember how now, but thought it was neat to share.

 A true Sierpinski Pyramid would have a hole in the middle, but that is not easy to construct in actual life. Here is one I found on-line.


"Sierpinski pyramid". Licensed under Public domain 
via Wikimedia Commons.

Finally if you want to learn more about fractals in nature and the importance of them, check out this video of the Yale professor, Michael Frame, who taught the courses I took on fractals. He ends it with a story about the amazing late Benoit Mandelbrot (last week I shared a picture of myself with Mandelbrot) and about how fractals are about storytelling as is most math and science. It is really worth watching. On that note a talk about fractals is not complete without looking at the Mandelbrot Set. The math involved is more advanced, however it is beautiful to look at.

"Mandelbrot set with coloured environment". Licensed under  
CC BY-SA 3.0 via Wikimedia Commons.

For some more on fractals check out:

• Happy Pi Day
• What Are Frozen Fractals?
• My Math Pinterest Board

What Are Frozen Fractals? A Lesson on Fractals to Go with Disney's Frozen Movie and Let It Go Song

Hazel and I LOVE Disney's movie Frozen. We are constantly listening to the soundtrack in our car. One of our favorite songs from the movie is Let It Go. If you have not heard the song or seen the movie you can
check out the song from the actual movie (with movie scenes) from Disney's website.