Fractal Snowball
Project description
The focus of this project is to give students an opportunity to display varied understandings of fractals and explain this new mathematics to the community!
Fractale (first draft)
Driving question: "Where do we see fractals and how do we recognize them?"
If you’re anything like me, you have a deep seeded love of plants. The various fragrances, colors, and designs make each plant unique and inticing. So much so, that everything around the plant your eyes have rested upon becomes a blur, consequently creating a clear line of sight into the seemingly miniscule details of nature’s beauty.
While engrossed by the artistry of such a creation, your eyes may have stumbled upon a system of complex branching that is caused by the splitting of the main stem. Now, take a moment to scan your memory bank back to the days when learning how to draw a tree using crayon was the most crucial thing on your to-do list. “First, you draw the trunk! No, Megan, you need to use brown!” I recall my very authoritative pre-school teacher saying. “Now, draw a ‘V’ at the end of the trunk. Then draw two more ‘mini-branches’ coming off of the V,” she would say, as if it were that easy to visualize at the age of four. These branches split into smaller branches, the smaller branches split into even smaller branches, so on and so forth. This berserk splitting continues until you reach the smallest branches.
More in depth attention to detail will reveal that the tree branch looks very similar to the tree. If you’re unlike me, having a keen eye for mathematical patterns, there is a possibility that you’ve interepreted this seemingly identical branching as self-similarity. “What could self similarity possibly be used for,” you ask? Why, it is one of the most important properties of a fractal! What they don’t tell you in pre-school is that there are numerous ways branching can be achieved geometrically.
The most popular mechanism for modeling various plant-like structures is known as the L-system. An L-system provides mathematical formalisms to describe the growth of plants. They are very often self-similar, and thereby fractals. An L-system is a rule-like description of an object or image. The system contains descriptions of the various parts and how to assemble them, much like you would use in the assembly of an Ikea lamp. This list of “rules” is applied to itself a number of times, making fractal and recursive forms easy to describe. It is for this reason they are commonly used to create images of plants!
While engrossed by the artistry of such a creation, your eyes may have stumbled upon a system of complex branching that is caused by the splitting of the main stem. Now, take a moment to scan your memory bank back to the days when learning how to draw a tree using crayon was the most crucial thing on your to-do list. “First, you draw the trunk! No, Megan, you need to use brown!” I recall my very authoritative pre-school teacher saying. “Now, draw a ‘V’ at the end of the trunk. Then draw two more ‘mini-branches’ coming off of the V,” she would say, as if it were that easy to visualize at the age of four. These branches split into smaller branches, the smaller branches split into even smaller branches, so on and so forth. This berserk splitting continues until you reach the smallest branches.
More in depth attention to detail will reveal that the tree branch looks very similar to the tree. If you’re unlike me, having a keen eye for mathematical patterns, there is a possibility that you’ve interepreted this seemingly identical branching as self-similarity. “What could self similarity possibly be used for,” you ask? Why, it is one of the most important properties of a fractal! What they don’t tell you in pre-school is that there are numerous ways branching can be achieved geometrically.
The most popular mechanism for modeling various plant-like structures is known as the L-system. An L-system provides mathematical formalisms to describe the growth of plants. They are very often self-similar, and thereby fractals. An L-system is a rule-like description of an object or image. The system contains descriptions of the various parts and how to assemble them, much like you would use in the assembly of an Ikea lamp. This list of “rules” is applied to itself a number of times, making fractal and recursive forms easy to describe. It is for this reason they are commonly used to create images of plants!
Final Fractale
snowflake
Snowflake description
Never in my wildest dreams did I think I would be cutting out snowflakes as an eleventh grader, but of course such a seemingly simple thing has math involved, everything does! If you look closely at a snowflake, you will see that it has fractale patterns on it. So as Math III students at High Tech High, we were provided with the oppurtunity to create our own fractal snowflake patterns. Since a fractal is an object that displays self-similarity, I decided to create a row of self-similar diamonds on each side of my snowflake. In other words, I have rows of diamonds that are all the same shape, but not necessarily the same size (fig 1). I created them by first folding the traditional snowflake starting folds. Then I proceeded to make triangular shaped cuts along the backbone of the folded paper. Each cut would grow smaller and smaller, while still holding the diamond shape. With just a few cuts, you can create a snowflake fractal of your own!
exhibition reflection
1. How can we display “new math” for others to learn about?’
We can display “new math” in artistic ways to interest others. Once they’re hooked,
they’ll want to know all about the reasoning behind designs!
2. Where do we see patterns and how are "fractal patterns" different?
We see patterns everywhere! They are seen in nature, in business, even in the makeup
of our bodies. Fractal patterns differ from regular ol’ every day patterns because they are
infinite. They start with something seemingly simple and are reproduced with inifinite detail.
3. Of the pieces you contributed to our exhibition, which are you most proud of?
I am most proud of my Fractale. I enjoy writing, but only when I am able to choose the
style. It makes everything seem more relaxed, and when I’m relaxed, I produce much better
work. I also feel that I got a pretty interesting topic to write about, as I am a nature fan. Not as
hardcore as I come across in my fractale, but still, it interested me and sparked my imagination.
I am proud of my final Fractale.
4. What was your best contribution in the presentation exhibition night?
During exhibition, I feel that I was most helpful by simply “manning my booth.” I was a
greeter, and that is exactly what I spent my time doing. I was focused on grabbing viewers
attention and sharing an overview of the student’s work.
5. What could you have done more to prepare for explaining our exhibition to others?
I should have spent more time reviewing the script sheet. I knew all of the information
that I needed to explain, I just had a hard time explaining everything in an order that made
sense and was interesting.
6. What did you enjoy most about this exhibition process (if it was just decorating the room, be
clear as to why you thought this was important)?
I most enjoyed writing the Fractales! As I said before, I enjoy writing, probably because it
comes naturally to me. Math, however, does not come naturally, so writing about it is a bit of a
challenge. I spent a good amount of time trying different approaches, and I feel as though I vercame an academic challenge and produced something that I am proud of.
7. Give me some KSH on what could be improved with this unit if I was to use it with another
class.
I really enjoyed the project, and I’m having a hard time coming up with any critique. If
anything, I would just say to give the documentation groups from each class time to meet up
and discuss what the vision for the video should be. Each group had a different idea about the
filming and it made the video a little scattered. Other than that, jolly good!
We can display “new math” in artistic ways to interest others. Once they’re hooked,
they’ll want to know all about the reasoning behind designs!
2. Where do we see patterns and how are "fractal patterns" different?
We see patterns everywhere! They are seen in nature, in business, even in the makeup
of our bodies. Fractal patterns differ from regular ol’ every day patterns because they are
infinite. They start with something seemingly simple and are reproduced with inifinite detail.
3. Of the pieces you contributed to our exhibition, which are you most proud of?
I am most proud of my Fractale. I enjoy writing, but only when I am able to choose the
style. It makes everything seem more relaxed, and when I’m relaxed, I produce much better
work. I also feel that I got a pretty interesting topic to write about, as I am a nature fan. Not as
hardcore as I come across in my fractale, but still, it interested me and sparked my imagination.
I am proud of my final Fractale.
4. What was your best contribution in the presentation exhibition night?
During exhibition, I feel that I was most helpful by simply “manning my booth.” I was a
greeter, and that is exactly what I spent my time doing. I was focused on grabbing viewers
attention and sharing an overview of the student’s work.
5. What could you have done more to prepare for explaining our exhibition to others?
I should have spent more time reviewing the script sheet. I knew all of the information
that I needed to explain, I just had a hard time explaining everything in an order that made
sense and was interesting.
6. What did you enjoy most about this exhibition process (if it was just decorating the room, be
clear as to why you thought this was important)?
I most enjoyed writing the Fractales! As I said before, I enjoy writing, probably because it
comes naturally to me. Math, however, does not come naturally, so writing about it is a bit of a
challenge. I spent a good amount of time trying different approaches, and I feel as though I vercame an academic challenge and produced something that I am proud of.
7. Give me some KSH on what could be improved with this unit if I was to use it with another
class.
I really enjoyed the project, and I’m having a hard time coming up with any critique. If
anything, I would just say to give the documentation groups from each class time to meet up
and discuss what the vision for the video should be. Each group had a different idea about the
filming and it made the video a little scattered. Other than that, jolly good!