Mat 335 Essay Instructions Spring 2020
You’re going to write a series of short essays that will, in the end, be combined into a longer essay
which explains Chaos, Dynamics, Fractals, and their interconnections to a general audience.
Your essays should:
Have a perspective: Do you think fractals are fundamental facts of nature? Do you think the
“Butterfly effect” is a misused term in pop-culture? etc.
Introduce and explain the topics at hand (fractals, chaos and dynamics).
Not get into the nitty-gritty mathematical details and/or proofs.
In general, you will have to do independent research to get a
oader perspective on each essay topic.
In class we dive deep into specific examples, but there’s a lot more out there, and you don’t need to
write about the same stuff we’ve done in class!
The hardest part of the final essay will be developing a common theme. The goal is to write a single
coherent essay on chaos, fractals and dynamics—having a theme will provide a means to relate the
topics and unify your three short essays. Some examples of themes (in the form of essay titles)
include:
“From simple definitions, complex behaviour can arise”
“To study the clouds, we need new ways of measuring”
“In the 1800’s, we thought we knew it all—but that didn’t last long”
Your essay may feature equations, though they should be simple and used sparingly. You may (and
should) include figures to aid your explanation. Any sources you use (includingWikipedia) should
e cited with footnotes or endnotes.
Your essays must be typed in LATEX and should be aimed at a non-math audience. You are expected
to incorporate the feedback you receive on your short essays into your final essay.
Audience
The main focus of these essays is to communicate sophisticated mathematical ideas to a non-
mathematical (but intelligent) audience. To get an idea of what this means, start by reading Keith
Devlin’s essay “What is a mathematical proof?”
https:
mathvalues.squarespace.com/maste
log/what-is-a-mathematical-proof
After reading the essay, think about the following:
Did the essay have a message? If so, what was it?
Did the essay use technical math terms? If so, were they defined or described the same way
they would be in a math class?
Who do you feel the essay was targeted towards? Was it you? Your younger sibling? You
professor?
Another excellent, albeit much-longer-than-you-should-write-for-this-course, example of technical
math writing for a smart but non-technical audience is Scott Aaronson’s“Who Can Name the Bigge
Number?” (This one comes with some translations for non-native English speakers.)
https:
www.scottaaronson.com/writings
ignumbers.html
1
https:
mathvalues.squarespace.com/maste
log/what-is-a-mathematical-proof
https:
www.scottaaronson.com/writings
ignumbers.html
Project
Timeline
In total, this writing project is worth 45% of your grade for this course. Below is a short description
of each part of the project, along with its deadline.
Fractals Essay
9 %
You will write a short essay on fractals. It should address some combination of the following:
What are fractals?
Where do they come from?
Where (and why) do fractals appear in the real world?
How can they be measured or classified?
How are they different from figures in Euclidean geometry?
The Fractals essay should be ∼2 pages, and is due by Fe
uary 2 11:59PM.
Dynamics Essay
9 %
You will write a short essay on dynamical systems. It should address some combination of the fol-
lowing:
What is a dynamical system?
Where do you encounter dynamical systems?
How do you classify dynamical systems?
Are some dynamical systems easier to understand? Why?
The Dynamics essay should be ∼2 pages, and is due by Fe
uary 23 11:59PM.
Chaos Essay
9 %
You will write a short essay on chaos. It should address some combination of the following:
What is chaos?
Where does chaos appear?
Are chaos and randomness the same thing?
What challenges/solutions does an understanding of chaos offer us (as a society)?
The Chaos essay should be ∼2 pages, and is due byMarch 22 11:59PM.
Title & Abstract
2 %
You will submit the title of your final essay, along with a one-paragraph abstract explaining:
The theme of your final essay (based on your title).
How each of topic (chaos, fractals, and dynamics) relates to your theme.
How you will unify your small essays into a single coherent piece.
The title & abstract is due by March 29 11:59PM.
Final Essay
12 %
You will submit a final essay combining your three short essays into one. You may need to signif-
icantly re-work parts of your previous essays to get them to flow together. You can put the essays
together in any order (it doesn’t need to be Fractals→ Dynamics→ Chaos, and probably shouldn’t
e in that order!)
The final essay should be ∼6 pages, and is due by April 13 11:59PM.
Reflection
4 %
You will submit a few paragraphs reflecting on the following:
How did you incorporate the feedback on your short essays into your final essay?
How has your understanding of mathematical communication changed over the term?
How has your perception of maths and definitions changed over the term?
The reflection is due at the same time as the final essay.
2
Grading
The focus of this writing project is mathematical communication. To this end, your essays will be
graded on the following criteria:
Writing for your audience: You are trying to explain ideas to a non-mathematical audience.
(Imagine you were writing a feature for a magazine like New Scientist.) This means you should
avoid overly technical descriptions, but you should also explain why your audience should care.
Why are fractals interesting? How will knowing about dynamical systems impact their day-
to-day life?
Mathematical understanding: You need to talk about the specific ideas you’ve learned in the
course. An essay with little mathematical content is not acceptable. It should be clear to a
mathematician reading your essay that you know what you’re talking about.
Quality of writing: If your essay is hard to read, it’s not going to communicate anything
ecause people won’t read it! Make sure to write in complete sentences and use effective
paragraphing. Your essay should also flow logically from one paragraph to the next.
In addition, the final essay will also be graded on two extra criteria:
Cohesion of topics: The end goal is a single 6-page essay, not three 2-page essays! You should
elate each topic to your common theme and have a natural progression of ideas.
Creativity: Do your best to make your essay your own. Have you used diagrams in you
essay in a novel way? Have you used any interesting outside sources? Have you related math
topics to your personal experiences?
Pay careful attention to the page count: the short essays should be ∼ 2 pages, and the final essay
should be ∼ 6 pages. If your essays are much shorter than this, you probably haven’t explored
enough facets of your topic. If your essays are much longer than this, the TAs will only mark up to
the page count!
Advice
Take the 2-page essays seriously! Your short essays should be coherent and have a logical flow of
ideas, just like the final essay. Make sure to write in complete sentences and use effective paragraphs.
Remember that each draft is worth 9% of your final grade.
Read the TA feedback. The course TAs will provide specific, constructive feedback on each of you
drafts. Read their comments carefully and keep them in mind when working on the final version of
your essay. Any general advice that they give can also be useful when working on the other short
essays.
Make use of UofT’s writing centres.
Your college has a writing centre where they can provide support on writing both the drafts
and the final essay. You can find more details here:
https:
writing.utoronto.ca/writing-centres/arts-and-science
The faculty runs a handful of mini-courses for English language learners designed to give stu-
dents experience in writing formal, academic English: https:
www.artsci.utoronto.
ca/cu
ent/academic-advising-and-support/english-language-learning
UofT also has a writing advice site that covers everything from planning to revising you
writing: https:
advice.writing.utoronto.ca
3
https:
writing.utoronto.ca/writing-centres/arts-and-science
https:
www.artsci.utoronto.ca/cu
ent/academic-advising-and-support/english-language-learning
https:
www.artsci.utoronto.ca/cu
ent/academic-advising-and-support/english-language-learning
https:
advice.writing.utoronto.ca
MAT335
Draft 1 (Fe
uary 3, 2020)
Due: 11:59pm January 2
Shuaihao Song XXXXXXXXXX)
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark
is not smooth, nor does lightning travel in a straight line.” This statement, attributed to Benoit
Mandel
o, underscores the fact that the things of nature do not have a definite shape. Fractal
geometry can be used to describe the fluidity of shapes in nature (Mandel
ot, XXXXXXXXXXFractals
are often only recognized as useful and beautiful pictures as backgrounds on computer monitor o
on poster. But the question that needs to be answered is “what really are fractals?” Behind the
eautiful and the elegantly looking patterns, “Fractals is a
anch of mathematics and art”. Systems
of nature, especially the physical systems and human artifacts do not often come in regular geometric
shapes as derived from Euclidian geometry. Through fractal geometry, there are almost infinity ways
to describe and measure natural phenomena. This paper seeks to understand and describe the useful
eauty of fractals, their origins and how they appear in nature.
There is no doubt that many people are addicted by the beautiful images named as fractals.
Fractals go beyond the typical perception of mathematics as a body of complicated, boring formulas.
Instead, fractals combine both the mathematics with artwork by demonstrating that equations are
more than just a collection of numbers (Lewis, XXXXXXXXXXPerhaps factors makes fractals particularly
interesting, because they are the ways to give mathematical descriptions to existing shapes. There
is a close connection between fractal geometry and computer techniques. However, working with
fractals started way before by technique of computers. The British Cartographers first encountered
fractals when they faced the problem of measuring the length of the British coast. The cartogra-
phers realized that the length if the coastline as approximated using detailed map di↵ered with the
length obtained using a large-scale map. Even though it presented itself as a challenge, the British
cartographers did not realize that they had just discovered one of the main properties of fractals.
Fractals have certain features that distinguish them from Euclidian geometry. Two of such
properties include the fact that fractals always exhibit self-similarity and non-integer dimension.
“what does self-similarity mean?” Perhaps the best example of this is the symmetry of fern leaves:
every smaller leaf is part of the larger one and has the same shape as the larger leaf, and ultimately
the larger fern (Lewis, XXXXXXXXXXThe same property manifested by the fern is also evident in most
fractals. Every fractal can be magnified by many factors, every time maintaining the original shape.
Besides the property of self-similarity, fractals also have non-integer dimensions. In most classical
geometry (Euclidian geometry), objects are measured in terms of integer dimensions, for instance;
zero dimensional points, one dimensional line, two dimensional planes and three-dimensional solids.
While Euclidian geometry may be about exact shapes and figures, this is not always the case things
of nature. Instead, it is better to describe most