Research project

Due dates: November 28, December 5, December 12, December 19

Two of our learning goals for this course are:

1. Gather, interpret, evaluate, and apply information discerningly from a variety of sources.
2. Acquire tools for lifelong learning.

Learning and writing about something new is a great way to achieve these goals. To this end, your final output for this research project will be a short essay on a topic related to geometry that is of your choosing.

Essay details

1. Minimum length: Equivalent of around 2-3 pages, single spaced (around 1000 words) including pictures.
2. Type: Expository with narrative elements. 
   1. An “expository” essay simply aims to explain an idea, concept, or topic to the reader. 
   2. A “narrative” essay includes personal storytelling. Your narrative elements may include an explanation of what drew you to the topic and/or how you came to understand it.
3. Audience: You are writing something that your classmates could understand easily. You may wish to write something that someone who has not taken this class can understand.
4. Style: Your essay can be as formal or as informal as you wish. Whichever style you choose, you must write in complete sentences and paragraphs. Make sure your own voice is expressed.
   1. If you want to write a serious academic paper, go for it! 
   2. If you want to structure your essay as though it’s a section of a textbook with theorems and definitions, that’s okay! 
   3. If you want to write an informal blog post that includes gifs or links to TikTok videos, that’s wonderful! 
5. Submission: You will submit your final essay as a post on the OpenLab.

Requirements

1. It must be clear which geometric facts you are discussing; is there just one? More than one? Are you referring to a concrete elementary geometric fact or are you just trying to give your audience a notion of something more advanced? Are different facts being used together?
2. Your main point must be obvious—what is it you want the audience to get from your essay?
3. Your work must be clearly organized; what are the different sections?
4. Include pictures to support your writing whenever necessary.
5. The writing must be your own and your own voice must come through; cite your sources when you are pulling ideas from them.
6. Bibliography: list around three references (Wikipedia is okay, but can be only one of your references; blog posts are okay)

Grading

Your overall grade will reflect whether you met the intermediate deadlines; your final essay will be graded according to this rubric.

Dates

• November 28: Step 1 due (proposal; OpenLab post)
• December 5: Step 2 due (outline; OpenLab post)
• December 12: Step 3 due (draft and feedback due May 16 (Dropbox upload and Google survey)
• December 19: Step 4 due (final essay due)

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The following four steps should take you around one to three hours each.

Step 1: Perform preliminary research, choose a topic, and submit a proposal – due November 28

You have a lot of freedom for your topic choice. The only major requirement that your topic must satisfy is that it must involve geometry at some level. There are lots of amazing online resources so once you’ve chosen a topic, narrow the resources down to around three references that you will draw from for your essay (chances are you will rely mostly on one; that is okay).

Some suggested topics are below. To choose a topic that is not on this list, get approval from your instructor. If you need help choosing a topic, contact your instructor.

Your proposal consists of

1. your topic choice
   • Example: Linear perspective in drawing and painting
2. about three sentences describing what you’ve learned
   • Example: Artists use linear perspective to create two-dimensional renderings of three-dimensional space. There are three different type of linear perspective: one-point, two-point, and three-point. The three types have some elements in common but are used for different purposes. For each type, geometric constructions are used to represent certain three-dimensional configurations in two dimensions.     
3. what approach you plan to take in your essay, 
   • Example: I will describe the important elements of linear perspective, describe how each of the three types of linear perspective work and when to use each type, and give examples of constructions for each of the three types.
4. links to around three references.

You are not bound by what you propose; your final essay may differ from your initial proposal. If your essay will differ greatly (for example, if you want to choose a completely different topic), get permission from your instructor. 

• Submit your proposal as a post on the OpenLab. Title it Research project proposal — [your topic choice] and select the category Research project proposal before publishing.

Step 2: Write an outline – due December 5

1. Decide what your thesis statement is—what is the point you are trying to get across to your audience? This might be in the form of a question.
   1. Example: How can three-dimensional space be represented realistically in three dimensions?
2. Break your outline into sections. 
   1. Include an introduction (providing context and motivation) and a conclusion (summarizing how the body supports your thesis statement).
   2. Your other sections will depend on your topic.
      • Example: 
        1. Introduction
        2. One-point perspective
        3. Two-point perspective
        4. Three-point perspective
        5. Choosing a type of perspective
        6. Conclusion
3. For each section, sketch what points you will make (this may be a bullet-point list).
4. Decide which parts need examples; include descriptions of pictures you will create as necessary.

• Submit your outline as a post on the OpenLab. Title it Research project outline — [your topic choice] and select the category Research project outline before publishing.

Step 3: Write a draft and get feedback – due December 12

Chances are your first draft will require some editing and you may end up producing multiple drafts. Share your first draft with a “reader;” A reader is your peer, your family member, or a friend. You may share it with more than one person if you wish.

• Share the requirements below and ask your reader to complete this form (based on the rubric above). Make sure your reader provides you with this feedback as well as filling out the form; they may also share more detailed feedback with you.
• Upload your draft here. (Note that you are not posting your draft on the OpenLab—you may post it if you wish, but that’s not required.)

Step 4: Write your final essay – due December 19

Incorporate the feedback you have received to write your final essay. Don’t forget to include a bibliography (cite your sources in any style).

• Submit your essay as a post on the OpenLab. Title it Research project essay — [your topic choice] and select the category Research project essay before publishing.

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Suggested topics

Pure math

• Symmetry
  • Group theory—dihedral groups
  • Tilings
  • Polyhedra for algebra/combinatorics (associahedra/permutahedra)
  • Fractals
  • Scissors congruence
  • General regular polyhedra (e.g., https://www.math.harvard.edu/media/AllenLiuTheStarsAboveUsThesis.pdf)
• Topology
  • Surfaces; curves on surfaces
  • Orientable and non-orientable surfaces (e.g., Möbius strips and Klein bottles)
  • Euler characteristic for polyhedra and genus of surfaces
  • Graph theory; Seven bridges of Königsburg and other problems
  • Knot theory
  • Configuration spaces
• Higher-dimensional geometry
  • Higher-dimensional polyhedra
  • Manifolds
  • Curvature
  • Singularities
  • The Shape of Space (book by Jeffrey Weeks)

Applications

• Geometry of gerrymandering
• Map projections; Gauss-Bonnet theorem
• Spherical geometry and classical astronomy
• Geometry of spacetime
• Shapes of molecules
• Crystallography
• Shapes of viruses/bacteria
• Optics
• Sphere packing and the geometry of pandemic reopening (e.g., https://www.quantamagazine.org/the-math-of-social-distancing-is-a-lesson-in-geometry-20200713/)

Geometry and art

• Linear perspective
• Proportion; the Golden ratio
• Islamic tessellations; 17 “wallpaper” groups
• Geometry of Native American art (e.g., https://in.nau.edu/wp-content/uploads/sites/101/2019/12/Chischilly-2019.pdf)
• Navajo weavings (e.g., https://archive.bridgesmathart.org/2012/bridges2012-581.pdf) 
• Geometry in pop culture
• Knots in art (e.g. Celtic knots)
• Indian Rangoli
• Flatland (book by Edwin Abbott Abbott)
• MC Escher’s work
• Richard Serra’s work (e.g., http://archive.bridgesmathart.org/2013/bridges2013-287.pdf)
• Vi Hart’s YouTube channel (e.g., Hexaflexagons; Wind and Mr Ug)

Geometric thinking

• Proofs without words
• Mathematical visualization
• Geometry and the imagination (e.g., https://math.dartmouth.edu/~doyle/docs/gi/gi.pdf)
• Geometry in ethnomathematics

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Somewhat similar examples

I don’t have example essays that are just like the ones that you will be writing, but I can show you something somewhat similar. I gave an assignment in my Differential Equations class one semester where students wrote about an application that interested them. Their instructions were more constrained than yours and their minimum length was shorter, but these examples can give you a flavor of the type of essay you might write:

1. The Differential Equations of Counter-Insurgencies by Oscar
2. Tacoma Narrows Bridge Collapse and Differential Equations by Brian