One of my favorite books is Flatland by Edwin Abbott, which is incredibly rich in geometric concepts especially relating to dimensionality. I would really like to assign it to my secondary geometry class to read but I am struggling with a final project/assessment for them to do after they finish reading. I would like to stay away from just answering questions about the reading/writing an essay. Even if that was the route I went, I can't seem to come up with a solid essay topic or list of questions that go beyond regurgitation of what they read. Does anyone have any suggestions on creative ideas for projects or assessments I could use with the novella Flatland?
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$\begingroup$ You might go in the other direction and introduce them to Rudy Rucker's The Fourth Dimension: A Guided Tour of the Higher Universes. $\endgroup$– shooverCommented Jan 29, 2015 at 21:56
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$\begingroup$ Look at AK Dewdney, The Planiverse, and compare the 1884 and 1984 books. Which do you enjoy reading or thinking about more? Which issues from the Planiverse could be resolved differently in Flatland? $\endgroup$– user173Commented Feb 2, 2015 at 12:13
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2 Answers
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The book goes into detail giving the two-dimensional cross-sections of a sphere. You could ask the class to find the cross-sections of a cube or of another 3D object more complex than a sphere. It seems hard to find all possible cross-sections of a cube, but you could assign points for each possibility they do find. (It would be interesting to see how many students would find the regular hexagon.)
You could also ask for other ways to represent a cube in two dimensions. Students could think of a square inside another square with segments joining the corners--a kind of perspective drawing. They could also try to find all the ways to "flatten out" the cube: cut along some edges to fold the cube down to the plane. The most common way results in the Latin cross, but there are other ways.
answered Dec 28, 2014 at 19:56
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2$\begingroup$ I like the "cross sections" idea. You could pitch this as "How would a xxx look to a Flatlander?". Interesting 3D shapes to put in the place of xxx would be a cube, a cone, a triangular prism. You could even get them to make animations of what it looked like as it moved past the plane of Flatland - a physical flip book would really bring home the point of the cross sections. $\endgroup$ Commented Dec 29, 2014 at 8:29
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$\begingroup$ Those are excellent comments on my answer: I should have said all that myself (but didn't). $\endgroup$ Commented Dec 29, 2014 at 11:00
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$\begingroup$ In the same line of ideas, one can show a set of polygons and ask students to guess from which polyhedron they are cross-sections. $\endgroup$ Commented Dec 30, 2014 at 19:04
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$\begingroup$ Tougher: ask what they think would be the 3D cross-sections of a 3-sphere, and try to give them a sense of what a 4D world can offer (in terms of polyhedrons) through these types of questions. $\endgroup$ Commented Dec 30, 2014 at 19:06
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$\begingroup$ @RoryDaulton you can edit your post to include my comments. The purpose of comments is to help improve the answer anyway, right? $\endgroup$ Commented Jan 8, 2015 at 16:11
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Thank you for the great suggestion @RoryDaulton ! Below is the project description that i decided upon for this project. The students can pick from one of the below options for their project, or have the opportunity to come up with their own idea. Figured I would post this for anyone who would like to get ideas for similar project ideas.
Family Tree A Square provides very detailed descriptions of the social ladder in Flatland and how Flatlanders proceed are able to slowly climb from generation to generation (Ch. 3, Ch. 7, Ch. 11). For this project, you will construct a family tree, starting from an isosceles triangle all the way to a circle. Here are the requirements for this project: Include at least 7 family members from distinct generations, including the very first isosceles triangle and the final circle
-For each family member provide a picture, basic measurements, and a paragraph of biographical information, making sure to include what generation they are in and who their parents and children are. Be as creative as you would like!
-Provide your “rules” for your family tree. A Square, while thorough, does not provide us with actual numbers, likely because every family is slightly different. For your family, I would like to know the starting angles on the original isosceles triangle, how much that angle increases from generation to generation, at what generation it is considered an equilateral triangle, how the side numbers increase from generation to generation, how many sides constitute a circle, and how many total generations it took to make it to a circle (This list is not exhaustive. Make sure to provide ALL information that is needed to understand your family tree)
Worldbuilder In Chapter 11, “Concerning Our Priests”, A Square begins the chapter by apologizing to the readers that he will not be able to explain the entirety of Flatland, including such topics as “our method of propelling and stopping ourselves,...the means in which we give fixity to structures of wood, stone, or brick...our Alphabet and method of writing, ” Your job for this project is to pick three aspects of Flatland that A Square does not provide an explanation for and come up with (they do not have to be the ones that he says in Ch. 11) a description of how you think they would work for Flatlanders.
-Provide a brief background to each topic and why Flatlanders cannot perform the task or experience the phenomenon in the same way that we Spacelanders can
-For each topic give a complete description as to how the topic is handled in Flatland. Each topic must be at least a page in length, with a title and date
-You can be as creative as you would like, providing your description is consistent with A Square’s own description (You can’t say “The Flatlanders just pick it up with their hands” because polygons do not have hands, at least as we know them. If you would like to give your Flatlanders hands then you need to describe exactly what you mean by “hands” and how it fits into Abbott’s Flatland)
-Creatively frame your topics by introducing yourself as a citizen of Flatland and your motivation for writing the three passages, just like Abbott did with A Square
Phlatland: A Derivative Work For this project, you will use Flatland as an inspiration for your own creative work. Your project does not necessarily need to be a piece of fiction, it can be a movie, a song, an epic poem or any creative mode of expressing a mathematical story. It does not necessarily need to be structured the way Flatland is, however, if you decide to tell your story in a different world or universe you need to provide some level of explanation so that the reader/listener/watcher knows what is going on.
-A creative work of sufficient length that ties together storytelling and mathematics
-You may produce a recreation of Flatland in a different media than written fiction (comic, short film, song) provided it is consistent with the universe and story of Flatland
-It must somehow reference Flatland or an aspect of Flatland as laid out by A Square
Other Ideas
-A Map of Flatland: Create a map of an entire Flatland city including exact measurements and a brief history of the city and its context within Flatland
-Flatlander Profiles: Detail the lives of a handful of various Flatlanders, including detailed biographical information, sketches, and different possible cross sections
-Flatland Infographic: Create an “Everything Flatland” infographic that explains all of the details of the Flatland Universe and A Square’s journey