Assignment Question
Portfolio requirements: At least 10 items that represent high-quality work that you are proud of and that showcases important and/or interesting things you have learned in this this course. For each entry, comment briefly in what ways it is representative of your learning in this course. At least three of the items should include complete, correct, and rigorous proofs. At least two of the items should include GeoGebra constructions with detailed explanations (this could be a video or a written narrative) At least one of the items should be a history essay (about 1 page). The remaining items could also be a mix of proofs, constructions, or history essays, or you could include discussions of anything that you have found interesting or intriguing. If you have identified any K-12 connections, that could also make a great entry. Both Euclidean and non-Euclidean content should be represented the portfolio. For example, it could be interesting to discuss a fact that is true in Euclidean geometry but not true in hyperbolic, explain why, and illustrate in GeoGebra. Or you could have separate entries discussing Euclidean and non-Euclidean facts. Another type of entry to include could be a meta cognitive reflection about your learning in this course. For example: describe how you have improved as a mathematician and student of geometry this semester. Do this by writing 2 or 3 paragraphs that describe your growth, both in terms of geometric knowledge and in terms of your overall understanding of how mathematics works (the role of axioms, proofs, historical context, etc.) Include a few examples of work that illustrate this growth. Questions: (1) Compare the Euclidean and spherical geometries. What properties do they have in common and what properties are different? You can get ideas from the video https://www.youtube.com/watch?v=6D-MEb-599A What do you find interesting or intriguing about the spherical geometry? (2) Explain how to calculate distances on a sphere. You can present the examples we did in class (from Quito, Ecuador (0˚ N, 78˚W) to Kampala, Uganda (0˚ N, 32˚E) and from Saint Petersburg, Russia (60˚ N, 30˚E) to Anchorage, Alaska (60˚ N, 150˚W) ) or make up your own. Include illustrations and calculations. (3) State and prove the “midpoint quadrilateral theorem”. Be sure to carefully state any facts or assumptions you are using. Illustrate with a Geogrebra sketch. You can also include an analysis of student work (at the end of 1.2 outline.) (4) Choose 2 or 3 compas/straightedge constructions (equilateral triangle, any part of major exercise 1 from Chapter 1, major exercise 4 from Chapter 1). Perform the constructions using only a compass and a straighedge (or “create line between two points” and “create circle” commands in GeoGebra.) State the steps of each construction (like in Euclid Proposition 1) and prove that your construction results in the desired outcome. List any assumption/facts you need to justify your proofs. (5) Historical essay (about 1 page): what is the significance of Euclid’s contribution to geometry and mathematics as a whole? (You would probably want to find other sources in addition to our textbook.)
6.[Proof] In your own words, explain the classic proof of the fact that the square root of 2 is irrational.
7 [Histrorical essay] How and when was the existence of irrational numbers discovered? What was the significance of that discovery? (About 1 page.) You will probably need other resources in addition to our textbook.
8 [Proof] Describe what incidence geometry is and prove a few propositions that are true in incidence geometry (for example, the ones on p. 51).
9 [Proof] Show that the incidence axioms don’t imply any specific parallel property (Euclidean, hyperbolic, or elliptic.) To do this, for each parallel property give an example of a model of incidence geometry where that property holds. 10 History Essay about Farkas Bolyai (mathmatician) explain their attempts to prove the parallel postulate and the significance of their work. Farkas Bolyai (Hungarian, 1775-1856) Dedicated his life to proving the parallel postulate. Was “ready to become a martyr who would remove the flaw from geometry and return it purified to the mankind”, but then despaired. Warned his son János not to work on this problem, for “it can deprive you of all your leisure, your health, your rest, and the whole happiness of your life.” 11 In GeoGebra, open one of the hyperbolic models, for example https://www.geogebra.org/m/eJAw9rcQ (but feel free to search for other ones if you would like – it could be a good project to compare different models and evaluate their pros/cons) . Use the ideas from Euclid’s Proposition 1 (which doesn’t use the parallel postulate and therefore is valid in hyperbolic geometry) to construct an equilateral triangle. Measure the sides of your triangle to confirm that it is in fact equilateral.. In hyperbolic geometry, is it still true that the angles of an equilateral triangle are congruent? If so, how to prove this?. Measure the angle sum of your triangle. What happens to the angle sum as you change the length of a side of a triangle? Is it possible to get the angle sum to 180 degrees, or close?. What angle sum corresponds to a side length of one unit?. Construct another equilateral triangle and measure its side lengths and angles. Can you manipulate your two triangles to make them similar but not congruent? 12 explain how to make a physical model of a hyperbolic plane. Or even better, make your own! Here are some resources: • Using heptagons: https://blog.doublehelix.csiro.au/hyperbolic-paper-craft/
• Using triangles: see p.2 of https://math.berkeley.edu/~kpmann/DIYhyp.pdf
• How to crochet a hyperbolic plane: https://static1.squarespace.com/static/5754f47fcf80a16bffa02c45/t/60a2dd27e2d1 e010d14216b6/1621286201908/Handout-Hyperbolic_Crochet.pdf This comes from the book by DainaTaimina, “Crocheting Adventures with Hyperbolic Planes”; see also her TEDx talk: https://www.youtube.com/watch?v=w1TBZhd-sN0&t=957s Here is another visual for a hyperbolic surface (“saddle”): http://www.math.sc.edu/~filaseta/courses/Math241/animatedsaddle.html
