Useful Links

I will add more links to this page as the semester progresses.  If you find a site that you think should be included, let me know and (if I agree!) I will put it up.

General Information

• In June 2010, Wisconsin adopted the Common Core Standards for Mathematics and English Language Arts. These standards are therefore a description of what students in Wisconsin are expected to know and be able to do by the time they graduate from high school. Those of you who are intending to become high school mathematics teachers will need to become familiar with these standards at some point, though certainly not during this course.
• You can download a PDF version of the standards at the Common Core website, but there is also a very useful hyperlinked version available at the Common Core Tools website.
• The State of Wisconsin Department of Public Instruction developed a nice graphic to remind teachers (and others) of the importance of the Standards for Mathematical Practice.
• The Wisconsin Mathematics Council is our state branch of the National Council of Teachers of Mathematics. You should consider attending the WMC annual conference at Green Lake in May; see the WMC website for conference and registration information.

Chapter 6

• It is not possible to build an isometric model of a full hyperbolic plane in 3-dimensional Euclidean space, but it is possible to build a model of a part of one. Daina Taimina came up with the idea of doing this with crochet. In this article you can even see a picture of a (portion of a) hyperbolic plane with some hyperbolic lines stitched in.

Chapter 5

• Not Euclidean geometry, of course, but here are the two websites we saw in class that explain the argument leading to the area formula for a spherical triangle in terms of its angle sum:
  • A Math Stack Exchange page (if you read this one, you should concentrate on the first answer);
  • An explanation by Caroline Seaman and Stephen Szydlik from UW-Oshkosh.

  You might also be interested in this site by the same UW-Oshkosh team, which allows you to explore lunes and spherical triangles with Geogebra applets.

• Dynamic geometry software is a powerful tool for carrying out geometric explorations. One quick way to use this tool is the online GeoGebra calculator. (You can also download a free vewrion of the program to run on your own computer.)

Chapter 4

• From the excellent Cut The Knot website, here is a page on Euclid’s Fifth Postulate (the Euclidean Parallel Postulate in our terminology). If you scroll down, you will find a list of propositions which are equivalent to that postulate in neutral geometry. Amongst the links at the bottom of the page is one to a proof that the Pythagorean Theorem can be added to that list; I have not found this equivalence in our text, but many of the proofs on that page will be familiar to you. (By the way, Cut The Knot also has a page full of different proofs of the Pythagorean Theorem.
  It follows that every one of those proofs must use the Euclidean Postulate in some essential manner; it is a nice exercise to identify where in each proof that postulate is needed.)

Chapter 1

• The Babylonians knew the Pythagorean theorem long before Pythagoras. Unfortunately, from the evidence that has come down to us, it is hard to tell just how much they knew, but the famous Plimpton 322 tablet provides a list of Pythagorean triples that are unlikely to have been discovered by guesswork. (David Joyce provides more explanation of the tablet on this page, if you are interested.)
• The Egyptians may also have had some knowledge of the Pythagorean theorem, though again we do not have the documentation to be able to say just how much they knew.
• If Thales was the first person to introduce logic into geometry, as stated in our text, then what is Thales’ Theorem?
• David Joyce has a wonderful Euclid’s Elements website. Almost the entire text is there, with commentary, and many of the diagrams are “live” java applets.