This is a course I've been teaching at the Urban School every other year since 1992. It is an advanced geometry elective, with two interrelated components.
1. Symmetry and Transformations
Symmetry is a great entry point into abstract algebra, an area of pure mathematics. Transformations provide a different approach to geometry than the one students usually experience in Geometry class. These two topics are intimately connected, and we study them in depth.
- Transformations of the plane, especially the four isometries, and their interrelationships.
- The structure of symmetric designs -- around a point, along a strip, in the plane. The seventeen wallpaper groups. Creating and analyzing designs.
- Introduction to group theory -- symmetry groups, other examples of groups (finite and infinite, commutative and non-commutative.)
- Some helpful review of complex numbers, which helps to lay the groundwork for...
- Computation of geometric transformations with the help of complex numbers at first, then matrices -- this is the mathematics that underlies all computer graphics.
- Connections to art and design. Tiling. Escher.
- Students are assigned symmetry groups, and are asked to create designs for bulletin board display.
2. Dimensions, from 1 to 4
This is not so much a topic as an arena for exploration. We mostly work in three dimensions, based on hands-on model building with the Zome System and with Cabri 3D:
- The Platonic and Archimedean solids; other polyhedra. Duality.
- The golden ratio
- Euler's and Descartes' Theorems
- Review of various topics from geometry, trigonometry, and algebra as applied in three dimensions
- Some basic theorems of solid (3D) geometry
Towards the end of the course, we do some work on visualizing the fourth dimension, through reading, model building, discussion, and analogy -- in part based on Abbott's Flatland, a classic of mathematical fiction.
You can download a presentation about the class, which includes among other things photos of student-created designs, Zome buildings, and symmetry in every day life. See also a shorter and more recent version of the talk, and finally a 2D version for teachers of grades 7-10. (Warning: these are large files.) The Cabri files below are also part of these presentations.
Here are some materials I use in the course. Alas, I can't always do everything outlined below as time tends to run out.
From Algebra: Themes, Tools, Concepts by Anita Wah and Henri Picciotto, several lessons on Abstract Algebra, which work well in combination with the material in Brown's book:
Transformational Geometry by Richard G. Brown (Dale Seymour Publications -- out of print but findable on the Web). Pretty much the whole book.
Handbook of Regular Patterns by Peter Stevens (MIT Press). An extraordinary multicultural mathematical source on symmetry. I use it as a source of designs and ask students to identify the symmetry groups. I would do more with it if I had time.
Scott Kim's Inversions (Key Curriculum Press), a book where symmetry interacts with a demented typography to create entertaining and thought-provoking images.
Flatland by Edwin Abbott. Extremely cheap from Dover Books, and free on line. (Some students read the whole thing, but the math is mostly in chapters 1, 2, and 13-22.) There are several excellent sequels, but I don't have time for them in this class. I have not seen the Flatland movies, but I have sometimes shown part of a fun Halloween episode of The Simpsons where Homer accidentally goes 3D.
Geometry Labs by Henri Picciotto. Labs 5.1, 5.5-5.8, 6.5, 7.3, 7.4. Some of those involve pattern blocks.
The Zome system and Zome Geometry by George Hart and Henri Picciotto (where to get it). Lessons 2.1, 2.2, 3.3, 4.1, 6.1, 6.2, 7.1, 7.3, 9.1, 9.2, 10.1, 11.1, 11.2, 21.1, 21.2, 21.4, 24.1. I would do more with this too if I had time! (See also George Hart's site).
Jovo, another building toy which is based on faces (as opposed to Zome's focus on edges and vertices). Jovo is excellent for introducing the Platonic and Archimedean solids.
Cabri software (two- and three-D) is a crucial tool. (In two dimensions, Geometer's Sketchpad would also work, of course.) Once I started using Cabri II+, the transformational geometry part of the course became vastly more accessible. See the attached Cabri files for some examples of teacher demos I use when presenting this material to teachers, but of course the main benefit of using this kind of software is when students build their own figures. I have only scratched the surface with Cabri 3D, which has tremendous potential for the latter part of the course.
The TI-89 calculator is wonderful for the work with complex numbers and matrices. One can create functions that output matrices, so for example after defining those functions, one can enter tr(2,3)*ro(45)*tr(-2,-3) to create a matrix that will translate by the vector (-2,-3), then rotate 45 degrees around the origin, and finally translate by the vector (2,3). See the worksheets on Computing Transformations and the TI-89 files.
Adobe Illustrator is a very mathematical program, very well-suited for creating symmetric designs.
vZome: software which I have yet to use with students, but which provides a way to create Zome constructions and more on the computer.
How proof figures in the course:
Transformational Geometry is largely about formally proving what Brown calls the fundamental theorems of isometries in the plane (that there are only four, that one is uniquely determined by the images of three non-collinear points, that each can be expressed as a sequence of no more than three line reflections.) This is rather epic, and takes a few weeks, but it is very satisfying. He follows up with proofs of various results about isometries, some of them geometric, and some algebraic. The latter are often very elegant, and a testament to the power of group theory.
Zome Geometry includes proofs of why there are only five platonic solids, as well as Descartes' and Euler's theorems about polyhedra.
If you are interested in offering a version of this course at your school --or some parts of it-- please get in touch!