# Space

An advanced geometry elective after Algebra 2
by Henri Picciotto

## Course Description

This is a course I taught at the Urban School Math Department every other year between 1991 and 2013. 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 a traditional 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.

## Resources

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. The Cabri and Geogebra files below are also part of these presentations.

The sequence of lessons on the transformational geometry and symmetry component of the course, the last time I taught it, is here. That page also includes some discussion of the impact of the Common Core State Standards in bringing transformational geometry to the mainstream.

Isometries of the Plane is a substantial unit on transformational geometry, largely based on worksheets I used in the class.

Computing Transformations is a related unit focused on how to calculate the coordinates of the images of points using complex numbers, and then introducing matrices. (I used this GeoGebra file to create a figure. It could be useful on a projector.) I outlined that part of the course in this blog post: Matrices. (After retiring from the classroom, I created a GeoGebra version of these worksheets, but I have not had the opportunity to use them with students.)

I created a Symmetry home page with material for grades K-12. Much of it originated in this course. See also this unit for an online exploration of symmetry.

Here are some student-created symmetric designs: Rosettes | Friezes

Here are some materials I used in the course. Alas, I couldn't always do everything outlined below as time tends to run out.

### Books

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. The book is made much more accessible with the help of interactive geometry software. (See below.) In particular see the online version of the core argument in the book, with interactive Geogebra figures, here. (I believe this is both simpler and more complete than Brown's approach.)

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.

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 online. (Some students read the whole thing, but the math is mostly in chapters 1, 2, and 13-22.) There are several excellent sequels, and two movies, but I don't have time for them in this class.

### Manipulatives

Geometry Labs by Henri Picciotto. Labs 5.1, 5.5-5.8, 6.5, 7.3, 7.4. Some of those involve pattern blocks, some involve mirrors, and some involve my Geometry Labs template.

Triangle paper is useful for some of the lessons on symmetry.

Here are starter layouts for wallpaper, using pattern blocks. Students can extend the designs by using only the indicated reflections. There is more along these lines on my Symmetry home page. I also use pattern blocks in a catalog of the 17 wallpaper groups.

Zometool 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. (See also George Hart's site).

One possible side trip when studying polyhedra is the Schlegel diagram (see Zome Geometry 6.2). Pent is a two-person strategy game created by George Mills. It is played on the Schlegel diagram of a truncated icosahedron.

I used Jovo, another building toy which is based on faces (as opposed to Zome's focus on edges and vertices). However it may no longer be available in the US. An alternative for introducing the Platonic and Archimedean solids, in advance of all the work with Zome, is the Polydron Frameworks Archimedean Solids. The class set allows the simultaneous building of all the Archimedean Solids.

### Electronic tools

Interactive Geometry: Cabri software (two- and three-D) was a crucial tool. In two dimensions, Geometer's Sketchpad would also work, of course. And now Geogebra includes both two- and three-dimensional options in a single free software package. That's what I would use if I were still teaching this class.

Once I started using Cabri, the transformational geometry part of the course became vastly more accessible. See these Geogebra or Cabri files for some examples of teacher demos I used 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 3D geometry software, which has tremendous potential for the latter part of the course.

The TI-89 calculator was 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, but it is pricy. A cheaper commercial alternative is Affinity Designer.

vZome provides a way to create Zome constructions (and more) on the computer.

## How proof figures in the course

This is a course for students who are more mathematically mature and developmentally ready for proof than students in a ninth or tenth grade geometry class.

Transformational Geometry is largely about formally proving what Brown calls the fundamental theorems of isometries in the plane (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, and that there are only four distinct isometries.) This is rather epic, and takes a few weeks, but it is very satisfying. (See an online version here.) 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.

Related pages on this site:
Symmetry
Complex Numbers
Geometric Transformations
Proof | Math 2
An introductory lesson about knot theory.
Some notes on Rubik's cube.
Pent, a two-person strategy game
Infinity, another advanced elective for high school.