When one studies group theory in mathematics classes one is told groups are used by crystallographers. The math texts then go on to talk about other problems. Some abstract algebra books prove there are just 17 possible wallpaper groups, some even have short proofs. Short proofs are the result of building up a lot of notation and lemmas, not something to share with non-math majors. So I wanted to find out what crystallographers did with groups and perhaps find out how it is shown that there are just only so many ways molecules could be arranged to form crystals.
I wanted to find opportunities for students to read on their own, even to do independent projects. This is an introduction of some topics that lend themselves to independent study. Crystals are interesting by themselves, students can grow the crystals and will be moved to ask questions. Finding answers to those questions might motivate some students to become scientists. Mechanical drawing is a skill that one can teach oneself, a skill that can teach one problem solving, and a skill that one becomes proficient at by independent work. There are books on group theory that claim to be accessible to high school students. By working on this unit I hope I have made an introduction that will help students to work through the references. The books I consulted say a great deal more than what I was able to report here. There is more in those books than I got out of them from my first readings of them, so I intend to go through some of them again. Especially Elizabeth Wood’s
Crystals and Light
.
This unit gives a starting point for my students to find projects for themselves. By starting with this material we will have some common vocabulary. My students can assume that I know the vocabulary and concepts in this material, but anything they discover that is not in here will be news to me, that is, something for them to teach me.
This unit was an opportunity to find problems and concepts to introduce or allude to when teaching the standard math curriculum. Answers to the question: “What good is this stuff?” The law of constancy of angle points out the value of trig functions over just angles. The tangent is the “obvious” variable not the size of the angles.
Matrix multiplication is taught and students need time to practice matrix multiplication before they use it in larger problems. One early practice activity would be to multiply all the 3 x 3 matrices having a one in each row and column and zero in all other positions. No two ones in the same row or column. It would be a chance for the students to see closure.
If you want a book list I have a data base of the books I looked at. It tells what libraries they may be found in. The bibliography is books I believe others can easily enjoy.
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