Teaching Symmetry
From Prep08Wiki
Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. - Hermann Weyl[1]
| Presentation Slides | |
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| File:Symmetry-part1.pdf, File:Symmetry-part2.pdf |
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Mathematical Background
What is a symmetry? In the most general form, it is an object that is preserved by a transformation. Both 'object' and 'transformation' can be many things, but for simplicity we focus on an object which is a subset of the Euclidean plane, and a transformation which is an isometry, or rigid motion.
A classification of plane isometries:
- Reflection
- Rotation
- Translation
- Glide reflection
The symmetry group of a set is the group of isometries which preserve the set. We focus on discrete groups, which have no isometry within epsilon of the identity. There is a complete classification of plane symmetry groups:
- Rosette groups. The only finite plane symmetry groups. Cyclic groups have only rotations, C1 is the trivial group, then C2, C3, C4,... have 2, 3, 4,... elements. Dihedral groups have reflections. The dihedral group Dn has n reflections, and n rotations, so 2n elements. D1 is also known as bilateral symmetry.
- Frieze groups. The frieze groups (also known as border groups, strip groups, or one-dimensional groups) have one axis of translation. There are seven of these.
- Wallpaper groups. These have more than one axis of translation (the translation subgroup forms a lattice). There are 17 of these.
Patterns often have coloring with two or more colors (for example, the yin-yang). This adds considerable subtlety to the classification of possible symmetries.
Teaching Symmetry
How deep to go?
- Discuss the classification of isometries? Prove it?
- Discuss composition of isometries? Discuss the group structure of a symmetry group? Multiplication tables? Abstract groups?
- Prove the classifications? Rosettes? Frieze groups? Wallpaper groups?
- Deal with non-discrete groups? Deal with color symmetry?
In a liberal arts math course, it's even possible to get by without discussing 'isometry' as an abstract idea, but only in the context of a pattern. Use canonical markings for symmetries, and then the markings are the group.
Classification can be used as a unifying theme, here and throughout the course. For example, there are many accessible classification problems involving tessellations.
Rosette Patterns
Students can almost complete the classification with no introduction to the subject (escherwiki:Symmetric Figures Exploration). Reflections call for mirrors, folding, and a vertical orientation. Rotations can be described using angle or order. Doing reflections or 90° rotations on graph paper is hard for (our) students. The Java Kali applet is useful throughout symmetry explorations.
This is a good place to introduce the idea of composition of isometries, for example escherwiki:Composition Exploration.
Frieze Patterns
Introduce translation, beware of 'direction' vs. 'axis'. Glide reflection is very subtle. Strips copied onto transparency can be helpful. There are many naming conventions for Frieze groups. The classification of seven frieze groups is feasible.
Wallpaper Patterns
The classification is out of reach, unless you're doing a whole semester of symmetry and teaching a higher level course. Students can use a flowchart to identify patterns. Beyond that, they can mark all symmetries on a pattern. Can introduce the lattice of translations and the fundamental domain. Escher Web Sketch is useful here, or purchase something like Tess.
Sources
- Washburn & Crowe, Symmetries of Culture, 1988, University of Washington Press.
- Isometrica A Geometrical Introduction to Planar Crystallographic Groups, by George Baloglou
Sources for examples? Escher, science, culture, and the world around you. For example, escherwiki:Celtic Art Exploration, escherwiki:Frieze Group Exploration.
To take home: Work alone or in groups on the Developing Symmetry Sources project to practice developing sources into course materials.
