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Quick Pix

Add to Studio It is easy to fill a cube up with smaller cubes, but in Euclidean geometry, dodecahedra don't fit together so nicely. However, it is still possible if you allow yourself infinitely many smaller and smaller dodecahedra. This picture, created by Chiam Goodman-Strauss and Dan Krech, shows one step in the procedure.

Today at Science U

How do triangle tilings turn into polyhedra?

Find out in Triangle Tilings and Polyhedra in the Science U Geometry Center

Focus on Math

The Math Forum at Swarthmore is one of the premier math sites on the Web. With an old and extensive collection, the Math Forum offers materials for nearly everone, from students to teachers, parents and researchers. Some of the perennial favorites at the Math Forum are:

Ask Dr. Math
where students can post questions and search for answers
Teachers' Place
with lesson plans, activities and discussion forums
Steve's Dump
a vast collection of math software and other internet resources.

Five-Minute Seminar
Introduction to Isometries: The Holy Grail

Let's review our story. We began by thinking about creating computer animations. To do an animation of a teapot, for example, we first constructed a model of the teapot out of polygons, since polygons are relatively easy to draw. To animate the teapot, then, all we had to do was figure out how to move around a polygon.

For a computer to effectively move around a polygon, we needed a formula for computing new locations for the coordinates of the vertices of the polygons. However, coming up with the proper transformations is difficult, since in general, transformations will distort the shapes of the objects they move. So we began looking for transformation that introduced no distortion, often called isometries.

A bit of thinking revealed four different kinds of isometries. If you think of picking up a paper square from a table top, and setting it back down, the result will either be a reflection, translation, rotation or glide reflection. We also concluded that these transformations are all fundamentally different by considering whether or not they have fixed points and how many, and whether or not they are orientation reversing, i.e. producing mirror images.

Finally, we convinced ourselves that these are the only isometries of the plane by asking how they combine. Doing one transformation followed by another always produced a net effect which was again a transformation already on the list. For eaxmple, two reflections produce the net effect of a rotation or a translation, depending on whether their mirror lines cross.

So, all that remains is to produce the formulas for each kind of isometry, the holy grail of our quest. A bit of an anti-climax, perhaps, since typically, the actual formulas for isometries are buried deep in the software we use, and of relatively little general interest. However, understanding about isometries and how they work pays you back, whether you are just looking in the mirror, or sitting down to watch the latest special effect extravaganza from Hollywood.

And just in case you are a programmer or a mathematician, a compendium of useful isometry formulas is available in the Science U library.
Complete Seminar Series available in the Science U library.

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