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Quick Pix

Add to Studio This oddly-shaped object is a smooth version of a threebolite tile. These pieces fit together to fill out three-dimensional space in such a way that the tiling pattern does not repeat. This image was created by Chiam Goodman-Struass to illustrate his work on higher-dimensional aperiodic tiling.

Today at Science U

Why is it hot in the summer and cold in the winter?

Find out in Seasons Reasons in the Science U Observatory.

Focus on Math

At HyperGami, the "pineapplehedron" is just one of many delightful polyhedral creations you will find. HyperGami is Macintosh software that bills itself as a CAD system for paper sculpture.

The HyperGami software is freely available for non-comercial use upon request, and a wealth of kits, crafts and educational materials are available from HyperGami Crafts.

For more information and activities involving polyhedra and paper folding, visit the Science U Geometry Center.

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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