## Symmetry @ Otterbein

The image represents a molecule of chloropentacarbonylmanganese or ClMn(CO)_{5}. For the last half-hour or so, I have been playing around in the Symmetry Gallery put together by Otterbein. Lots of lovely molecules with high degrees of symmetry. It is remarkable (to me) that such things even exist in nature. I encourage you to go and check out the lovely interactive demos on the site.

## Three-Dimensional Kaleidoscope

My highschool student, Lukas Boelling, made this three-dimensional icosahedral/dodecahedral kaleidoscope with his dad, @eric_boelling. Lukas based his models off this excellent paper: Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes by Roe Goodman. Stay tuned for more models!

## Homework #5 Question 4

Consider a solid ball of radius . Cut a cylindrical hole, through the center of the ball, such that the remaining body has height . Call this the donut . Use Cavalieri’s principle to calculate the volume of . Calculate the volumes of and .

Several students have asked what looks like. Here are some pictures that I found to illustrate the concept. The donut is the region between the red sphere and blue cylinder. The golden balls below show various views of the donut. The donut should fit between the two planes and , so that it has total height .

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