## Three-Dimensional Kaleidoscope

Posted in Math by pgadey on 2019/05/05

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!

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## Homework #5 Question 4

Posted in Math by pgadey on 2018/07/20

Consider a solid ball of radius $R$. Cut a cylindrical hole, through the center of the ball, such that the remaining body has height $h$. Call this the donut $D(R,h)$. Use Cavalieri’s principle to calculate the volume of $D(R,h)$. Calculate the volumes of $D(25,6)$ and $D(50,6)$.

Several students have asked what $D(R,h)$ looks like. Here are some pictures that I found to illustrate the concept. The donut $D(R,h)$ 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 $z=h/2$ and $z=-h/2$, so that it has total height $h$.

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