Spin a Rectangle

thingiverse

I had a fantastic time spinning the cube both physically and mathematically, and I was eager to explore other shapes. Professor Fardin brought up Erickson's piece on resolving credit in the MAA book Aha! Solution (p. 64), which sparked my interest in examining the problem from a design perspective. It turned out to be quite fascinating—a cone-rich puzzle! In short, if we take a rectangle and spin it around one of its diagonals, what 3D shape do we get? What is the volume of the resulting 3D solid? The 3D models provided here are based on a 30mm x 40mm rectangle. Several pieces are included for students to perform the physical experiment and appreciate the outcome. Algebraically, the volume of the 3D solid is a function of the two dimensions of the rectangle. One can derive the formula using the cone components of the shape (see the figure for the formula, typeset in LaTeX). There are several ways to look at the 3D shape: spin the rectangle, examine the whole 3D shape (smooth whole or cage), or look inside the solid. A touch of super glue is needed to make a whole out of two halves. Among the Files: 1. Rectangle (30mm x 40 mm) 2. Stand 3. Spinning rectangle cage (12 copies) 4. One-piece half 5. One-piece half with hole in the middle 6. Three-piece half 7. Three-piece half with hole in the middle Reference: Erickson, Martin. (2009). Aha! solutions. Washington, DC: Mathematical Association of America ("Revolving Credit," p. 63-64)