Cut a Sine Wave with One Straight Cut
You can cut a strip of paper along a sine wave with one straight cut. Does this sound impossible?
- A sturdy cardboard tube about three centimeters (an inch or so) in diameter. Cardboard tubes in toilet paper rolls are too flimsy to hold their shape when you cut through them. If you don't have a sturdy tube, you can use a candle, or even a relatively cylindrical carrot.
A strip of paper long enough to wrap around the tube a few times.
- Roll a long strip of paper around the tube a few times.
- Cut through both the paper and the cardboard tube with a knife at an angle i.e. not perpendicular to the cylinder's axis.
- Unroll the paper to see a sine wave cut.
- At first you might think that a slanted plane cuts a cylinder at an ellipse, not a sine wave. It does cut the cylinder in an ellipse, but not from the perspective of the paper that wraps around the cylinder. To prove that the cut you made is a sine wave along the paper, combine the parametric equations for a cylinder: x = cos t and y = sin t with the equation for a slanted plane: z = A y.