Moving Squares
This problem is inspired by a picture created by the artist Bridget Riley entitled "Movement in Squares" (1961)
Take two pieces of squared paper and colour alternate squares, giving a checkerboard pattern.
When you look straight at the two cylinders, the squares appear as rectangles getting narrower and narrower as the page curves away from you:
How could you represent this effect on a flat piece of paper?
What is the significance of the black and white sections at the bottom of the image?
Robert from Bishop Tonnos High School in Canada sent us the following solution:
In the picture viewing the cylinders from above, take each line that forms a radius of the circle, and draw a horizontal line between the left edge of the page and its contact point on the circle. The radial divisions of the circle are equal; we'll call this angle $\theta$.
Radius of the circle is $R$, and square size of the $n^{th}$ square will be denoted $S_n$.
It's clear that for the first square, $S_1 = R\sin\theta$.
For the second square, $S_2 = R\sin(2\theta)-R\sin\theta$
What this problem reduces to is finding the difference between horizontal lines, since it is the "overhang" which will determine the apparent size of a square. The difference between lines (and therefore the width of each square) is:
$$S_n=R\sin(n\theta)-R\sin((n-1)\theta)$$
Louis from Eltham College considered what would happen with a cylinder with $n$ divisions around it. You can read his solution here.
Why do this problem?
Possible approach
Once learners have devised a way to work out the measurements needed to create the image, allow plenty of time to actually make the images - an excellent opportunity for a classroom display!