Turning the Place Over

As part of Liverpool08 European Capital of Culture there were a huge number of events and displays. One of the art installations was called "Turning the Place Over". Can you find our how it works?

Where Art and Maths Combine

In this article, Rachel Melrose describes what happens when she mixed mathematics with art.

It Depends on Your Point of View!

Anamorphic art is used to create intriguing illusions - can you work out how it is done?

Moving Squares

Stage: 4 Challenge Level:

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.