Activity 1-1: Geodesics

A spider spots a fly on the other side of the room.

The room is a large cuboid, 30m by 12m by 12m. Task: what is the spider’s shortest route to reach the fly? The fly is one metre from the ceiling in the middle of the opposite end wall. The spider is one metre from the floor in the middle of one end wall.

It helps here to create a net of the room and lay it flat. So clearly a 42m journey is a possible solution. But can we do better?

If we draw our net a different way... Our shortest path here is m Can we do better still?

This time our net gives a length of 40m. Note that our best solution crosses five of the six walls!

A geodesic is simply the shortest distance between two points. On the plane, the geodesic between two points is the straight line that you can draw with a ruler. What the geodesic is varies according to the surface that you are working on, and the idea of distance that you are working with.

On the sphere, the geodesic between two points is an arc of a great circle (a great circle is the largest circle you can draw on a sphere, with its centre at the sphere’s centre). Above, a geodesic triangle ABC on a sphere, made up of sides that are arcs of great circles.

There is a principle in physics; That light always takes the quickest path from A to B. If light travels from A to B via reflection in the blue surface, what path does it take?

This principle was explored by Heron, Ibn al-Haytham and Fermat.

The spider slowly follows. Let’s get back to our spider and fly. Just before the spider reaches the fly, it flies off and settles on the side of a glass. The glass is cylindrical, 4cm high and 6cm in circumference. The fly is 1 cm from the top of the glass, on the inside. The spider moves to a point 1 cm from the base of the glass on the outside, directly opposite to the fly.

Task: what is the spider’s shortest route to reach the fly now? The answer is 5cm.

With thanks to Pixabay, and Wikipedia. Carom is written by Jonny Griffiths,