### How Many Geometries Are There?

An account of how axioms underpin geometry and how by changing one axiom we get an entirely different geometry.

### When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

### Flight Path

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

# Weekly Challenge 47: Weird Universes

##### Stage: 5 Challenge Level:

Four identical robots and their enemies, the stick men, live in small universes based on squares with the opposites sides identified (i.e. corresponding to the same points in space) in various ways, as shown in the diagrams below:

Consider these universes. In which ways could the stick men shoot the robots? Which create a geometry which can be made in 3 dimensions out of a sheet of paper? Perhaps you might like to experiment with other universes?
Did you know ... ?

String theory suggests that each point we see in 3D space is in fact a higher dimensional surface wrapped up in very specific ways.