Non-Euclidean geometry

problem

Flight Path

Age

16 to 18

Challenge level

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

Pythagoras on a Sphere

Age

16 to 18

Challenge level

Prove Pythagoras' Theorem for right-angled spherical triangles.
problem

Spherical Triangles on Very Big Spheres

Age

16 to 18

Challenge level

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.
problem

Weird Universes

Age

16 to 18

Challenge level

Consider these weird universes and ways in which the stick man can shoot the robot in the back.
problem

Over the Pole

Age

16 to 18

Challenge level

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.
problem

Torus Patterns

Age

16 to 18

Challenge level

How many different colours would be needed to colour these different patterns on a torus?
article
Geometry and Gravity 1

This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.
article
Curvature of Surfaces

How do we measure curvature? Find out about curvature on soccer and rugby balls and on surfaces of negative curvature like banana skins.
article
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.
article
How Many Geometries Are There?

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