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.

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

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

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

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.

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. . . .

Prove Pythagoras' Theorem for right-angled spherical triangles.

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.