The idea that the angles of a triangle add up to 180° is so well engrained that it comes as a shock to some students that it isn’t true in other geometries. These geometries don’t have to be obscure or abstract since the angles of a spherical triangle drawn on the earth’s surface (assuming it is a sphere) always add up to more than 180° by an amount proportional to its area.
It’s a shame that like much else, spherical trigonometry has long since disappeared from the A level syllabus. The 2-dimensional sine rule
is (or ought to be 8-)) well-known but how many are aware of the 3-dimensional version
or the fact that the great circle distance (the shortest distance) in nautical miles between points with latitude & longitude is given by
where is measured in minutes? You could use sites such as Surface Distance Between Two Points of Latitude and Longitude but it’s not the same as sitting down and proving the general result.
Without this introduction to three dimensions how is one going to start visualising geometry in four or more dimensions? Of course, reading Flatland would be an excellent start. Phoenix-Library has some excellent versions of this book in a number of online formats.
This is perhaps a suitable place to publicise my all time favourite puzzle:
- A hunter leaves his house one morning and walks one mile due south. He then walks one mile due west and shoots a bear, before walking a mile due north back to his house. What colour is the bear?