Season 9 Episode 10

Maybe you've studied geometry on the plane or "Euclidean geometry". We live on a (roughly) spherical planet, and so in this episode of the Oxford Online Maths Club, we're looking at geometry on a globe or "spherical geometry".

Watch on YouTube

Further Reading

Previous Geometry Episodes

Last year we had two adventures with non-Euclidean geometry on the OOMC;

• Projective Geometry
• Two Routes to Hyperbolic Space

I’m saying “non-Euclidean” because these geometries don’t necessarily obey all of Euclid’s Postulates. See this page for a list!

You might also like “taxicab geometry”, which you can read about in this article on the maths careers website or in more detail in this free online textbook. For the full book Models of Non-Euclidean Geometry, see this page.

Spherical triangles

Wolfram MathWorld has statement of the area fact in terms of radians, and in terms of the concept of “spherical excess”.

There’s a related concept of “angle deficiency” for polyhedra made with planar faces (like a dodecahedron, or a triangular prism). See this NRICH activity and then perhaps read this NRICH article afterwards.

Spherical triangles aren’t the same as planar triangles, but you’ll be pleased to know that there is a version of the cosine rule for spherical triangles. You can find an overview of lots more facts at Wikipedia. Some of this has been around for longer than you might guess; mathematicians in the medieval Islamic world worked on a version of the sine rule for spherical triangles in the 10th and 11th centuries.

The study of spherical geometry is essential if you want to understand shape and space on a globe, perhaps because you’re measuring the size of a country, or planning your route across the ocean, whether you’re a 10th century mathematician or a modern pilot. The flight path image in the livestream was taken from Flight Radar 24, where you can track live flights and check for yourself what the shortest distance between two points looks like on the surface of a sphere.