The elderly-styleed Greek geometer Euclid conshort-termed a catalog of five axioms he held to be self-evidently real. They are (or are equivalent to):
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You can draw a line between any two points.
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You can extfinish lines indefinitely.
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You can draw a circle at any point with any radius.
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All right angles are equivalent.
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All triangles have inner angles that comprise up to 180 degrees.
Euclidean geometry is what we lget at school, and only applies to flat surfaces. The inner angles of a triangle on a curved surface do not comprise up to 180 degrees – the topic of today’s confincludes.
1. Right, Right, Right.
Assume the Earth is a perfect sphere. Imagine dratriumphg a straight line from the North Pole to a point on the Equator. Can you draw two more identical lines to produce a triangle where all the inner angles are right angles (i.e. they comprise up to 270 degrees overall)?
2. Full circle
Next, let’s go hugeger, angle-teachd. Can you discover a way to cover the Earth with equipostpoinsistral triangles that have inner angles of to 120 degrees (i.e. they comprise up to 360 degrees overall)? These triangles must all be the same size and there must be no overlaps or gaps between them.
(Hint: skinnyk about dratriumphg triangles side by side.)
3. Tasty triangles
Now envision a donut instead of a sphere. Can you draw two identical right-angled triangles on the donut so they perfectly cover its surface? And what will the sum of the six inner angles of these two triangles comprise up to?
(A donut is a ‘torus’, a cylinder that curves and joins itself in a loop, as in the image above.)
I’ll be back at 5pm UK. PLEASE NO SPOILERS. Instead talk your favourite axioms.
Today’s confincludes were set by Adam Kucharski, who is a maths professor at the London School of Tropical Medicine and a famous science author.
In his intelligent recent book Proof: The Uncertain Science of Certainty Adam tells the story of how nineteenth century skinnykers began to dispute Euclid’s self evident truths – and how this shaped the history of mathematics. It’s a fantastic read that covers many fields, including history, politics, statistics, computer science and epidemiology, which is Adam’s area of professional expertise.
Proof by Adam Kucharski is out in the UK on Thursday and useable at the Guardian Bookshop.
I’ve been setting a confinclude here on alternate Mondays since 2015. I’m always on the watch-out for fantastic confincludes. If you would enjoy to recommend one, email me.
