Season 5 Episode 1
An equilateral triangle with side length 1 fits inside a square box; what's the minimum possible side-length of the box? What's the best way to cover a disc with circles? We explore both problems in this episode!
Triangle in Square
For an alternative derivation, consider rotating the triangle by some angle, and find the size of the rectangle aligned with the axes which fits around the triangle. The area of this rectangle is minimised (skipping all the geometry and calculus) when the triangle is tilted by 15 degrees and (it turns out) when the rectangle is a square.
Thanks to Hitham for writing in to the email address below with this approach.
More packing problems
Erich’s Packing Center has lots more problems, and progress towards solving them.
Circles covering a circle is more of the problem that we covered on the livestream.
Triangles in a square is more of the problem that we managed to fit into the livestream.
Squares in squares went viral on Twitter in between when the episode was broadcast and when I wrote this further reading!
Covering circles with circles as a fairground game; see this YouTube video (this is from a TV show called The Real Hustle about how to scam people… it was a thing in the late 2000s / early 2010s).
For a discussion of this game as a fairground game (no mention of any maths), see this page. I think it's interesting to see people discuss this without talking about the weird mathematical trick that (in my opinion) helps to make the trick work.
For fans of Physics, the phenomenon of Symmetry Breaking is not really related to the problems discussed on the livestream, but it does have a similar flavour; minimising something by taking a solution that is not symmetric.
For fans of Philosophy, see this page for a different take on symmetry breaking, from the fantastic Stanford Encyclopedia of Philosophy.
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk