Season 7 Episode 12
In this episode, the last before we take a short break for Easter, James presents two views of hyperbolic space, with an applied maths traffic puzzle and a pure maths heptagon problem.
Further Reading
Hyperbolic space
Some of the properties of hyperbolic space are outlined at the top of this page.
The issue of the parallel postulate is particularly noteworthy (and, indeed, we noted it on the stream!)
Crochet
Dr Daina Taimina at Cornell University has developed a way to crochet models of hyperbolic space. There’s a description on her Wikipedia page with the story of how she invented this, there are some pictures in this PDF. Daina has also written a book which includes activities you can do with your crochet hyperbolic plane to learn about hyperbolic space; Crocheting Adventures with Hyperbolic Planes Tactile Mathematics Art and Craft for all to Explore.
Heptagons
If you’d like to recreate the heptagon model, but you don't want to draw your own heptagons, fear not! You can print your own heptagons with this handy PDF that I've made; heptagons.pdf (2 KB)
M. C. Escher
You can see more artworks by Escher here.
Escher worked with Roger Penrose; you can read about that here.
For more on Escher himself, see this page.
Calculus of Variations
Towards the end of the episode, I tried to describe a way to find the function that minimises an integral. This was my attempt to introduce Calculus of Variations. Judging from the reactions in chat, this is perhaps something that I do too often!
There are some Oxford lecture notes on this technique here https://courses.maths.ox.ac.uk/course/view.php?id=3990 (I’ve picked the lecture notes by James Maynard, because they’re the ones I was most recently reading).
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.