Thank you to BetterHelp for sponsoring this video! To get 10% off your first month of therapy, go to https://betterhelp.com/standupmaths to sign up today. If you are need of urgent mental health support please check for crisis help lines available in your country. USA: https://www.mentalhealth.gov/get-help/immediate-help UK: https://www.nhs.uk/mental-health/advice-for-life-situations-and-events/where-to-get-urgent-help-for-mental-health/ Do check out Jordan Ellenberg's book Shape. This video was inspired by Chapter 2. https://www.nytimes.com/2021/06/21/books/review/jordan-ellenberg-shape.html https://www.jordanellenberg.com/book/shape/ Support me on Patreon and I will post you a free torus balloon. I hope I don't regret this. https://www.patreon.com/standupmaths Here is Sam Hartburn's geogebra file: https://www.geogebra.org/m/yjtjxr2n Trouser animations thanks to: Bentley Davis https://BentleyDavis.com/ Em Raymond https://twitter.com/techcavy Kalina Panteleeva https://www.instagram.com/kalinadoescg/?hl=en Thanks to Visual Topology & Geometry for "Punctured torus homeomorphism". https://www.youtube.com/watch?v=tz3QWrfPQj4 Me cutting two joined cylinders in half way back in 2015. Good times. (At 29:29.) https://www.youtube.com/watch?v=1wAaI_6b9JE CORRECTIONS: - When talking about the Euler Characteristic of the torus at 16:29 I say the value of zero is "only on our friend the torus" which is technically wrong. In the moment I was just talking about the sphere and the torus but loads of other shapes have an Euler Characteristic of zero. Like the cylinder. - I'm undecided if the string method of thinking about holes at 17:04 was helpful. I could have talked about the dimension of the boundary manifold, but that felt like I was repeating the previous bit too much and wanted a different way to think about it. Not sure if this counts as a 'correction' per se, but I just needed to talk to someone about it. Thanks for listening. - Let me know if you spot any other mistakes! Filming and trouser sewing by Alex Genn-Bash Editing by Michelle Martin Graphics by Sam Hartburn Doughnuts later eaten by Matt Parker Music by Howard Carter Design by Simon Wright and Adam Robinson MATT PARKER: Stand-up Mathematician Website: http://standupmaths.com/ US book: https://www.penguinrandomhouse.com/books/610964/humble-pi-by-matt-parker/ UK book: https://mathsgear.co.uk/products/5b9fa76f230ffa140094dc43 Nerdy maths toys: http://mathsgear.co.uk/