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Today’s video is about plane shapes that, just like circles, have the same width in all possible directions. That non-circular shapes of constant width exist is very counterintuitive, and so are a lot of the gadgets and visual effects that are "powered" by these shapes: interested in going for a ride on non-circular wheels or drilling square holes anybody?
While the shapes themselves and some of the tricks they are capable of are quite well known to maths enthusiasts, the newly discovered constant width magic that today’s video will culminates in will be new to pretty much everybody watching this video (even many of the experts 🙂
Here are a few links that you may want to check out:
Drilling a square hole with rounded corners using a Reuleaux triangle (click on the video !!)
http://www.etudes.ru/en/etudes/reuleaux-triangle/ Same (wonderful) Russian site. An animated intro to shapes of constant width.
http://www.etudes.ru/en/etudes/wheel-inventing/ An animation of the cart with non-circular wheels that I talk about in the video.
http://www.qedcat.com/articles/waterwheel1.pdf Preprint of my write-up of all the stuff I talk about in this video. This was published in the Mathematical Intelligencer.
https://demonstrations.wolfram.com/DrillingASquareHole/ An interactive demo illustrating how a perfect square hole (NO rounded corners) can be drilled using a special shape of constant width.
Probably the most accessible intro to shapes of constant width is the chapter on these shapes in the book "The enjoyment of mathematics" by Rademacher and Toeplitz.
This article which I also mention in the video is behind a paywallMasferrer Leon, C. and Von Wuthenau Mayer, S. Reinventing the Wheel: Non-Circular Wheels, The Mathematical Intelligencer 27 (2005), 7–13.
Just found a Japanese toyshop the other day that sells wooden Nothing grinders https://global.rakuten.com/en/store/good-toy/item/c-006?s-id=rgm-top-en-browsehist and a wooden Reuleaux triangle that can be rotated inside a square https://global.rakuten.com/en/store/good-toy/item/c-018?s-id=rgm-top-en-browsehist
I didn’t mention them in the video but there are also 3d shapes of constant width which are also very much worth checking out. All the touching stuff I talk about in this video generalises to these 3d shapes.
The tune you can hear in the video is from the free audio library that YouTube provides to creators. https://www.youtube.com/audiolibrary/music . It’s called Morning_Mandolin and it’s by Chris Haugen.
As usual thank you very much to Danil for his Russian translation and to Marty for all his help with the script for this video.