The Klein Bottle and the 4th dimension [4D Toys v 1.7] « mtbdesignworks Blog [Miegakure, 4D Toys, Marc ten Bosch]

One of the shapes people requested the most for 4D Toys is the Klein Bottle, so I finally added it!

I originally wasn’t very keen on adding the Klein Bottle for a strange reason: because it’s a 2D shape when sliced it would give thin 1D lines, which are obviously too thin to be interesting to look at. But people kept requesting it, and after thinking about it I figured I could give thickness to the lines by sweeping a circle across the 2D surface to get a 3D surface. This would look way better than simply extruding the Klein Bottle in 4D.

I also wanted to use this update as an opportunity to make a great video about Klein Bottles because the current ones on YouTube are in my opinion unnecessarily confusing. For example I don’t think it’s good to start with how the Klein bottle is two Möbius strips glued together. And I knew I could make better illustrations of the constructions.

Also, and importantly, none of the previous Klein bottle videos on YouTube properly explain the 4D part of it (the fact that a Klein Bottle has to self-intersect in 3D, but not in 4D). I was uniquely suited to fix that, ahah.

In the end, I am very proud of the illustrations I came up with. For example: showing the exact same perspective / construction for the Möbius Strip and the Klein Bottle really makes it clear what is going on! And as a bonus you get how the Klein Bottle is made out of two Möbius strips in an incredibly straightforward way…

It was obvious to me that the same construction could be done in higher dimensions to make what I call a “Spherinder Klein Bottle” which is a 3D version of the 2D Klein Bottle. Well, technically the “thickened” 2D Klein bottle in 4D Toys is also three-dimensional, but in a less interesting way since it’s just the Cartesian product of the 2D Klein Bottle with a 1D circle… Anyway, interestingly it looks almost simpler when sliced than the 2D Klein bottle.

Relatedly, because people have asked me before: yes there is a Klein bottle in Miegakure, and it shows up in a yet another unexpected way.

As always, thank you for your patience and enthusiasm!

By the way, I’ve decided to more seriously update the Patreon I originally made for my quaternion article/video. In many ways Miegakure is more than a game: It is revolutionizing how people explore scientific concepts with games, its YouTube channel is an educational tool in its own right with millions of views and many thousands of subscribers, scientific research developed for the game is published alongside it, etc… If you would like to support us, the Miegakure team would be very grateful. I will also post exclusive previews of what we are working on!

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on Tuesday, December 14th, 2021 at 7:15 pm and is filed under 4D Toys, Mathematics.
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