4D Chain and Sproket

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

4D Chain and Sproket

Postby ICN5D » Sat Apr 19, 2014 10:34 pm

I was thinking about this today, while at work, obviously. Using the property that a glome can have two non-intersecting ortho circles, the powers of extrapolation, leads me to the idea that this glome can be bristling with pointed sprocket teeth. This would allow for two chains to be wrapped around like a cassette ( cluster of cogs ). This allows for the possibility of two drivetrains to operate as one, but at 90 degrees of each other. I'm sure there's some more phenomena going on I'm not seeing right away. What the applications/implications would be, I don't know. It's just something cool that translates to a familiar system for us trionians. It's similar to the polytwister gear drive idea, in a bicycle setup. It seems like the glome sprocket can rotate at different speeds for each chain, which would have an interesting turn out of effects.
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Re: 4D Chain and Sproket

Postby quickfur » Sat Apr 19, 2014 11:37 pm

I'ts not just similar to the polytwister gears idea, it's actually the same thing. :D The decomposition into two orthogonal circles is just the Hopf fibration of the digon, which is just the simplest case of a polytwister gear. :)

The interesting thing about this decomposition is that rotation in one ring is equal to twist in the other ring. That is, if you attach teeth in two orthogonal great circles around the glome, and lock gears into either ring, then rotating one gear causes the corresponding ring to spin in its plane, and that causes the orthogonal ring to twist in-place (i.e., rotate around the great circle) -- the teeth don't move but they will spin. Similarly, if you make the teeth ridged such that you can twist them in place, then doing so will induce a rotation in the orthogonal ring.

Making the teeth smooth (i.e. they can freely spin in-place) will allow you to use the glome as two independent gears simultaneously. In this case, one ring can rotate independently of the other; the other can stay put, but the teeth will experience wear as they spin in place. :)

But my thinking is that the rotation-twist correspondence can probably be exploited in other ways, such as making a gearbox, or something similar.
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Re: 4D Chain and Sproket

Postby wendy » Sun Apr 20, 2014 7:40 am

For a bycicle in 4D, you really don't want to go for hopf fibulation unless you enjoy being spun in the saddle.

What is called for is a good old prism of 3d chain and sproket. Nothing more, nothing less.
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Re: 4D Chain and Sproket

Postby anderscolingustafson » Mon Apr 21, 2014 1:55 am

Each tooth can extend through all the circles of rotation on a glome cogwheel.
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