Designing a 4D Oxcart and Wagon

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.

Designing a 4D Oxcart and Wagon

Postby PatrickPowers » Wed Dec 30, 2015 2:28 am

It seemed like a good idea to start with the simplest thing, so I designed an oxcart. It took a week of fumbling around. I think the main problem besides acute beginneritis stemmed from using projections as a guide. Things improved once I switched to Cartesian products.

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The body of the 4D oxcart is a hyperprism. It has a top, bottom, and six sides. There are four wheels, one each for the ana, kata, left, and right sides. It is steered by a bar in the front that pivots only up and down.

Perhaps I spent most of the time worrying about the axle upon which the hyperprism rests. One axle or two? What shape should those axle(s) be? Should it be a one axle then it is the Cartesian product of a planar object -- a square would do -- with a small disk. I think that would be all of heavy, weak, and difficult to make, so I went for two perpendicular axles. A trouble I had was trying to use as an axle a spherinder, the Cartesian product of a small sphere and a line. Such has many possible planes of rotation is so is not a stable mount for a wheel. It is better to have as an axle a cubinder, the Cartesian product of a disk with a long skinny rectangle. Then there is only one possible plane of rotation. (The axle doesn't rotate, but it must be possible for a hub to rotate around a portion of it.)

The two axles could be merged into one to make an X. But I think that would be difficult to make and less durable to boot. Better to have one axle slightly closer to the front of the cart than the other, so that the two axles will be at the same height but not intersect.

The hub of the wheel is the Cartesian product(CP) of an annulus with a small square. The spokes needs to fit tightly in holes. This is easier with round holes, so a spoke would be a spherinder, the CP of a sphere with a line.

Once this was done, a wagon is not far away. Use two of the wheel assemblies, forward and back, for a total of eight wheels on four axles. The forward assembly is on a yoke that steers independently of the body. The yoke pivots on an axle that is a spherinder so that it may move both left-right and kata-ana. That's all.

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Re: Designing a 4D Oxcart and Wagon

Postby PatrickPowers » Sat Jan 02, 2016 5:31 pm

If one wants to scale the oxcart up to N dimensions, then intersecting axles are the way to go. You can't have, say, 100 axles because there are only two dimensions to work with to avoid intersections. Eventually the offset of the axes gets out of hand.

So how to make a single axle for an ND oxcart? Get a fairly small spherinder for the intersect. Make cubinder half-axles that can be pounded into and/or welded onto the spherinder, quite like the hub of a wheel.

As N grows large -- more than a dozen? -- the surface-to-volume ratio of a hyperprism grows large. It could be that it becomes worthwhile for the standard package to be a spherinder or even a hypersphere to save on packaging materials. Further study seems indicated.
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Re: Designing a 4D Oxcart and Wagon

Postby PatrickPowers » Fri Jan 08, 2016 10:14 pm

Second thoughts on the N-dimensional oxcart:

It is practical to have N-2 axles in N dimensions. Displace each axle slightly in some other dimension so that they don't intersect.

Label the dimensions as d1, d2, d3... dN. The first two dimensions are up/down and forward/back, so there are no transverse axles in these dimensions. For 2 < i < N, displace the di dimension transverse axle in the di+1 dimension. Displace the dN axle in the d3 dimension.
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