## Rolling things in 3d and 4d.

Discussion of shapes with curves and holes in various dimensions.

### Rolling things in 3d and 4d.

Instead of looking at the rolling thing itself, we can look at its foot-print on the ground.

In 3d, the ground is 2d. A cylinder laid on the ground produces a line, and rolling this cylinder moves the line across the plane, vertical to the line. Motor-car wheels are short lines that are easy to change the direction of motion. Stood on its end, the ground-section becomes a circle, and the thing does not roll.

A sphere produces a point, and this is perfectly capable of rolling in a great variety of directions.

A cone produces a line, but this line acts as a radius of a circle.

In 4d, the ground is 3d. So if the section is 2d, it rolls perpendicular to that. A 1d section gives a two-directional movement, and a point gives a 3d range. Such are the outcome when the vertical section of the prism is a circle, sphere, and glome. Note that the floor of a car cabin would be the 2d section times the forward motion. So in the case of a duocylinder, the cab becomes a cylinder-section, the foot-print is the end of such, we don't want the occupants to get giddy, so we should not rotate the non-rolling part of the cylinder.

A swirl-prism has a foot-print in the shape of one of its faces, such as a pentagon. Under rotation, the pentagon advances on the ground vertical to its hedrix (2space), but in the process, it rotates. The whole swirl-prism is rotating, which means that if drive is passed across another face of the swirlprism, it would turn the 'wheel' at a 1:1 ratio of the drive. The cabin would need to be supported on a different set of bearings, but such bearings, breaks etc, can be applied to a surface different to the one that contacts the ground.

The spheric pyramid still presents a line, but the rotation becomes the radius of a sphere.

The bicircular tegum presents a line, this can freely move around, because the two different ends can be advanced at different speeds.

A sphere-line cone (ie a cone-pyramid), produces a triangular foot, one side is stationary, while the other can rotate around it.
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wendy
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### Re: Rolling things in 3d and 4d.

Ooooh, this is unexpectedly interesting! Please, tell me more about the cones. I see how spheric pyramid works. It's a direct analogy from a cone. But, can things like a cone prism roll? Or a duocylindric pyramid? Or a cylconinder (cone*circle)?

--------------

Later thoughts:

Now that I think about it, I can see some intuition. For some of these shapes, we can start with the rolling property of a 'base shape' , i.e. a cone = footprint of a line revolving on a fixed face (a 0D point). From here, whether we extrude, taper, or bisecting rotate the base-shape cone, we can apply the same construction to the footprint. But there are exceptions, like the bisecting rotation of a cone to generate a spheric pyramid. There are two ways to rotate a cone into 4D. One way makes a spheric pyramid, the other makes a type of self-intersecting cone torus.

Cone Prism IO>I : square revolves on fixed line edge in circular path

Duocylindric Pyramid IOIO> : cone revolves ...... around fixed apex point in spheric path?

Cylconinder IO>IO : cylinder revolves around fixed circle face in circular path
in search of combinatorial objects of finite extent
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### Re: Rolling things in 3d and 4d.

A cone prism (or cylinedr || line), would have a square footprint, and roll around the line-end, rather like a flag. If you stand the cone-line prism on its base, then it has conic base, and like any slab, it doesn't roll really well.

A duocylinder pyramid has the footprint of a cylinder in 4D, and rolls around one of the ends of it.

As to the bicircular tegum, you have to think of it as a bar with two axles at right-angles. So it can rotate end over end, so its progression is a series of half-circles touching and at right-angles. That is, if it's going in the z-direction, you have semicircles in zx, touching at (0,0,2n) and in the zy plane has semicircles touching at (0,0,2n+1). There are drawings of light, where the E and H field travel perpendicular directions, each a set of circles forming an ordinary chain. The bicircular tegum can be made to do this. If both the wheels are turning, i imagine the direction of propagation would be perpendicular to the line, in the direction set by the two speeds independently. So if one is going at speed 3, and the other is at speed 5, the direction it would pass through is 0,0,0 to 3,5,0, where its footprint is in the z-axis.

The really cute one is the swirl-prism. Different faces of it all travel at the same speed around the centre, but only one face touches the ground. If you think of a dodecahedron, imagine it's standing on a pentagonal face. That's where the rubber is. Now any other combination of faces can be used for things like brakes, the mounting race (ie a circle representing the hub on a bicycle wheel), the power train (turning the wheel through a contact-wheel), etc. It sort of gets you thinking about alternatives to cylinder-wheels that do not rotate the cabin.
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wendy
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### Re: Rolling things in 3d and 4d.

Could it be an option to have a spherical cabin around which you'd have "legs" that would have dodecahedronical "wheels" at their tips? This way, the vehicle can "grip" any sort of surface or object they can use as a support, no matter what sort of dimension it belongs to, and the "legs" can move along the surface of the spherical cabin to reach the place where they're needed.
Sure, there's no way we could achieve something like that with our technology, but could be adapted to a 4d world.
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