Corkscrews and Duocylinders

[anderscolingustafson]

I was thinking about corkscrews and in 3d a corkscrew is a line drawn along a cylinder. The x and y coordinates of a corkscrew repeat an infinite number of times but a corkscrew does not curve back in on itself because the z coordinate never repeats. Because the Doucylinder is basically a cylinder that curves back in on itself it is possible to have a kind of corkscrew along its surface that does curve back in on itself. Not only do the x and y coordinates repeat for a corkscrew drawn on a duocylinder but the z and w coordinates also repeat. The equation for a corkscrew drawn on a duocylinder is

x=sinat
y=cosat
z=sinbt
w=cosbt

[Polyhedron Dude]

I'm quite familiar with this. If we used the various corkscrews to carve shapes out of glomes, it generates a cool group of shapes that I call "coiloids". Coiloids are transitive on the contact regions (part of facets that touch a "table's" surface) - so I include them among the fair dice. You only need two numbers to designate a coiloid. For all pairs a and b (a,b positive integers) there is an a,b - coiloid (except for 1,1 which is degenerate). The 2,2 - coiloid [which can be written as coiloid (2,2)] is the duocylinder and coiloid (a,b) = coiloid (b,a). If a and b are relatively prime, then the coiloid is formed by the corkscrew that winds a times and circles b times. Those that are not relatively prime are carved by a number of evenly spaced corkscrews - for example, coiloid (10,4) is carved by two corkscrew (5,2)'s and coiloid (7,7) by 7 corkscrew (1,1) which is 7 circles swirling about each other. Coiloid (n,n) = dyster (n).

[anderscolingustafson]

I was just wondering if this would at all effect orbits in 4d. I know the orbits we're use to tend to be in a plane because the up and down motion in a solar system tends to cancel out. So while the orbits we're use to have the physics of 3d they only require a planet to have two degrees of freedom to move around in because they orbit along a plane. In 3d a corkscrew is a stable way for something to spin and it requires three dimensions but it's impossible for a planet to have a corkscrew orbit because stars are spheres and corkscrews keep moving up or down without curving back in on themselves so it's impossible for a planet in 3d to have an orbit that requires three degrees of freedom. The fact that in 4d there's something like a corkscrew that curves back in on itself means that a planet wouldn't need to orbit in a plane like a 3d planet but could instead orbit along a doucylinder. I wonder if this would effect how stable the orbits would be as it seems the instability of 4d orbits seems to be based on the assumption that planets in higher dimensions would still orbit along a plane rather than being able to have a none planar orbit.

[ICN5D]

Well, much like the way a glome wants to rotate, I imagine a 4D planet wants to orbit along both ortho 2-planes, since they are most stable. But, in this case, there would likely be two ways to be ecliptic, and two ways to be elongated, in these planes. So, we would have two elongations and two ecliptics to describe the 4D orbit, which starts to sound a lot like the shape of a duoring. This is the 'duocylinder orbit' you are referring to, in this case it's the margin between curved cells that is the orbital path. This also means a 4D planet orbits along a 2D plane, having the dimensions of a duoring. Interesting.....

[Klitzing]

Sad to disappoint you here. Your assumption still requires a 1/r² potential. Whereas within 4D it would be likely a 1/r³ one instead. And that one does not allow for elliptical (or any other) oscillations any more...

--- rk

[ICN5D]

Oh no! Oh well. Hmm, then I suppose that if we cannot have elliptic or ecliptic circles, then perfect circles are the only way. If this is the case, then maybe the circles can be stable if one were to orbit along both simultaneously, in equilibrium. Trying to travel along only one, or mostly one, will be unstable, leading to the double rotation. Rather than oval vs circular orbits, in 4D we would have single vs double circular orbits. Does that work out in any way possible?

[quickfur]

This doesn't work. First, an orbiting object can only follow a linear path. Having an orbit in the shape of a 2D manifold means it is shattering into a million pieces and following different paths over time.

Second, when the rates of rotation in the two orthogonal planes are equal, it's an isoclinic rotation with an infinite number of stationary planes. This means that every combination of orbital paths is actually identical to the circular orbit -- the only difference is the orientation of the orbital plane relative to the star's reference frame. So this doesn't actually produce anything new.

[ICN5D]

Hmm, well, if it's unstable to rotate around a star, would it be possible to rotate in place, around a centrally located star? Like a 'Dyson Tiger' world that orbits ( or more precisely rotates ) along a 2D duoring. During star formation, the glome-shaped collapsing cloud formed a star, which began to rotate, generating Hopf fibration gravity waves, causing a tiger-shaped accretion zone to form, and eventually settle into landtubes and asteroid zones, along a tiger-shaped 'orbital zone' ring. Just pure speculation .....

[quickfur]

Hmm. Gravity waves in the shape of a Hopf fibration? That's a new one. What consequences would it have?

[ICN5D]

I'm imagining it forms an accretion zone of material, that settles into a tiger-shaped world of 4D. This whole thing would have the double rotation, and cover a 2D plane as the land tubes slide horizontally along the duoring orbital path.

[quickfur]

I'm not so sure about the tiger shaped part. Assuming things eventually settle into isoclinic rotation as I've mentioned, then there would be no distinction of a tiger in one orientation vs. in any other orientation within the same isoclinic rotation. So assuming that something funny happens with gravity to make gravity wells at non-zero radius from the star, you'd end up with a Dyson-sphere like accumulation of matter, undergoing isoclinic rotation. It would be a literal Hopf fibration. I suppose you could have inhabitants living on the inside of this Dyson sphere.

An alternative scenario is that matter accumulates along distinct fibres, so you'd have numerous interlocking ring-worlds that rotate around each other in intertwining ways, but never colliding.

[ICN5D]

How would echoing gravity waves in 4D fit into this? Would this property supply a non-zero Lagrange zone, long enough to accumulate and solidify into landforms?

[quickfur]

Honestly, I have no idea. The kind of math involved is way above my head.

[wendy]

The calculations in three dimensions, tell us that if there are circles for oceans to circle in, like the Southern Ocean, you tend to get great masses of ice. So a planet with literal hopf fibulation land-masses are likely to be fairly cold places.

While the maths-theory of our world, extended to four dimensions, tells us that many of the things are not possible, (as Richard correctly notes), it does not mean that we cannot pretend that they work as we want them, until we find something that does.

Twelfty works. The current mathematical model of bases tells us we ought learn 14400 tables, but a different model cuts this down to 144. So we *could* suppose an orderly plan of planets, if we pretend some different model of physics works. That's how it goes.

The tiger *is* a literal hop-fibulation, of a torus.