The Strong Force

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.

The Strong Force

Postby anderscolingustafson » Fri Apr 04, 2014 4:16 am

In our universe there are 8 different kinds of gluons. 8 is the same as 23. So in our universe the number of gluons is the same as two raised to the power of the number of dimensions we live in. There are also 3 different color charges for the strong force.

In 4d would there be 16 gluons considering that 24=16 and would there be 4 color charges for the strong force?
Last edited by anderscolingustafson on Fri Apr 04, 2014 3:51 pm, edited 1 time in total.
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Re: The Strong Force

Postby quickfur » Fri Apr 04, 2014 2:52 pm

Be careful with this analogy. The 3 color charges of quarks are not tied to the 3 dimensions of space; rather, they arise from the special unitary group SU(3), which are a group of certain matrices with complex entries. SU(2) corresponds with the electroweak theory (the unification of electromagnetism with the weak force) and SU(3) corresponds with quantum chromodynamics (i.e., the strong force). The fact that it's SU(3) and space happens to be 3D, is a coincidence; there is no direct connection between the number 3's in either case.

For more information, see the wikipedia article on special unitary group.
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Re: The Strong Force

Postby wendy » Sat Apr 05, 2014 7:38 am

Did you spot the catalan matrix there is what we call the Dynkin matrix. It's hardly supprising, since Lie groups derive the catalan matrix from the dynkin graph in the same way we do, except there are some differences.

1. They don't have non-crystalgraphic (ie 5 branch)

2. The 4 and 6 branches are written as two and three bars between the same nodes, and these have directions.

3. The catalan eliminates square roots, by putting a_ij * a_ji in i,j and j,i = 1.

If the SU groups represent the rotation of a sphere, then SU4 would be six-dimensional, and SU5 would be 10 dimensional.
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Re: The Strong Force

Postby anderscolingustafson » Sun Apr 06, 2014 12:32 am

How would the strong force work in 4d?
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Re: The Strong Force

Postby quickfur » Sun Apr 06, 2014 12:48 am

We haven't even figured out 4D electromagnetism yet, and you're asking about the strong force? :lol: At this point, I don't think we can even say much about it beyond pure, wild, unfounded speculation. :P (Assuming there's even such a thing as the strong force in 4D in the first place... and that's a pretty big assumption)
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Re: The Strong Force

Postby ICN5D » Sun Apr 06, 2014 1:45 am

Ahhhh, yesss. The Realm of Pure, Wild, Unfounded Speculation .... my favorite :) Considering the intellectual journey we take during a rant of RoPWUS, there is nonetheless little gained in doing so. Most times, at least. Sometimes, though, strange new connections are made, and I think I might contribute here:





Image


" The structure of the very fabric of the universe has a specific geometry: an infinite array of perfectly interlocking spheres and tetrahedrons. The seed of this geometry from any given point is what Buckminster Fuller called the Vector equilibrium, otherwise known as a cube octahedron, pictured here in black with spheres around each of the 8 tetrahedrons that make up this perfectly balanced geometry. Eight tetrahedrons pointing inward to a single point (a singularity) creating 12 radial vectors, all of equal length, forming edge vectors all also of that same length.

A cube octahedron is 8 tetrahedrons pointing inward.
A star tetrahedron is 8 tetrahedrons pointing outward.

You need both the male (tetrahedrons) and the female (spheres) to be able to understand how the universe is able to create and self-organize itself on all scales. Thanks to the ancient cultures of the planet, this information has been left for us in the form of an ancient symbol found all over the world: The Flower of Life "

- Nassim Haramein




So, perhaps a CRF, like the 24-cell, has some manifested symmetry capable of describing a working 4D standard model....
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Re: The Strong Force

Postby quickfur » Sun Apr 06, 2014 2:46 am

If we're gonna go all-out on wild unfounded speculation, then we might as well shed all pretense of following any resemblance of 3D physics and go for radically different physical systems... like one founded on discrete symmetries rather than spherical symmetries? :P

One of the ideas I posted about a while back is an alternate universe where the fundamental law isn't any strong/weak/electromagnetic force, but the Law of Symmetry. Basically, "energy" (in a loose, analogical sense -- it has nothing to do with energy as we understand it on earth) has an inherent symmetry to it, a kind of "vibrational mode" that takes the form of some polytope. The higher the degree of symmetry of the polytope, the more stable the energy. So an icosahedral quantum of energy is more stable than a tetrahedral quantum of energy, but a tetrahedral quantum of energy is more stable than a polygonal quantum of energy, and so on. The Law of Symmetry dictates that energy (including matter) prefers to be in a "vibrational mode" with a high degree of symmetry. If something deviates from that symmetry, the Law of Symmetry causes an opposing force to attempt to pull it back into that symmetry. So a planet prefers to orbit in a perfect circle, because when it deviates from a circle (highest symmetry) the Law of Symmetry will pull it back. Thus, stable orbits are a given in a 4D version of this alternate universe. :D

So "atoms" in this alternate universe is not made of protons or electrons, but of quanta of energy in some kind of symmetry. In a 4D universe, a 600-cell symmetry will be very stable, and atoms in that state will be highly unreactive and inert, whereas something in the state of a 5-cell will be less stable, and more likely to react. Something in the shape of a 4D Catalan will be somewhat stable, because of the overall global symmetry (e.g., a Catalan derived from the 600-cell symmetry would inherit the global symmetry of the 600-cell) and the facet transitivity. But the non-transitivity of lower-dimensional elements within each cell reduces this stability somewhat. Similarly, a CRF polychoron of low global symmetry will be inherently unstable, but the regularity of its polygons somewhat counteracts this instability so they won't instantly disintegrate or rearrange into more stable pieces. So basically, rather than being driven by a scalar "energy", physical processes in this alternate universe are driven by the interplay of symmetries.

In our chemistry, for example, chemical reactions are driven by atoms wanting to attain a stable electronic configuration, but counteracting that, are factors like extra energy introduced by charge separation, or lattice energy, or existing bond energies that must be broken, etc.. In this alternate universe, the driving factors are not electronic configuration, but the give-and-take of compromising one form of symmetry in order to attain another. To use a 3D example, a rhombic dodecahedron has the stability of cubic symmetry, but its rhombus faces destabilize it because they are non-regular. So a rhombic dodecahedral atom would like to rearrange itself to something with regular faces, but that may require breaking cubic symmetry, so there's an energy barrier to that reaction. Only if you supply enough energy to overcome the loss of stability in the breaking of cubic symmetry, will the atom rearrange itself into something more stable. Similarly, a snub disphenoid atom has low stability because of its low symmetry, but it does have regular faces. Rearranging itself into something more regular will require (temporarily) breaking the regularity of its faces, so that serves as a barrier (admittedly a not very high barrier) against spontaneous decomposition. Something like an icosahedron, in contrast, has basically no tendency to rearrange, because it is stable both globally (icosahedral symmetry) and locally (faces are regular). But if placed in a highly irregular environment (e.g., many atoms around it have irregular shapes), then the global tendency of the system would prefer to break the icosahedral symmetry if it can reduce the irregularity of the other components of the system. Of course, this would require a lot of initial energy to overcome the stability of the icosahedron, but once broken, it will react with the other irregular atoms to form (relatively) more stable polyhedra.
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Re: The Strong Force

Postby anderscolingustafson » Sun Apr 06, 2014 4:03 am

I was thinking in 3d because something can only have one direction of rotation it is impossible for two planets to be the same distance from their star without having their orbits intersect. In 4d it's possible to have a double rotation with two independent directions of rotation and an infinite number of other directions of rotation between them. In 4d if two planets were the same distance from their star without having their orbits intersect like they do in 3d. If there were multiple planets orbiting at the same distance from their star in none intersecting orbits then each planet would be influenced gravitationally by the other planets orbiting at the same distance. If there was a group of planets orbiting at the same distance from their star in which none of the orbits intersected each other would they mutually stabilize each others orbits with their gravity? What effect would having multiple planets orbiting at the same distance from their star in none intersecting orbits have on the orbits?
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Re: The Strong Force

Postby quickfur » Sun Apr 06, 2014 4:28 am

In 3D, it's possible to have two planets in the same orbit, 120° away from each other, and they actually mutually stabilize. This is actually witnessed, for example, in some of the shepherding satellites in Saturn's rings.

The 4D situation you're describing amounts to the Hopf fibration of a 3-sphere with the orbital radius. While I haven't looked into the general Hopf fibration from this angle yet, I did look into the simple case of two planets in two orthogonal orbital rings (i.e., orbiting along the rings of a duocylinder). This particular case is actually unstable, because the rings are mutually orthogonal, so each planet would pull the other in a sinusoidal way outside the plane of their orbit. So rather than stabilize each other, they actually destabilize each other by pulling each other out of their respective orbital planes. Furthermore, since these orbital planes are completely orthogonal, the net direction of the force felt over many orbits is an inward pull toward the central star, so the additional planet increases the effective mass of the star. Unfortunately, that doesn't really change the curvature of the gravity well, so the inherent instability of 4D orbits still applies (except much more pronounced now that an extra body is there adding disruptions to the orbital plane).

Now an interesting situation is if there are 4 planets in two orthogonal planes of a duocylinder, at 180° from each other. Then the gravity of each other cancel out in the orthogonal direction of the other pair, so the other pair no longer feels an out-of-plane pull. But it does still add to the effective mass of the star without really changing the curvature of the gravity well, so while it stops being destabilizing, it doesn't seem to add any more stability either.

I haven't studied the case with more than 2 rings in the Hopf fibration; but it seems unlikely for such an orbital structure to be self-stabilizing. But somebody would have to go solve orbital motion equations for the Hopf fibration in order to be sure... so good luck solving the many-body problem in 4D. :lol: (It cannot even be solved analytically in 3D for more than n=2, so don't expect easy answers in 4D!)
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Re: The Strong Force

Postby ICN5D » Sun Apr 06, 2014 11:02 pm

This still makes me think about toratope shaped worlds. If it requires four to negate instability, are you assuming they are glomes in those places? They could be any of the five available toratopes in 4D. Perhaps instead of four separate structures at 180 degrees of separation, it is one structure, that has a crosscut of four locations. A tiger has this property, and is a ring-world shaped toratope that shares the Hopf fibration characteristic. If separate structures are unstable, then maybe one single structure is. This type having a close interplay with the fibration.
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Re: The Strong Force

Postby quickfur » Sun Apr 06, 2014 11:36 pm

This probably falls outside of the current definition of toratope, but I'd really love to see a toratope that exhibits toroidal holes in icosahedral Hopf fibration symmetry. :P Like the 12 swirling rings of prisms and antiprisms in spidrox:

Image

Or the 20 swirling rings of square pyramids in the same (with the additional bonus that they have a secondary inner 3-fold twist aka the Boerdijk-Coxeter Helix):

Image

Now imagine if these aren't prisms or antiprisms or square pyramids, but smooth toroidal surfaces that wrap around the 3-sphere, interlocking each other. That would be a toratope to behold!
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Re: The Strong Force

Postby ICN5D » Mon Apr 07, 2014 12:05 am

Nice!! Now you're talking!! I've wondered this stuff myself. In fact, I previously explored the concept of non-circular tigroids, such as the cyltrianglintigroid. It would be the inflated margin of a cyltrianglinder, and have strange midsections of both a hollow circle and hollow triangle. Keiji says it doesn't exist, but if it does, then this shape is a potential elementary beginning to that sequence. Imagine an icosahedral tigroid, with 20 holes? Not as in a decatiger ((II)(II)(II)(II)(II)(II)(II)(II)(II)(II)), which would have this property, but a geometric figure manifested in this tigroid symmetry. These shapes would be the inflated ridges of common CRF's that you all have been discovering. More pure, wild, unfounded speculation....
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Re: The Strong Force

Postby quickfur » Mon Apr 07, 2014 12:57 am

It's actually not as unfounded as you think. Jonathan Bowers has studied what he calls the "regular polytwisters" before. These are basically smoothed-out versions of the above, where you have this polygonal torus (i.e. torus with a polygonal cross-section) that is twisted around its axis and then wrapped around and linked into a toroidal strip in 4D. Take some number of these and glue them up in 4D and you get a closed shape, a polytwister. :) There are a number of regular polytwisters -- where each twister is transitive with all the others. All of these are expressions of the underlying Hopf fibration, as applied to a regular polyhedron. :)

My idea, though, was to take these polytwisters and make toratope-like shapes out of them. So instead of a bunch of interlocked twisters, you take the 2-manifolds where they touch each other and inflate it, which produces a shape with holes in the shape of these twisters. The non-hole parts constitute the toratope. The tiger would just be the simplest of them, corresponding with the Hopf fibration of the digon.
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Re: The Strong Force

Postby ICN5D » Mon Apr 07, 2014 1:33 am

Well, that's interesting: hopf fibration of a digon. So, then what would be the HF of the triangle? Or square? But, let's not stop there! Is there any research on HF's of toratopes themselves?
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Re: The Strong Force

Postby quickfur » Mon Apr 07, 2014 2:03 am

ICN5D wrote:Well, that's interesting: hopf fibration of a digon. So, then what would be the HF of the triangle? Or square? But, let's not stop there! Is there any research on HF's of toratopes themselves?

The Hopf fibration applies only to the 2-sphere, so it cannot be applied to toroidal shapes, and it only applies to the 3D sphere (or its tilings thereof). (However, it does produce toroidal shapes, like the duocylinder! :)).

The digon I mentioned is actually the digonal tiling of the 2-sphere (i.e., two hemispheres). The Hopf mapping transforms the hemispheres into the two toroidal surfaces of the duocylinder.

You can, of course, also have a trigonal tiling of the 2-sphere (i.e., 3 lunes stretching from pole to pole). I'm not sure what kind of shape it will produce. It will certainly be interesting to find out, though! :)

The applications to various regular polyhedra are known, and correspond to certain cell groupings among the 4D polytopes. For example, in my duoprism page (linked in previous post) you see that the tesseract can be decomposed into two rings of 4 cubes each, which actually corresponds with the two toroids of the duocylinder. Similarly, the duoprisms exhibit the same kind of decomposition into two orthogonal rings. The 24-cell has two possible decompositions, corresponding with the Hopf fibration applied to the tetrahedral and cubical tilings of the 2-sphere. The 120-cell can be decomposed into 12 rings of 10 dodecahedra each, and they correspond with the Hopf fibration applied to the dodecahedral tiling of the 2-sphere. Similarly, the 600-cell can be decomposed into 20 rings of 30 tetrahedra each, corresponding with the Hopf fibration applied to the icosahedral tiling of the 2-sphere.

Oddly enough, none of the regular 4D polytopes correspond with the Hopf fibration applied to the octahedral tiling of the 2-sphere: that honor goes to the unusual biicositetradiminished 600-cell, aka BXD, which consists of 8 rings of 6 tridiminished icosahedra each.

Spidrox, whose projections I gave earlier, has two sets of rings, as I mentioned, 12 rings of alternating prisms and antiprisms corresponding with the dodecahedral tiling of the 2-sphere, and 20 rings of square pyramids corresponding with the icosahedral tiling. You could say that, considered together, these rings correspond with the icosidodecahedral tiling of the 2-sphere.

The Hopf fibration mapping preserves the relationships between the elements of the 2-sphere tiling in the resulting structure, so generally it's most interesting when applied to objects of high symmetry, since the toroidal rings will likewise be symmetrical with each other. But it's also possible to apply the mapping to an irregular tiling of the 2-sphere, say a snub disphenoid, and you'd get an irregular bundle of interlocking toroids that exhibit the symmetries of the snub disphenoid. :lol:
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Re: The Strong Force

Postby ICN5D » Mon Apr 07, 2014 4:17 am

That tiling is some pretty wild stuff. I've previously learned from your posts about the two ring characteristic of 4D duoprisms. It's an interesting new ability above 3D, the two orthogonal planes. It's very unexpected and bizarre to see how a clump of square pyramids can have a spiraling twist to them.
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Re: The Strong Force

Postby quickfur » Mon Apr 07, 2014 2:33 pm

ICN5D wrote:[...] It's very unexpected and bizarre to see how a clump of square pyramids can have a spiraling twist to them.

It's known as the Boerdijk-Coxeter helix.
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Re: The Strong Force

Postby ICN5D » Tue Apr 08, 2014 4:29 am

quickfur wrote:You can, of course, also have a trigonal tiling of the 2-sphere (i.e., 3 lunes stretching from pole to pole). I'm not sure what kind of shape it will produce. It will certainly be interesting to find out, though! :)


Well, now that you mentioned it, what in the heck would this thing be??? Is it some undiscovered shape? Perhaps the phantom cyltrianglintigroid?!? We must investigate...
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Re: The Strong Force

Postby quickfur » Tue Apr 08, 2014 5:04 am

ICN5D wrote:
quickfur wrote:You can, of course, also have a trigonal tiling of the 2-sphere (i.e., 3 lunes stretching from pole to pole). I'm not sure what kind of shape it will produce. It will certainly be interesting to find out, though! :)


Well, now that you mentioned it, what in the heck would this thing be??? Is it some undiscovered shape? Perhaps the phantom cyltrianglintigroid?!? We must investigate...

Jonathan Bowers aka Polyhedron Dude would probably be the best person to ask. ;) He may even have studied this case before. Maybe.
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Re: The Strong Force

Postby Polyhedron Dude » Tue Apr 08, 2014 6:15 am

Those are the regular dyadic polytwisters - or "dysters" for short - that one in particular is the trigonal dyster - it has three twister sides which have lune shaped cross sections.
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Re: The Strong Force

Postby quickfur » Tue Apr 08, 2014 3:08 pm

Ah, so that's what dysters are. :P So I'm guessing they exist for every n-gon?

What about bipyramids, which are face-transitive? Or antipyramids, which are also face-transitive? I suppose they would produce polytwisters with transitive (non-regular?) twisters?
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Re: The Strong Force

Postby Polyhedron Dude » Tue Apr 08, 2014 9:33 pm

Yes there are dysters for every polygon - also for the star polygons. The bipyramid and antibipyramid (which I like to call gems and crystals) also form face-transitive polytwisters, therefore they will make fair dice. There are also Catalan based polytwisters.
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Re: The Strong Force

Postby quickfur » Tue Apr 08, 2014 9:55 pm

Whoa. Polytwisters can serve as die?? Heh, didn't think of that one. :P It'd be a weird kind of die, though, 'cos no matter which surface it lands on it can still roll, but you can pick out which of its surfaces it's rolling on, and map that to a number. :D

On second thoughts, this just occurred to me: polytwisters would be ideal gear shapes!!! Take a polytwister with n twisters, and cut grooves with different spacings on different twisters. Now cut another polytwister (or maybe just plain ole cubinder) with grooves that can fit into the polytwister's grooves. Now the two can fit together like gears. Depending on which twister the cubinder is fitted into, it will act as a gear of a different grade. The neat thing about this, is that since the polytwister is isometric under rotation in the plane of its twisters, the adjacent twisters will always be aligned with the current one, so gear-switching requires only the rotation of the polytwister so that the cubider's grooves switch to an adjacent twister. The neatest thing about this, is that you can reserve one of the twisters to be where the polytwister gear connects to the parts that hold it in place -- say you cut a deep groove in it, and lock it into a ring-shaped socket. Then this ring-shaped holder will serve both as the socket to keep the polytwister gear in place, and to switch gears, since rotating it out-of-plane (e.g., by turning the socket via a gearbox controlled by the driver) will cause the cubinder to realign to a new twister, i.e., switch to a different gear.

An octahedral polytwister would give you 7 gears + 1 twister reserved for the gear switch / socket. If you're going for a fancier gearbox, an icosahedral polytwister would give you 19 gears + 1 socket. :P

A polytwister would also serve as a nice way of multiplexing gears: 1 of the twisters can be connected to the engine, say, and (n-2) other twisters to multiple shafts, suitably grooved, and the last twister reserved for the socket. You can then groove each twister with different spacings, to give each shaft a different speed. :)
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Re: The Strong Force

Postby Polyhedron Dude » Wed Apr 09, 2014 5:35 am

Polytwisters not only can select a labeled number, but could also choose a color on the color wheel - imagine painting each twister to make it look like a color wheel and labeling numbers on each twister. When you roll the icosatwister, you might land on 5 and then it will roll on the 5 twister until it stops at bluish green. A sphere could be used to randomly select a spot on the Earth's surface. A spindle (dual of cylinder) can be used to select a real number between -1 and 1 - one side is negative, the other positive, and the equatorial curve can be marked to label numbers from .000 to .999. My definition of a fair die would be a convex shape which is transitive on the "contact regions" which are the part of the facet in contact with the table. An egg shape wouldn't pass since some contact regions (which are points in this case) are closer to the center of gravity than the others. There are a smorgasbord of interesting curved dice in 4-D including the spiral shaped coiloids, the roly duoc, polytwisters, duocylinder, duospindle, and others. In 8-D, the very bizarre polyswirlers will join the party, they can roll like glomes (which take on Hoff fibration symmetry) and their symmetries mimics the polyteron symmetries, i.e. there is a penteract swirler.

Polytwister gears - that's a cool idea - I would love to see a 4-D or 5-D machine with them in action.
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Re: The Strong Force

Postby ICN5D » Wed Apr 09, 2014 7:47 pm

Sounds like a good addition to the 4D driving thought experiment. Now, you have a good working model of a 4D manual transmission. Another thought would be an automatic that uses sun and planetary gears. Perhaps some sort of polytwister toratope could be the planetary ring gear? Hmmm.....
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