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EDIT by EricStucky: The full text of original post below for reference, but I have talked with the OP in chat and believe that this is the mathematical core of the question.

Suppose that we have objects with the following shape:

Red Blue Blur Ball

(In the comments, Alan suggests $r = 10 + .2(cos(5\theta) + cos \frac{4\phi}{2})$ as a possible parametrization for this surface.) (Edit: Feb 2021 I have found out the formula to create the shape...I do not know if it matches the equation given)

enter image description here

If we create a face-centered cubic arrangement of these objects, the ridges will "settle into" one another.

If we now think of them as physical objects, so that they are never allowed to intersect, is it possible to rotate them?

(For example, if we have infinitely many gears in a line, they can be rotated. But if we have three arranged in a triangle, it is not possible to rotate them.) Yes. The question I seek to answer, is... In a friction-free environment, what might naturally emerge when each of the components is rotating... enter image description here And how to calculate it.

I realize it is much more complicated (the math) than I had originally thought. I am not really any longer expecting a concrete answer to this question as it stands.

Hello mathematicians of advanced skills and intuition.

I'm a visual guy, who needs a little help describing what I see.

I see the ball presented below, as a sort of "2-sided" sphere. But also a a 12 sided sphere, because I think that two overlapped tetrahedrons inside of it, could spin and rotate in a toroidal pattern, about a common fixed barycenter, to create this shaped sphere. But my visual math exceeds my written math abilities, I think.

Maybe has something to do with two paired equally overlapped tetrahedrons, of different size, by a factor of 12/13, with their different volumes, describing a common empty barycenter for all of the enclosed space, with the two paired tetrahedrons rotating in toroidal and counter toroidal directions, until their tips have painted the patterns seen with red for the larger, and blue for the smaller, and purple where they balance. Or something like that??

Could this shape infinitely co-exist in a pattern of balanced counter-rotation, in which gaps accumulate, and dissipate in a balanced fashion where there are changes in speed in any of three paired intersecting coil like balances, patterns of energy transfer, and creation of balance points in the accumulation of matter, from a mathematical perspective.

Is this shape stackable and expandable "geodesically"?

Please help me decipher what I see.

Red Blue Blur Ball

Added: It is my understanding that the source of the shape is derived from this pattern of oscillation I think called e mode and b mode polarization. The source pattern is from the oscillation in gravity as derived through detection of magnetism in the cosmic microwave background.

Oscillation in the Primordial Gravity Wave, as derived from the cosmic microwave background (gravity coming from the direction of the oldest known point in the universe)

I am trying to determine if it CAN stack tetrahedrally, and counter-rotate in unison, in a field, mathematically, in 3 pairs of offsetting orientation.

Pattern of potential axis orientation if counter-rotated in a field, if the intersection point of 12 of them is always balanced: enter image description here

Alistair Riddoch
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  • I don't think the second picture matches the first. Notice that when you rotate the second picture you have these creases that form along certain lines of latitude, but in the first picture, there are only smooth curves. Did you mean to highlight some other property in the second picture? – Eric Stucky Aug 23 '14 at 19:00
  • Where are you getting this shape from? Might help us decipher it. – Alexander Gruber Aug 23 '14 at 19:03
  • I agree it's not the best picture to add. just the best I had. – Alistair Riddoch Aug 23 '14 at 19:06
  • sure, it came from a pattern of oscillation in the cosmic microwave background radiation, I think it is the intersection of e mode and b mode polarization or something like that. the mathematical symbology goes past my level of familiarity. – Alistair Riddoch Aug 23 '14 at 19:09
  • I think I might understand the question. Are you asking whether it is possible to stack these objects as one would stack oranges/tennis balls/spheres, interlocking the ridges such that rotating any one of them causes the entire arrangement to rotate, without jamming, and any two adjacent objects rotate with opposite orientation? – Eric Stucky Aug 23 '14 at 20:06
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    OMG Yes, and it is like you are the first person to ever understand the thought. thank you for that!! – Alistair Riddoch Aug 23 '14 at 20:11
  • Tennis balls is a good example for getting your mind around two axis counter rotation by rolling them against one another along their lines. Not perfect, but a good start. – Alistair Riddoch Aug 23 '14 at 20:12
  • The finished shape I imagine is rigid, because it can be, so if at any point the universe has "rigidity", it could be at this point. Perhaps. – Alistair Riddoch Aug 23 '14 at 20:13
  • Let us continue this discussion in chat. – Eric Stucky Aug 23 '14 at 20:15
  • In a rigid particle universe, one would expect retention of momentum. I am unsure whether friction is a necessary consideration or not, since it is a composite of many things that may not apply at every level. It could be only "squeeze/squit", "gap/fill the gap" scenario. If the surfaces are uniform and perfect. – Alistair Riddoch Aug 23 '14 at 20:16
  • Marc Kamionkowski uses the first picture in one of his papers on CMB Polarization, as has been pointed out above. It would be interesting to have the article to construct the surface , Link: http://arxiv.org/find/hep-ph/1/au:+Kamionkowski_M/0/1/0/all/0/1?skip=25&query_id=98a7c1da48da0537 – Alan Aug 23 '14 at 20:25
  • My guess is we are looking at a certain sequence of spherical harmonic functions on the sphere . An example would be $ r = 10 + .2(cos(5\theta) + cos \frac{4\phi}{2})$ we are warping the sphere just a bit to get the pretty picture. – Alan Aug 23 '14 at 21:19
  • @ Alistair Riddoch No please, that was just an example of how to warp a sphere using spherical harmonic basis functions. That is not the way it is being warped in the picture. I would really have to read that article very carefully to get it right. – Alan Aug 23 '14 at 21:48
  • I am actually pretty confident of my personal visual construct, but I also do consider the mathematical mind at work a beautiful thing and appreciate any contribution or comment, no matter how small! Or large. I don't think marking a correct positive answer with EUREKA, will be uncalled for, when someone gets to that point, and goodness bless you all who consider or try. :-) – Alistair Riddoch Aug 23 '14 at 23:08
  • Thank you Alan for the Arvix link. I like several of the titles there. Is there a public access to view content? – Alistair Riddoch Aug 23 '14 at 23:09
  • Also, it may be that some of the mathematics are consistent with string theory, maybe 12D, but I can'd decipher the different versions mathematically. If one of the versions creates three balanced counter-toroids? – Alistair Riddoch Aug 23 '14 at 23:13
  • If it helps, this discussion may "kind of" relate..http://math.stackexchange.com/questions/405714/nets-of-geodesic-spheres?rq=1 – Alistair Riddoch Aug 23 '14 at 23:29
  • Would someone please tell me the polite process to invite a specific member to review a question, if his insights and logic in another question suggest some level of prior thought on the subject? – Alistair Riddoch Aug 24 '14 at 02:37
  • @ :) Willie Wong Hi Willie, I noticed your answer on a similar but not entirely related question and wondered if you would review this one for comment?? – Alistair Riddoch Aug 24 '14 at 20:42
  • American Mensa "Liked" the link to this question on their Facebook page, which I think means all 60,000 of their followers may see the link in their facebook feed. Fingers crossed it gets the attention necessary to get answered. :-) It will explain so much. – Alistair Riddoch Aug 25 '14 at 16:24
  • @AlistairRiddoch: (1) You cannot summon other members like that. Please read this document on how comment-replies work. You are lucky that I am telepathic (not really). (2) I have no comment to your question, mostly because I don't see how EricStucky got his interpretation of your question and also that I don't understand what you are trying to ask. Sorry. – Willie Wong Aug 26 '14 at 14:58
  • @WillieWong Thank you for your aspiring Atriedes attribute. (Dune). Specifically, my thought is, the red-blue sphere above, is capable of being a single cog in a 3 dimensional field of cogs, that can counter-rotate. I saw your calculations regarding a field of geodesic spheres, and think, therefore you have abilities in this area specifically. – Alistair Riddoch Aug 26 '14 at 21:05
  • @WillieWong Further, if the above mentioned field, CAN exist. Then it is extremely likely that it DOES exist, and that this is the lowest common denominator of all energy and matter in the universe. The Final Theory. If it CAN exist, it CAN mechanically explain the four fundamental forces, and CAN provide us magic-less, time-rate-static universe, which is much easier to swallow, than the current belief set. – Alistair Riddoch Aug 26 '14 at 21:20
  • @anyone I would like to offer, that if confirmed this theory IS Nobel Prize material, as would be mathematical proof it CAN exist. If there are not individual prizes for such, I glady commit half of any prize I may be granted for asking the question, to the provider of the mathematical proof. So one doesn't wonder why not 100%, it is because I officially commit my share to ALS, in advance, because I have nothing else to give them, and have been challenged to contribute. I consider it extremely likely the field can exist. Sincerely. Cheers. Alistair – Alistair Riddoch Aug 26 '14 at 21:24
  • However I can only claim that my offer is valid in my intent to carry it out. If my theory is wrong, or non-confirmable, it will be worthless effort, for the joy of mathematical and physical exploration. – Alistair Riddoch Aug 26 '14 at 21:25
  • and PLEASE, if you can "SEE" the potential validity, and/or the potential scale of recalculations the world will need if this theory is viable, speak up, and call your brothers to arms to tackle this. I think it may be the most important question ever asked. (a lucky guess on my part). – Alistair Riddoch Aug 26 '14 at 21:27
  • The edits over the last couple of days have served to make this question much less suitable for math.SE, as they've introduced wholly inappropriate amounts of wild speculation and theory into what is supposed to be a largely objective site Q&A site. I strongly encourage you to revert them. – Steven Stadnicki Aug 28 '14 at 00:04
  • I understand and have tampered my speculations, begging forgiveness for my over-enthusiasm. My purpose is only good. – Alistair Riddoch Aug 28 '14 at 00:19
  • You know how you can take a triangular notch out of one side of a square, translate the notch to stick out from the other side of the square, and what's left is able to be tiled per a square tiling? It seems like if you can get the rippled spheres to pack in the first place, you might be able to answer whether you can rotate the balls once packed. That tiling trick may give you a method for finding such a static packing. Then you could try finding the intersection points of an arbitrary pair of spheres taken from the packing, and monitor that description over rotation. – Loki Clock Jan 09 '15 at 06:23
  • Thank you Lori. I hadn't seen your post prior, I understand what you say. I have a thought about that too. I wonder if you or anyone will consider mathematically....were this sphere to be able to in a stack of 12 surrounding it, ...I think I better add it to the original question to use pictures, etc..... – Alistair Riddoch Jul 28 '15 at 02:50

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I don't know how to answer the question I have asked, but I think I can answer it by analogy, in the hopes that someone will see my pathetic attempt to explain a mathematical concept on a board where math is a specialty, take pity on me, and try to provide an answer that is more in keeping with the nature of this boards math section.

If one was to imagine two boys with their backs to one another, using their hands and feet to press outwards on the silo, and trying to crabwalk up it, but "mis-programmed their feet" to work in slightly opposite "phases of movement" where two limbs would purposely "short step" by the smallest touch, while their other two limbs would "longstep", then they would oscillate and the alternate limb do the "short step" or "long step". Then if in one boy, an cross mixed pair was also given "mis-programming" to "step up" adjust, and across the diagonal, the other pair was given a "step down".

Then because in a 3D array, there would be the relative motion of the walls, which are other balls in counter-rotation. The combined "up and down" and "left and right" and the cross mixed reversal of those actions, and the appropriate counter action from the "limbs" of virtual people inside the spheres, will create a 3 dimensional array of "square dancers". The interference pattern of their hands and feet, will equal the interaction pattern, of where "pressure" will be felt, and "released, while the spheres counter-rotate.

Alistair Riddoch
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