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:
(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)
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...
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.
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.
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:
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.