# Can we rotate a 3D lattice of deformed spheres?

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.)

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.)

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"?

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 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
• 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