4D ROTATION AND SPHERE PACKING

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In 2d space, i.e., in plane, around one sphere we can arrange six spheres of exactly the same radius.
In 3d space, around one sphere we can arrange 12 spheres of the same radius.
In 4d space, around one four-dimensional sphere we can arrange 24 spheres of the same radius.

On the bottom gif-animated picture you can see the example of the densest packing in the 4d space. For our convenience the spheres are shown with diminished radii, in order, we could see the behind spheres. The fourth dimension is associated with color. We are situated at the point "green" on the color coordinate.

The left group of spheres is rotating in the horizontal plane xy.
The middle group of spheres is rotating in the vertical-color plane zs.
The right group of spheres is rotating in both planes: xy and zs.

Four-dimensional rotation has an interesting property; - the right thing is constantly transforms into the left thing, and visa verse. Figure to speech, in order to make the right shoe from the left one, we must turn it in the plane zs on p radians. It looks as fantasy, and in reality we can not make such transformations. But the physicists can make such procedures with elementary particles.

In the bottom text you can see an example of coordinates of the centers of spheres under the densest packing. The first six points give the densest packing in 2d space; the first 12 points give the densest packing in 3d space; and at last, the first 24 points give the densest packing in 4d space.
Designation: the stroke means square root, for example: 3'=sqtr(3).
Let's designate the central sphere by the digit "0". Then its coordinates are (0, 0, 0, 0). The spheres, contacting with the central sphere we'll designate by digits: 1, 2, 3,.. 24. Let's name these 24 spheres by "shell spheres". Coordinates of the spheres are written in the round brackets. The numbers of shell spheres contacting with the given shell sphere is written in the rectangular brackets.

1 (1, 0, 0, 0) [2,6,7,10,13,16,19,22]
2 (1/2, -3'/2, 0, 0) [1,3,7,12,15,16,21,22]
3 (-1/2, -3'/2, 0, 0) [2,4,8,12,15,17,21,23]
4 (-1, 0, 0, 0) [3,5,8,11,14,17,20,23]
5 (-1/2, 3'/2, 0, 0) [4,6,9,11,14,18,20,24]
6 (1/2, 3'/2, 0, 0) [1,5,9,10,13,18,19,24]
7 (1/2, -1/(3'2), 2'/3', 0) [1,2,8,9,13,15,19,21]
8 (-1/2, -1/(3'2), 2'/3', 0) [3,4,7,9,14,15,20,21]
9 (0, 1/3', 2'/3', 0) [5,6,7,8,13,14,19,20]
10 (1/2, 1/(3'2), -2'/3', 0) [1,6,11,12,16,18,22,24]
11 (-1/2, 1/(3'2), -2'/3', 0) [4,5,10,12,17,18,23,24]
12 (0, -1/3', -2'/3', 0) [2,3,10,11,16,17,22,23]
13 (1/2, 1/(3'2), 1/6', 1/2') [1,6,7,9,14,15,16,18]
14 (-1/2, 1/(3'2), 1/6', 1/2') [4,5,8,9,13,15,17,18]
15 (0, -1/3', 1/6', 1/2') [2,3,7,8,13,14,16,17]
16 (1/2, -1/(3'2), -1/6', 1/2') [1,2,10,12,13,15,18]
17 (-1/2, -1/(3'2), -1/6', 1/2') [3,4,11,12,14,15,16,18]
18 (0, 1/3', -1/6', 1/2') [5,6,10,11,13,14,16,17]
19 (1/2, 1/(3'2), 1/6', -1/2') [1,6,7,9,20,21,22,24]
20 (-1/2, 1/(3'2), 1/6', -1/2') [4,5,8,9,19,21,23,24]
21 (0, -1/3', 1/6', -1/2') [2,3,7,8,19,20,22,23]
22 (1/2, -1/(3'2), -1/6', -1/2') [1,2,10,12,19,21,23,24]
23 (-1/2, -1/(3'2), -1/6', -1/2') [3,4,11,12,20,21,22,24]
24 (0, 1/3', -1/6', -1/2') [5,6,10,11,19,20,22,23]

fi = pi / 3
M = Sqr(2 / 3)
For t = 0 To 20
psi = 0 + t * pi / 30 'rotation in the plane zc
hi = 0 + t * pi / 30 'rotation in the plane xy
a(0) = 0 'coordinates of the central sphere
b(0) = 0
c(0) = 0
d(0) = 0
For i = 1 To 6 'coordinates of spheres 1, 2,.. 6.
a(i) = Cos(-fi * (i - 1) + hi)
b(i) = Sin(-fi * (i - 1) + hi)
c(i) = 0
d(i) = 0
Next
For i = 7 To 9
a(i) = Cos(-fi / 2 - 2 * fi * (i - 7) + hi) / Sqr(3)
b(i) = Sin(-fi / 2 - 2 * fi * (i - 7) + hi) / Sqr(3)
c(i) = M * Cos(psi)
d(i) = M * Sin(psi)
Next
For i = 10 To 12
a(i) = Cos(fi / 2 + 2 * fi * (i - 10) + hi) / Sqr(3)
b(i) = Sin(fi / 2 + 2 * fi * (i - 10) + hi) / Sqr(3)
c(i) = M * Cos(psi + pi)
d(i) = M * Sin(psi + pi)
Next
For i = 13 To 15
a(i) = -Cos(-fi / 2 - 2 * fi * (i - 13) + hi) / Sqr(3)
b(i) = -Sin(-fi / 2 - 2 * fi * (i - 13) + hi) / Sqr(3)
c(i) = M * Cos(psi + pi / 3)
d(i) = M * Sin(psi + pi / 3)
Next
For i = 16 To 18
a(i) = Cos(-fi / 2 - 2 * fi * (i - 16) + hi) / Sqr(3)
b(i) = Sin(-fi / 2 - 2 * fi * (i - 16) + hi) / Sqr(3)
c(i) = M * Cos(psi + pi + pi / 3)
d(i) = M * Sin(psi + pi + pi / 3)
Next
For i = 19 To 21
a(i) = a(i - 6)
b(i) = b(i - 6)
c(i) = M * Cos(psi + pi + 2 * pi / 3)
d(i) = M * Sin(psi + pi + 2 * pi / 3)
Next
For i = 22 To 24
a(i) = a(i - 6)
b(i) = b(i - 6)
c(i) = M * Cos(psi + 2 * pi / 3)
d(i) = M * Sin(psi + 2 * pi / 3)
Next

At every moment the distance between the central sphere and any of the shell spheres is equal to 1. At every moment the distance between any shell sphere and any of another shell spheres is equal or bigger than 1.
(I don't exclude that this piece of program can be written shorter.)

Earlier I have thought, that in n-dimensional space it is possible to pack m = 3*2^n-1 spheres around the central sphere. But it turned out to be, that this formula is correct only for dimensions 2, 3, and 4.

To see the rotation in pseudo-Euclidean space-time open my Electron-model program http://darkenergy.narod.ru/SR2007.exe

Look also the page 4D - World.

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My Curriculum Vitae.

Ivan Gorelik.

Four-dimensional Rotation of the Universe.

If our Universe is closed, then it can rotate. The last observations give several proves of this idea.

Non-Euclidean closed Universe can be imagined as three-sphere x²+y²+z²+q²=r², arranged into Euclidean 4-d space, where q is aditional coordinate, associated with color. That is slightly remind us the left figure, without the central ball. That will be the rotation in 4-d space. We live in pseudo-Euclidean space-time. That is why, we must change the q-coordinate into ict-coordinate.

In order to see the rotation in the pseudo-Euclidean plane (x,ict) you can open my program http://darkenergy.narod.ru/SR2007.exe

It is very interesting that the length of two hyperboles after dividing of every element of hyperboles by corresponding γ is iqual to 2pri. And more, - the quantable velocity was found. As a result the time-axis can by finite and closed.

Now unite the two pictures and imagine the rotation of pseudo-Euclidean hyper-sphere: x²+y²+z²+(ict)²=r². 

All points of the Universe can move with the same linear velocity c, and with the same angular velocity H.

The usual ball can not rotate such, that all it's point moved with the same linear velocity. In this sense the Universe is more homogeneous.

But we can find one dipole (axis-plane of rotation B) for every point A of such Universe. Moreover any point A has its proper axis-plane B and the point A is situated at B. The point A divides B on to rays. Let say, that this axis-plane is composed by axes z and ict. In every point of the Universe the square of 4-velocity (vg, icg), is equal to -c². At the point A v=0. Consequently its g=1 and 4-velocyty of this point has only time component (0, ic). Point A moves here with the velocity of light only relatively the time-coordinate. All other points of this axis-plane has a spatial component of 4-velocity. Consequently all galaxies visible from A along the axis-plane (z,ict) will be transverse red-shifted along z-coordinate. But all galaxies laying on this axis-plane does not move relatively each other, - they rotate together along circumference in the plane (z,ict). The plane-axis (x,y), perpendicular to axis (z,ict) is circular and embraces the axis-plane at the distance R. It is also rotating. And the galaxies, visible in that direction will also be transverse red-shifted.

Such rotation explains many-many things, I can not write all here. And only make a note.
Rotation gives and explains:
the spin of every particle of Universe;
particle's quantum ψ-function;
the right-left asymmetry in galaxies spirality;
why the conventional age of the Universe is constant and is 13,34 billion years, always!
abnormal rotational velocities of peripheral stars in galaxies;
the critical density in such Universe is less...

If the velocity along the axes are slightly different we will see the multipole structure in the cosmic microwave background radiation; the difference in red-shifts of galaxies.  

 Some additions from different sources.

^[3] Problems of arranging balls densely arise in many situations, particularly in coding theory (the balls are formed by the sets of inputs that the error-correction would map into a single codeword).
The most important question in this area is Kepler's problem: what is the most dense packing of spheres in space? The answer is obvious to anyone who has seen grapefruit stacked in a grocery store, but a proof remains elusive. (It is known, however, that the usual grapefruit packing is the densest packing in which the sphere centers form a lattice.)

The colorfully named "kissing number problem" refers to the local density of packings: how many balls can touch another ball? This can itself be viewed as a version of Kepler's problem for spherical rather than Euclidean geometry.

^[1] In mathematics, sphere packing problems concern arrangements of non-overlapping identical spheres which fill a space. Usually the space involved is three-dimensional Euclidean space. However, sphere packing problems can be generalised to two dimensional space (where the "spheres" are circles), to n-dimensional space (where the "spheres" are hyperspheres) and to non-Euclidean spaces such as hyperbolic space.
A regular arrangement (also called a periodic or lattice arrangement) is one in which the centres of the spheres form a very symmetric pattern called a lattice. Arrangements in which the spheres are not arranged in a lattice are called irregular or aperiodic arrangements. Regular arrangements are easier to handle than irregular ones—their high degree of symmetry makes it easier to classify them and to measure their densities.

^[2] The number of equivalent hyperspheres in dimensions n which can touch an equivalent hypersphere without any intersections, also sometimes called the Newton number, contact number, coordination number, or ligancy.

Exact values for lattice packings are known for n=1 to 9 and n=24 (Conway and Sloane 1993, Sloane and Nebe). Odlyzko and Sloane (1979) found the exact value for 24-D.

Exact values for general packings are known for n=1, 2, 3, 4, 8, and 24. Musin developed a bounding method in 2003 to prove the 24-dimensional case, and his method also provides proofs for three and four dimensions (Pfender and Ziegler 2004).
 

SO(4)

^[4] In mathematics, SO(4) is the four-dimensional rotation group; that is, the group of rotations about a fixed point in four-dimensional Euclidean space. The name comes from the fact that it is (isomorphic to) the special orthogonal group of order 4.

   Simple rotations
A simple rotation R about a rotation centre O leaves an entire plane A through O (axis-plane) pointwise invariant...

Half-lines from O in the axis-plane A are not displaced; half-lines from O orthogonal to A are displaced through α; all other half-lines are displaced through an angle < α.

      Double rotations
A double rotation R about a rotation centre O leaves only O invariant. Any double rotation has at least one pair of completely orthogonal planes A and B through O that are invariant as a whole, i.e. rotated in themselves. In general the rotation angles α in plane A and β in plane B are different. In that case A and B are the only pair of invariant planes, and half-lines from O in A, B are displaced through α, β, and half-lines from O not in A or B are displaced through angles strictly between α and β.

     Isoclinic rotations
If the rotation angles of a double rotation are equal then there are infinitely many invariant planes instead of just two, and all half-lines from O are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements. Beware: not all planes through O are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-line are invariant.

Links:

1. Sphere packing. Wikipedia
2. Kissing Number
3. Sphere Packing and Kissing Numbers
4. SO(4)