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
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.
In order to transit from the turn to rotation, we must
express the coordinates with the help of functions sin and cos. Here is
the piece of program, which helps us to express the coordinates after every
equal interval of time, corresponding to the turn at
p/30 radians at both planes of rotation.
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
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)
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)
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)
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)
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)
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)
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*2n-1 spheres around the central sphere. But it turned
out to be, that this formula is correct only for dimensions 2, 3, and 4.
And here is the 2-d rotation of elementary electric charge (electron) in x, ict coordinates, i.e. the rotation in pseudo-Euclidean space-time.
Four-dimensional Rotation of the Universe.
If our Universe is closed,
then it can rotate. The last observations give several proves of this
Non-Euclidean closed Universe can be imagined as
arranged into Euclidean 4-d space, where q is aditional
coordinate, associated with color. That slightly reminds us the left
figure, without the central ball. That will be the rotation in 4-d
space. But we live in pseudo-Euclidean space-time. That is why, we must
change the q-coordinate into ict-coordinate.
In order to imagine the 4d rotation in the pseudo-Euclidean space (r,ict) you must unite both bottom gif-rotaitions in your brain. Try it! It's easy!
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
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.
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
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.
 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
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
 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).
 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.
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.
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.
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.