1. Why do cells divide?
What's the principle behind this propagation?

In one sense it’s like the propagation of a light beam through space as it's proposed interpreted in file EM-waves: a quantum jump in an atom from a 0-pole completing itself with the complementary antimatter of "negative" energy in space to a new 0-pole and a new quantum jump, as a sine wave and cosine wave with a displacement of 90^° between magnetic (M) and electric (E) components:

Fig Ge-1-142-2

A cell is a quantum of energy. DNA resembles a light beam with its complementary bases as representing the E- versus M- factors (a chromosome as a "lumosome").
   Like the light beam the DNA-strands complement themselves from what is antimatter on their level in their surrounding, the manifold of all individual complementary nucleotides.

The L-type of waves, ← 00’ → 0’← 00’ →…, could illustrate this factor in propagation. It seems also expressed in all building of linear structures in a cell, proteins, even globular ones, fatty acid, collagen etc. Why this rather curious lining up of the molecules after one another? "Dipole" bonds are surely a factor, this concept taken in a wider sense of some complementary feature in the individual units like 0 - 00 in the L-wave.
   Elementary waves as one guiding principle?

However, this feature of propagation is only a part of an answer. The cell as a world of trapped motions is in itself, internally, an ongoing, repetitive process of breaking-down and reconstruction as outward and inward directions in a dimension chain regarded as a unit. In this sense the processes get the character of standing waves.

Total reproductionof a cell from 1 to 2 to 4 ... etc. is certainly something more.
   It's preceded by a total duplication of the chromosomes and other organelles before cell division. Some built-in program for the step gets activated.
   Certain signals from "anticenter", primary polarizing force in our model, could possibly trigger it. It’s said that in embryological development of multicellular organisms the first cell division is initiated from mothers RNA in cytoplasm, i.e. anticenter to the nucleus.
   Scientists talk about other conditions, using concepts as "maturity" and "saturation". It could point to a correspondence with processes of crystallization on the physical level.

Next part of an answer would be a deeper polarization, 4 ← 3, from dimension degree (d-degree) 3, the volume that the cell occupies in a 3-dimensional space, to d-degree 4, this in terms of the dimension model; the step from inner self-sustaining ("haploid" or partial) processes to a total duplication of higher d-degree, including both DNA strands (double direction, d-degree 4). Through saturation developed to two separate centers.

Fig Ge-2-139-1

The step inwards to d-degree 4 could be seen expressed by the spindles as opposite, radial vector fields.
   The centrosomes that seem to guide the development of these spindles move also to opposite poles of the nucleus at start of cell division, i.e. a polarity of 180°, assumed in our model as of d-degree 4. (They are not necessary for cell division but appear in animal eukaryotes as substantiations of an underlying geometry.)

Duplication of haploid chromatids at sexual reproduction (meiosis) includes over-crossings with exchange of genes. It has been said that these over-crossings in numbers are 1 up to 4 or 5, usually not more (Bg p. 142). It sounds astonishing few. Yet, if the statement is reliable, it suits well in the dimension model

     Fig Ge-3-141-4

In addition to the question of why reproduction it could be said that polarizations as such is a main principle in the dimension model. All organism can also be described as 1/2 on every internal level (see file Life) as centers in relation to the surrounding anticenter that they depend on. ("Saturation" becomes the expression for this dependence.).
    The inversion of ½ is 2 and in the mathematical depths of life one could perhaps suspect a relation between a 4-dimensional function and its conjugate?

2. Duplication leading to 2 centers:

If we imagine the complementary poles of a dimension chain as nucleotides of DNA, saturation becomes the completion from environment as 00-pole with all its individual nucleotides.

a) Duplication:
   

Fig Ge-4-139-1

b) Division, two "daughter cells":

Fig Ge-5-142-1

3. Do the "daughter cells" become exactly alike at cell divisions?
The usual answer is yes. However, the figure above with adding of "a-and b-poles" to the opposite strands, the complementary character of DNA and the opposite directions of the strands could yet indicate some underlying difference, at least at the very creation of two chromosomes? Following two data point to such a difference:

- Since copying of the two strands occurs from the same end but in opposite directions, copying of one of the strands needs at least two DNA-polymerase molecules in a discontinuous operation (Aph p. 96), while the other is continuously copied *.

      Fig Ge-6

*(There is a similarity in this difference with the opposition between whole mRNA chains and the tRNAs with anticodons representing the other strand. Perhaps the same principle applied on another level?)

- The other information: When stem cells divide, it’s said, one of the daughter cells becomes a stem cell, the other a functionally specialized one.

In such data it seems possible to find a first origin to the later developed sexual differentiation, a division on different individuals, characterized by the 0- and 00-poles.
   (A singularity and a multitude, outwards and inwards.)

4. The geometrical features at cell division are really noteworthy and could support the general hypothesis here about a geometrical scheme underlying all manifestations in Nature.
   Only to mention one special thing first: in the duplication of a centriole to a pair in a centrosome, the "daughter" cell, (probably better called son cell), growths out at straight angle to the mother cell: a curious fact that seems hard to explain with functionality (?). It looks like an angle step as if it could refer to the one between the radial spindle cones that develops from centrosomes and the equatorial plane of chromosomes?

Fig Ge-7-(139-2)

(The above mentioned fact that centrosomes not are necessary organelles for development of radial spindles at cell division obviously seems to underline that the geometries comes first, as in an architectural drawing, before their materialization.)

Spindles and equator plain with two sets of chromosomes:

Fig Ge-8-140-2

This arrangement can geometrically be seen developed from the opposite ends of a dimension chain:

      Fig Ge-9

a. The linear chromatids line up and get copied with homologues paired, d-degree 1 in macro-shape:
      Fig Ge-10-141-3

These pairs of chromatids get spiraled (a 3-dimensional motions in d-degree 2) in several steps, as a kind of substantiation and contracted to pairs of chromosomes.
   They get arranged in the equator plate (d-degree 2). Cf. circular geometry of pole 3a in the model. Chromosomes in anticenter position here.

b. The spindles develop from centers as the 0-pole to vector field outwards, pole 4b, as structures appearing as the radial pole 3b in the dimension chain.
   The outgrowth of the spindles and then the pull of chromosomes in anaphase towards centrosomes imply two phases of d-degree 4, vectors of outward (4b) and inward (4a) directions (outer poles of d-degree 3 in the dimension chain).    

The centromeres are the ring-formed protein structures that in some unknown way enclose and bind the chromosome pairs and with help of other proteins (kinetochores) get attached to the spindle threads. They appear as anticenters in terms of the dimension chain.

(The fact that it’s a meeting between to radial cones that here "define" the equator plane as a surface - or rather a double surface, and not complementary 3a-3b-poles, shows that a doubled center (0-poles of d-degree 4) already is defined.)

Fig Ge-11-140-1

Loop model: A sketch on the counter directed processes in first phases:

Fig Ge-12-141-1

5. Cell divisions to haploid mail and female gametes:

Motional moments increasing towards lower d-degrees according to the model.

Fig Ge-13-142-3

- Female cell: unequal cell divisions in complementary poles from higher d-degree outwards, as in a dimension chain step 4 →> 1 + 3 in numbers. The small cells, called polar bodies, may indeed represent virtual male cells. They "degenerate", which could be suspected as a description of how the 00-pole of d-degree 4 gets debranched and (as through another dimension) meets the other way around as pollen. (There is the appearance of 3 polar bodies, two sets of them, in ovulum of higher plants.)

- Mail cell: equivalent divisions 2⁰ →> 2¹ →> 2^2... as in a dimension chain from lower d-degrees inwards, with 2 as log-base. Cf. increasing motions at lower d-degrees and tail of the sperm.

Fig Ge-14-142-4

(About egg divisions after fertilization: 3 divisions can be done symmetrical. It’s said (old information) that these 8 cells all can develop to whole individuals. If so, it could probably or somehow depend on the fact that 8 quadrants all have direct contact with origin in a 3-dimensional coordinate system?)