1. A fundamental invention:

Centrioles are among the most beautiful and clear expressions for geometrical design within biology; an "invention" of eukaryotic cells on the level of organelles.

In cross-section the structure illustrates the center – anticenter polarity (0/00) in arrangement of microtubuli and radial fields illustrated by the spokes, in our model defining dimension degree (d-degree) 4 with outer poles 0 and 00, which meet in the d-degree of motions.

Fig Cil-1-157-1

Why the number 9 in the ring of tubules? Scientists don’t seem to ask such questions. For some speculative proposals, see the end of this file.

Cilia and eukaryotic flagella, built on this plan, appear as first "extremities" of cells. They serve a double function of sensory antennae and external locomotion in the environment. Nearly all mammalian cells have one cilium.

The inner centrosomes, consisting of two centrioles, have been called the "heart" of the cytoskeleton system in a cell, radiating through its cytoplasm.

Notice in the dimension model that d-degree 1 are debranched in each step outwards and may polarize into the "d-degree of motions".  

         Fig Cil-2

2. Evolution:

Evolutionary, such cell extremities have simpler forms in Archae and prokaryotic cells, and the evolution may be described as steps from center to anticenter or just to more complex c/ac-structures:
- from simple bundles of protofilaments in Archae (resembling pili), as an original center, to
- central but hollow, bigger, screw-shaped tubules (center widened to a circle) in prokaryotes, to
- the c-ac structure with explicitly anticentric manifold of tubules in eukaryotic cilia, including spokes.
This evolution has a character of d-degree steps from center to anticenter and simultaneously of geometrical steps of substantiation 3 ← 2 ← 1 in plain projection.

The differences include 2 polarities:

a) between Archae and the other: In Archae bacteria the microtubules are build from below, i.e., from inside the cell outwards (as a form of secretion). In pro- and eukaryotic cells they are built from the top furthest out. It reveals a pole exchange 0 – 00.

b) Next pole exchange of type 0-00 appears between prokaryotic and eukaryotic cells in the transportation of building material:
- in prokaryotic cells it occurs inside the hollow tubule,
- in eukaryotic cells outside the walls of the many microtubules;
a polarization c-ac besides the one between one central tubule and many smaller at anticenter.

Such features support a view in the development from center to anticenter, principally outwards, during increasing complexity, while simultaneously the direction in the dimension chain is inwards.

      Fig Cil-3

Fig Cil-4

(The fact that these organs are not homologous but built in different ways with different proteins, can support the presumption here that it’s geometrical principles that reign.)

3. Polarities in eukaryotic cilia:

Transportation:
In eukaryotic cilia there is further the polarity between outer and inner "dynein arms" of microtubules at anticenter:
- transportation upwards of building material occurs as said above on inner side of microtubules,
- transportation downwards occurs on outer side of them, which corresponds to the fundamental directions of d-degree 4 in the dimension model (inwards from anticenter, outwards from center).
   The outer and inner arms differ too in protein structures (KI-O), hence in some respect obviously polar. One reference mentions the proteins kinesin and dynein as such mutual polar proteins in directions of transportation.

The polarity resembles the arrangement of vessels in plants: outer phloem for downward transport and inner xylem for upward transport. There is actually a similarity too between the different structures of cilia in Archae, prokaryotic and eukaryotic cells and the development of steles (a, c, e in file Plants, No. 4). Plants as cilia of the Earth!

There is further the similarity that roots of trees lack pith, and basal bodies of cilia "lack" the central tubules.

Structural differences of anticenter:
The differences between centrioles in cilia, basal bodies and centrosomes may be regarded in terms of d-degree steps:

a) Inside the cell, in centrosomes and in basal bodies of cilia, the centrioles consist of 9 triplets, 9 x 3 tubules at anticenter. Outside the cell surface, in cilia, they consist of 9 doublets, 9 x 2 tubules The numbers 3 and 2 happen to correspond to outer poles in d-degree 2 and 1 in the model.
   Geometrically the difference corresponds to the step from membrane surface (d-degree 2) to cilia as d-degree 1 on a relative macro-scale.

b) The centrosomes (9 x 3 tubules) in the inner of the cell consist of 2 centrioles (mother- and daughter cell) in a curious 90° angle to one another, which geometrically implies definition of a 3-dimensional volume.
   When not taking part in cell division one of these centrioles migrates to the surface and transform to a basal body, thus illustrating a halving as a polarization step from a volume to a surface (d-degree 3 → 2).

c) The centrosomes as MTOC organelles, near nucleus, organize the radial protein skeleton of the cell. Their function as vector fields (d-degree 4a-4b as outer poles of d-degree 3) becomes obvious.

Eukaryotes appear as a new level in d-degree step 3 - 2 through meeting of directions according to the loop model. It’s possible to imagine centrioles in this step as a complex center (0’/00’), expressing the radial / circular polarity among proteins in this step:

0 → → 00'/0' ← ← 00, a haploid chain, which get a complex center in the middle, c₂

      Fig Cil-5

- In d-degree 3 it would give the triplets, function of center (MTOC) for radial outward vectors, the cytoskeleton and the spindles as vector fields at cell division.
- In d-degree 2 outwards it would give the doublets, function of basal bodies and further the structure of cilia.

Concerning forces, the electromagnetic force (FEM) has been assumed defined in step 3 – 2 in the dimension chain of forces. The radial cytoskeleton arranged by centrosomes have a direction outwards from minus-charge at center, plus-charge towards cell membrane as is assumed about charges from 0- and 00-pole respectively.

Centers and spokes:
Centers of centriole structures are of different kinds: in the inner centrosomes as in basal bodies there are no visible tubules but "something", not possible hitherto to identify. Hence called "9x3 + 0" structures.
   Cilia outside the cell surface are of two kinds: the primary, non motile one, without identifiable center structure, and motile cilia with 2 single tubules in the center. How explain this difference?

One guess is that the "something" in the center of inner basal bodies and centrosomes as 0-pole in the d-degree 4 of vectors are less substantiated, not yet have got the ring-structured mass through inward direction from environment as anticenter. Cf. also number 9 and 11, section No. below.
   Scientists would probably explain it as a question of need for stability in outer motile cilia or something like that? The function of these centers is not yet understood.

Spokes in the cross-section depart from tubules at anticenter and end in "heads" (some thicker structures) about halfway to the center. These spokes are unexplained too. It’s difficult not to presume some invisible spokes as vectors from the center too, and the picture can remind of the FG- and FA-forces in macrocosm: gravitation inwards to mass of galaxies and outward acceleration force (FA) of "invisible" Space.

4. Sensory and motile cilia:

The basic structure appears either as sensory antenna, called primary cilia, "9x2 + 0" structures, or as organelles for locomotion,"9x2 + 2" structures, the motile cilia.
   The types represent the fundamental opposition of the nervous system in multicellular organisms. (All developed neurons have one cilium and fibers for cilia are found on the neural plate.)

Primary, sensory cilia:
As the nervous system guides inner processes on a multicellular level, so seems primary cilia of cells in vertebrates and mammals guide a lot of internal structural relations, positions and processes in the body and coordinate a big number of signaling pathways (Wikipedia). Failures in their structure seems for instance to have a role in the position of heart to the right instead of left side in the body!

Maybe these regulating functions are just expressions for cilia as structures of d-degree 1 on the cell level and the very jumps or steps of polarizations in a dimension chain behind relations in the organism?

      Fig Cil-6-157-2

Each step:
      Fig Cil-7-161-1

Cf. that d-degree 0/00 (or 00 in a haploid chain) as anticenter represents first polarizing force in our model, and passive motions of these cilia are induced by environment. It’s also possible to think of poles 1a and 1b of d-degree 0/00 of motions in terms of insubstantial "field lines" in the environment.

The primary cilia are used in more or less specialized shapes in the primary senses of mammals, in photoreceptors in the eye, as kinocilia on hair cells in the ear and on olfactory neurons in the nose.

Cilia as primary receptors of sunlight in the eye have a certain similarity with chlorophyll on the molecular level in its circular structure, corresponding to the rotational construction of porphyrins to anticenter rings around a center (a Mg-ion in chlorophyll).

Motile cilia and motions:
Centrosomes have been called motional centers inside the cells, organizing the cytoskeleton. The motile cilia as first extremities serve external locomotion, d-degree 0/00 in our model.

Developed cilia contain over 600 different proteins (Wikipedia). However, it has been shown that 2 proteins is enough to bring about the beating motions of flagella (= cilia in eukaryotes). The 2 proteins could be illustrated by the poles 1b and 1a in the figure above. It’s added that many of the over 600 individual proteins can function as small "nanomachines".

Flagellum of human sperms is actually a modified cilium. (Cf. the mass number of side chain of amino acid Arginine = 101, said to be richly found in fish sperms. About number 101, cf. the ES-chain in the genetic code.)

The motion starts at the very tip of cilia furthest out according to observations. Curiously enough, since the "motor" is positioned at the base of cilia. However, in our model motions are the very end of a dimension chain outwards.

Tubulin makes up the building blocks of microtubules. It's a globular type of protein (like F-actin in muscles), which can illustrates quantification of a line in "half steps" into motions..

      Fig Cil-8

The polarization into "half steps": the poles 1a, "motions to each other", and 1b, "motions from each other" in the model, can give the pendulation as between two poles of convergence and divergence. The figure could illustrate motions along the tubules.

The cilia describe externally a 2- or 3-dimensional motion: a planar, wavy one – or in other cases a more 3-dimensional one with power and recovery strokes (Wikipedia). These d-degrees of motions are in the model associated with d-degrees 3 and 2 respectively (cf. No. 3 above). Which factors that decide the difference of motional degree seems still not clear.

Several internal motions in the tubular structures are described: a sliding or clipping force at positions of ATP, also an oscillating movement along the whole length (KI-O). Further, a rotating or twisting force (KI-O). The microtubules have an ability of contraction too (Zf). It’s the two dynein arms in eukaryotic flagella that get the microtubules in the doublets to slide against each other, which gives the external motion (Wikipedia).

Thus, with all these sliding – rotating or twisting – oscillating - contracting and clipping motions it seems as if they could represent all the presumed motional moments in different d-degrees of the dimension chain:

    Fig Cil-9-157-2

(D-degrees of motions in the model: Vibration 1, rotation 2, spiraling 3, pumping as expansion/contraction 4, and "pole exchange" as first embryonic germ to motion in d-degree 5.)

About details, there are at least 9 different types of dynein proteins (KI-D, KI-O), eventually differentiating the microtubuli in degrees of stiffness and sliding velocity as it sounds . Connected with different d-degree steps?

5. Number 9:

Why this number 9 (x 3, x 2) of peripheral microtubules in centrioles and cilia structure? As said above, scientists don’t seem to ask such questions. A random number?
   It’s a general hypothesis in this work however that such numbers can have a deeper foundation. We may repeat that this number has nothing to do with numbers of chromosomes or number of protein threads in cytoskeleton arranged by centrosomes.
   Naturally, following different suggestions are only speculations.

a) Reading each step in the dimension chain as 2-figure numbers, the difference outwards – inwards in directions = 9:

5 –|– 4 –|– 3 –|– 2 –|– 1 –|– 0/00

54-45, 43-34, 32-23… 10-01 = 9.

It would agree with the interpretation of motions in d-degree 0/00 as turns in main directions.

b) Compare number 11 = 54-43, 43–32 … 21-10: the difference between steps in outward or inward direction, i.e. in one-way direction.
   Note that 11 is the number of protofilaments in almost all flagella of prokaryotes (Wikipedia).

(If number-base systems (nb-x) should be connected with different d-degrees — cf. the genetic code, part III — 9 in nb-10 = 11 in nb-8. A d-degree step between pro- and eukaryotic flagella types - ? - simultaneously with a step from fiber level to first tube form, d-degree step 1 to 2 ?).

c) Number 2 in center of motile cilia: at each d-degree the difference in directions inwards higher d-degree and lower one is 2, also marking a difference of directions: 5 ← 4 | 4 → 3, read as 45-43 etc. A factor of opposite directed motions in each "quantum jump" between d-degrees?

   Fig Cil-10

d) Number 9 as inversion of a series of 10-power displaced 11111… ?

9 /\ 0, 1111111…

   
    (5 ten-powers as steps.)

Fig Cil-11-157-2

e) Doublets as 99?

101 /\ 99009900…

Cf. about side chain of Arginine above = 101 and last number in the ES-chain, the genetic code..

f) Five steps of 1 (degree) squared:

111112 = 123454321 = 9 figures:

Fig Cil-12-157-1b

9 figures from 1 to 5 inwards and back outwards. First and last "1" in the number as poles 1a and 1b in the dimension chain. This interpretation should imply that the total circumference of microtubules had opposite poles and identifiable half turns in some way. (Cf. the curious angled copying of the daughter cell in centrosomes?)

g) The d-orbitals for 10 electrons in the periodic system is the middle one in the 2x²-series (x = 5 - 0) behind this system. There is 1 central ring-shaped electron orbital and 9 electrons divided, 3 on coordinate axes, 6 on pairs of plane quadrants. That’s 9 polarized, more peripheral orbitals.

Could centrioles – as developed in the middle step according to the loop version of our model – have a deep correspondence in this d-orbital?

   Fig Cil-13

It’s said that the doublet of microtubules in cilia consist of 13 threads in the A-rings, 10 in the B-rings (http://www.solunetti.fi/se/solubiologia/varekarvat_1/2/Cilier):
   1/2 x 26 = 13 for A-rings with dynein arms and bridge to B-ring of next doublet with 10 threads?
   (9 x 23 = 207 = number for A-base in nb-8; an association to the genetic code.)

h) 9 triplets in the volume of cytoplasm = 27 microtubules = 3³

   9 doublets = 18, outside the cell membrane, d-degree 2 = 2 x 3²

   A step 3³ ← → 2 x 3² (18 in 2x²-chain above)

j) Simple reading of numbers in last d-degree step of the dimension chain:

Step 1 → 0 as 10: 10 -/+1 (as 1a – 1b?) = 9 and 11, difference 2?

1b-poles define a new anticenter, ~ the 9 doublets.

1a-poles define a new center, ~ + 2 central tubules ?

Whatever to believe about the numbers, centrioles and cilia structure seems to be the way of atoms to tell us a fundamental story about life.