Biology / An elementary 5-dimensional model applied in different sciences
- the animal kingdom -

1. Some general aspects:

It’s hard to see how only "random mutations" and "natural selection" could explain an evolution towards more complex organisms. All kinds of variations, yes. But hardly the increasing "capacities" of creatures from the enormously well adapted bacteria.

It has been said that even the simplest unicellular forms of life that have survived to our days have nearly all types of enzymes as higher animals (JB), the same number and kinds on different levels in the evolutionary chain. And all fundamental functions exist already in the unicellular organisms.
   If this still is valid after an immense amount of new knowledge during last decades, it has implications for the Darwinian view on evolution.

When it concerns multicellular organisms all phyla within the animal kingdom seem to have been differentiated already in Precambrian times (Ez). "In no case any transitional forms of today or as fossils are found."*

*If now this statement isn’t a kind of circular definition: the definition of phyla builds probably (?) on the lack of transitional forms. The classifications of phyla are also much debated.

For instance: fossil finds cannot confirm an evolution from 1- to 2- to 3-layer animals. On the other hand an evolution in the dimension of time seems proved within a group as chordates.

If we assume (irrespective of divisions into phyla) that all basic forms – with different degrees of complexity – appeared more or less "simultaneously" from the eukaryotic cell, we have to imagine that they got differentiated out of some underlying, basic differentiating principles, perhaps as through a series of more or less momentary ’quantum jumps’; the following differentiations within the basic forms along another coordinate axis of time. This as analogous to the first seconds of Universe after big Bang with following development through billions of years.


Fig Ev-1-177

Environment changes slowly. If crediting Darwinian "trial and error" mutations ’50 %’ of the truth, we have still 50 % to account for and ascribe to basic structural principles.
   Then, the longtime evolution should be connected with a hierarchy of genes of the same kind, with primary rows of genes differentiated into secondary, more specialized ones; the primary genes perhaps getting activated at certain threshold points of major changes in evolution?
   The different coordinate axes may also be seen as appearing in the relations egg → individual versus individual → environment: egg the 0-pole, environment the 00-pole. It’s a bond of dependence that implies mutual changes between the "poles".

2. Biologists’ primary bases for classification of organisms:

Sub-cellular level:

a. Autotrophic — heterotrophic organisms
    (= not depending – depending on organic nourishment as plants versus animals)

b. Prokaryotic — Eukaryotic organisms
    (cells without nuclear membrane as algae and bacteria – cells with nuclear membrane)

Eukaryotic organisms, animals:

- Unicellular – multicellular organisms

- 1-, 2-, 3-layer organisms

- Coelom (body cavity) types: degrees in development of a true abdominal cavity
(Acoela, Pseudocoela, Schizocoela, Enterocoela)

- Protostomia – Deuterostomia: oppositions in directions regarding formation of organs.
(Deuterostomia is the branch leading to vertebrates, mammals and Homo Sapiens.)

About classifications on the sub-cellular level, a) and b) above, see additional file.

In the last 4 points of classifications it’s easy to recognize the geometrical steps in dimension degrees (d-degrees) of the dimension chain, our background model. A rough, first description of the evolution in synthesizing direction inwards in the chain:

- from single cells as points to cell-contacts of multicellular organisms, a step 1 ← 0,
- to number of tissue layers as surfaces, ← 2 ← 1
- to separation of inner abdominal room in mass and space, step 3 ← 2,
- to the differentiations in directions inwards/outwards, front-back etc. in step 4 ← 3.

   Fig Ev-2-179-3

In the dimension chain in physical and geometrical terms:


Fig Ev-3-179x

a-poles with features from 00-pole, b-poles with features from 0-pole:
0-00 also a relation between higher and next lower d-degree.
   The level chain in a developed chordate corresponds rather naturally to these basic classifications:


 Fig Ev-4-179-1   


Fig Ev-5-179-2

The first three steps above of evolutionary classifications corresponds to the embryonic development of a 3-layer chordate:

Fig Ev-6-179-4

[A couple of following differentiations within Deuterostomia:
- Invertebrates – Vertebrates.
- Vertebrates: egg-laying — mammals.
- Mammals: without amnion — with amnion.

Subgroups within group Vertebrates
   Tunicates → Lancelets → Cyclostomes → (e.g. lampreys) → armored sharks (Placoderms) → Cartilaginous fishes → Lungfishes → Ray-finned fishes and bony fishes →
Coelacanths ("Old Four Legs") → Amphibians → Reptiles → Birds // Mammals → Homo Sapiens.

Biologists have divided living organisms in 5 kingdoms (Fc p. 95), a number 5 that hasn’t much to do with the 5-dimensional chain tested here:Unicellular prokaryotic – unicellular eukaryotic – plants – fungi (mushrooms) – animals. ]

3. Unicellular to multicellular organisms, step 1 ← 0/00:

The evolution from first prokaryotes to eukaryotes and then to multicellular organisms took billions of years. Changes of the environment and chemistry of the atmosphere of Earth are taken as main factors.
   However, there seems to be a lack in the explanations between this very slow, continuing change of the environment and produced, outer conditions on one hand and the clear, "sudden" steps on the level of the cells themselves on the other.
   It could indicate the existence of certain thresholds in values of parameters (light, oxygen etc. ), which function as borders for marked, new constructions.

As said before there are many intermediate forms with cell contacts (regarded as d-degree 1 in the level chain) forming colonies of individual cells, also among prokaryotes. It's however only eukaryotes that develop to multicellular organisms, obviously representing a new degree of integration.

Some aspects on this transition could hypothetically be
- inversions of some kind?
- new center defined at meeting between individual units of equal potential?
- a deeper center as start of cell division?
- just a reversed relation in relative amount of e.g. DNA and proteins versus lipids?

About inversions:

- Multicellular organism represent unity among cells - as guided by the integrating 0-pole in our model. Unicellular organisms express separation between cells - as out of the polarizing 00-pole. The multicellular represent unity, the unicellular the multitude - in spite of amount in their names.
   The relation unicellular egg - developed organism in the multicellular organism appear as a kind of reversion - or the previous opposition built in: it's the egg that has the high potential for differentiation, while the developed organism represent already differentiated cells, corresponding to start and end of a dimension chain as steps of polarizations.
   In terms of directions outwards / inwards it could be pointed to the circumstance that unicellular organisms have a tendency to seek themselves inwards multicellular organisms, to live inside these - as for instance algae in fungi and bacteria in animal stomachs - like viruses more explicitly. (A parallel to the theory (No. 11) of mitochondria and chloroplasts as immigrants in eukaryotic cells.)

Certain 1-layer species as calcareous sponges have a "curious" turn of the blastula inside-out at their embryonic development. Similar processes occur in colonies of flagellates as Volvox (Ez p. 57), hence among individuals referred to step 1 ← 0/00 in our interpretation here, single cells to cell contacts.
   Such "inversions" could be the expression for the "pole exchange" in last step 1 → 0/00 of our model (where motions outwards define a new anticenter, motions inwards a new center. (This last step 1 – 0/00 is represented in each higher d-degree step too.)
   The invagination of vegetative pole in embryology of chordates – as an antipositive curvature inwards – may perhaps be understood as a less drastic equivalence.

New centers defined through meeting of units ...?
It's from eukaryotic cells that multicellular organisms develop and it's eukaryotic cells that develop sexual cell divisions. (It's said too that sexual reproduction are initiated at changes in environment among organisms that have both alternatives.).
   One could imagine that in a certain step a duplication of DNA could occur, leading to meios, a halving as a polarization ← → at cell division, which the other way around gives the opposite meeting between two complementary units, defining a new, more highdimensional center: → ←.(Also a kind of inversion of directions, here in the process of reproduction.)
   Yet, sexual reproduction doesn't in itself lead to multicellular organisms. (A condition is perhaps that the small difference, assumed here in file Genetics, between "daughter cells" at simple cell divisions, has grown far enough to complementary poles that imply and stress the mutual dependence?)

Geometry in positions of cells may have been a primary factor, perhaps the most elementary, behind the evolution of multicellular organisms?
   There are species of bacteria (myxobacteria) that first spread radially outwards, then turn to grow in inward direction towards certain centers and superpose these, which through these superpositions become defined as spores (Bc p. 319). It’s an example that indeed seems to illustrate the views here on opposite directions of fields - and a kind of fertilization through meeting of opposite poles. There is already an underlying "we" in these groups, assumed as through some mutual signals.
   Cf. how the very relative position of single cells decide their development to different parts of a flower according to theories about plants.

In a field of first cells the one in the center could have been defined as 0-pole, the furthest out as anticenter, 00-poles, the opposition connected with a radial versus circular geometry; the central cell becoming defined by its position as outward directed and consequently with high differentiating potential, a condition for a multicellular organism. While cells at anticenter remain a multitude of similar cells. Cf. that this also characterizes ectoderm in multicellular organisms – and cells from the animal pole at "exogastrulation" (No. 4).

Now, as far as is known, the evolution of multicellular organisms from unicellular ones seems to have taken billions (?) of years. Then we have to count on three historical phases: 1) cell divisions and "radial" spread of the unicellular individuals over the surface of the Earth = divergence, 2) pole exchange to inward direction, 3) convergence leading to meetings with other cell groups. If so, it would be a parallel on another time-scale to what is mentioned about the myxobacteria above.
   We may compare the spread of human beings in small groups of kinfolks and clans out over the world and at certain densities a turn to convergence and confrontations with other kinfolks and clans, stepwise leading to more hierarchical structures and superposed organization of societies...

[Level development
One suggested view in the dimension model here is that a condition for level development is a meeting between equivalent units (of 5-dimensional types) with centers as 0-poles, whose mutual relation thus becomes opposite, defining new more complex centers.

            Fig Ev-7

The contrast to such level development would be just repeated propagation:
5 → 4 → 3 → 2 → 1 → 0/00, as propagation of a single 5-dimensional unit.]

A deeper center at start...?
Another hypothesis is the thought that the cells at divisions of unicellular individuals didn't get quite separated and retained a closer connection to one another on this fundamental level that concerns the inner of the cell with relative negative charge.
   The same thing could be described as a cell division that has reached a deeper level, e.g. the field level of d-degree 4 instead of earlier only of d-degree 3.
   It should imply a character of what is called neoteny, a branch of evolution starting from an earlier embryological stage of an individual.
   The field level - and strength of integrating 0-pole of d-degree 4 in terms of the model - is obviously incorporated to an internal property in multicellular organisms.

Relative amounts of proteins...?
In fundamental aspects the cell can be regarded as inversion of an atom. The relative negative charge inside membranes is mainly given by the proteins, which also make up the radial structure in cytoplasm and cell membranes with origin from center. As radial they are principally unlimited outwards - in opposition to the closed circular component of lipid membranes. (Cf. also the polarity the FA – FG forces outwards / inwards.)
   Thus, if the protein production became dominating over the one of lipid membranes, suddenly or not, it could have implied a new degree of communication and transports between cells - as materialization of field lines.
   The clear demarcation of a center through a nuclear membrane in eukaryotes could perhaps been a factor in such a change (?).

Light with its double character of particles and waves could be regarded as a ground plan for the opposition unicellular — multicellular organisms: in inward direction appearing as particles, to compare with individual, separate quanta as cells, in outward direction appearing as waves, corresponding to the primary, uniting, wavy proteins on the field level of multicellular organisms.
   (What should in that case among cells correspond to the certain conditions within quantum mechanics that gives these opposite aspects on photons?)

4. Classification in 1-, 2-, 3-layer animals, step 2 ← 1:

The division of animals according to number of tissue layers concerns dimensionally d-degree 2 in the sense that tissues represent cell contacts forming surfaces. We get the number chain 1 - 2 - 3 as outer poles of d-degree 2 and 1 and from 1 in the dimension chain. (Cf. that the 1-layer animals according one theory originates from 2-layer ones.)

The complementary poles 2a – 2b out of d-degree 2 in the dimension model are in elementary geometrical terms defined as outside/inside and/or convex/concave.
   1- to 2-layer species become more or less bowl-shaped around the gastric cavity in their outer design.

Fig Ev-8  

(The term species here used without regard to the scientists’ classification of levels into kingdoms, phyla, classes, orders, families, genus, species.)

There have been separate opinions between scientists on how to classify "1-layer" organisms as sponges since they also have an inner layer of cells, however more individual, not quite united to a tissue. They could be called 1.5-layer organisms. The same regards true Diploblastica, "2-layer" animals as Cnidarians (hydras, e.g. jellyfishes and corals), which have an extracellular layer between its two main tissues that can include spread cells, thus could be called 2.5-layer species.
   With reference to the figure above and "outer poles" as partial structures of each d-degree in the model such confusion or halfway steps are natural.

About differentiation of organs besides the gastrula the 1-layer species have only spicules as a kind of skeleton in the layer of extracellular material besides spread individual cells inside outer tissue. Among these however a kind of contractile cells appear as precursors to muscles.
   (About forms of spicules as basis for classification, see file Skeleton.)

2-layer species get further nervous and muscle functions but hardly as separated organs, only as a differentiation between single sensory cells besides muscle cells in the same epithelium (of d-degree 2) and ganglia that innervate the muscle cells.

First with 3-layer species, where the intermediate layer forms a real 3rd tissue of cells, 3-dimensional organs develop. Organs in the level chain as d-degree 3.
   In the level chain of organs we have regarded skeleton defined in d-degree step 2 ← 1 and muscles in step 3 ← 2. Thus, there is a natural correspondence between the classification of species and of organs, especially when counting with the outer poles in different d-degrees in the model.

      Fig Ev-9

(About d-degrees as numbers it could be added that with a 3rd tissue layer and development of a front – back axis in more developed animals the intermediate coelom material gets differentiated into 2-3 sections.)

The gastric cavity (the stomach) gets stepwise differentiated from 1.5- to 2.5-layer species and within these groups.
   The step from radial inflows in sponges to one-way directed inflow (the original embryonic mouth) in 2.5-layer classes are one example. (It's a drastic change similar to the one between the circular blastula and definition of a coordinate A-V-axis in embryology of chordates.) 
   The cavity in 1.5-layer animals is only a sac with inside and outside - as outer poles of d-degree 2. In 2.5-layer animals as hydromedusae the cavity is differentiated in a radial canal and a ring canal (Ez p. 62), the geometrical poles 3b and 3a in our model.

There are however transitional forms in shapes of the inflow canals: In 1.5-layer sponges the complexity of the inflow canals increases – from straight radial (ascon type) inwards to angled with side pores in membranes that have become wavy in convex – concave bows (sycon type) to a branched network of canals with widenings to globular chambers in the epithelium (leucon type) with flagella of the cells converging inwards (Ez p. 52, 56), i.e. a more 3-dimensional structure. It can be compared with how glands develop in chordates.)
   Cf. that these shapes of canal systems in structures geometrically correspond to d-degree steps 4 → 3→ in the dimension model, see No. 7 below about "Directions…".

In 2.5-layer species the gastric cavity becomes divided through walls, "septa", as a stepwise materialization of the inward direction. (According to reference Fc p.113 they radiate outwards from center of the gastrula, but according to the figures on the same page rather from the periphery inwards.) While hydras lack septa, jellyfishes have partial septa, corals whole septa.
   Numbers of septa are essential in the classifying system. Septa in the extinct tetracorals were shaped in 4 steps through three angle steps from a "vertical" axis which gives number of septa 2 – 6 – 10 – 14 (Fc p.113); compare number of electrons in s – p – d – f orbitals, intervals in the 2x2-chain behind the periodic system. Septa in stony corals (most corals of today) are typically "6-radiant", have 12 primary septa.
Another differentiation occurs in muscle functions on septa of 2.5-layer corals (Ez p. 75), which have "radial" muscles on one side of the septa walls with divergent, protracting function, and longitudinal muscles mouth - foot with contracting function along the vertical axis on the other side. Hence, a polarization both of coordinate axes to 90°, assumed angle step in d-degree 4 to 3, and of elementary directions outwards/inwards in function as well as on wall sides, poles of d-degree 2. Cf. d-degree 3 in the model with outer poles 4b – 4a of d-degree 4, Direction, and muscles referred to d-degree step 3 - 2 in the dimension chain of organs.

Other features separating these tissue classes concerns capacity of external locomotion, forming of colonies and symmetry, see No. 7 below.

5. Coelom — differentiations, step 3 ← → 2:

Coelom from mesoderm regards the 3rd real tissue layer (Triploblastica). Within the class of 3-layer animals the differentiation of this coelom is obviously as said above a question of polarizations Mass – Space, that’s out of d-degree step 3 - 2 in our model. It implies steps from species as Acoela with coelom as a whole mass to species with splits in coelom (Schizocoela) to species with coelom divided in outer and inner layers (d-degree 2) with real secondary body cavity (Enterocoela). We have the inward direction of gravitation (FG) connected with the property Mass, the outward acceleration force (FA) connected with Space in macrocosm as well as here, however, in a reversed relation: inner space, surrounding mass. (Cf. the cell regarded as inversion of an atom.)
   One can describe these polarizations either in terms of mass – space, physical poles 3a – 3b out of step 3 – 2, or just as a step from mesoderm as a volume (d-degree 3 in our model) to mesoderm as surface layers (d-degree 2).

6. Deuterostomia - Protostomia, step 4 ← 3:

Some characteristic polarities, although not general in all respects:

- nervous system ventrally
- heart dorsally
- archenteron becomes mouth
- mesenchyme from ectoderm (outer layer)
- exoskeleton
- schizocoel common
- spiral cleavage of egg often
- segmentation of body typical


- nervous system dorsally
- heart ventrally
- archenteron becomes anus
- mesenchyme from endoderm (inner
- endoskeleton
- enterocoel (real body cavity)
- never spiral cleavage
- not segmentation

As said in first paragraphs above this opposition concerns directions, d-degree 4 and poles 4a and 4b, inward/outward directions in our model:
- partly inwards - outwards from anticenter - center poles (ectoderm– endoderm) in such things as creation of skeleton, (exoskeleton versus endoskeleton) and mesenchyme,
- partly in positions of first neural and nutrition systems in relation to the two coordinate axes Animal – Vegetative poles (A-V) and Front – Back (F – B), in file Embryology interpreted as corresponding to d-degrees 4 and 3.)

The complementary polarity of d-degree 4 here implies also a further "centralization" of separate functions to organs, representing d-degree 3 in the level chain.

As for coelom in preceding paragraph, it’s enterocoel that implies a real secondary body cavity, internal secondary space. Again, we have the fundamental opposition Mass – Space:
   Space as such, representing divergence from 0-pole, here separating mass to tissues in both directions, seems as a factor behind the domination of outward direction in Deuterostomia, the properties that lead further to chordates.
   Mass of less split coelom represent convergence and inward direction, thus connected with Protostomia, including arthropods, the insects.

Why are insects so small and elephants so big?! Why this difference between enormous amounts of eggs and only a few? It seems as if it could have its origin in this opposition between contraction from the 00-pole and divergence from the 0-pole.

Further there were the differences
- in egg division: Protostomia often spiral cleavage, Deuterostomia never.
- segmentation: typical for Protostomia, in Deuterostomia no segmentation.

Spiral cleavage of first egg occurs among Protostomia in many groups of Schizocoela. It implies a kind of rotation, assumed as motional moment in d-degree 3 in our model.
   In Deuterostomia first cleavages of eggs are "bisymmetrical", vertical and horizontal, which means along crossing orthogonal coordinate axes that define a center. In such egg cleavage all 4 to 8 cells have principally direct contact with the center, the origin. (Cf. stem cells that separated can develop to whole individuals.)
   Thus, the opposite kind of egg cleavage reflects also this polarity between center - with "radii" - and anticenter as circular rotation.
   It corresponds to a view on the dimension model where a-poles of all d-degrees derive from end of the dimension chain, b-poles from its start:


Fig Ev-10

Spiral cleavage leads to early differentiated cells in opposition to the radial and bisymmetric one where the daughter cells keep a higher potential of differentiation.
   Compare perhaps that the first egg cell in mollusks (Protostomia) undergoes several internal polarizations in the cytoplasm before the total cell cleavage, while such polarizations in Deuterostomia as lancelets or frogs (batrachians) are only one or two.

Segmentation as a division in more or less equal parts of the body implies a division along the Front – Back axis, the axis which typically gets developed in 3-layer animals. It’s also the axis we have seen as representing d-degree 3 in embryology of chordates.
   Segmentation is typical for big groups of Schizocoela Protostomia such as ringed worms (Annelids) and Arthropods. Even mollusks are believed to come from originally segmented forms.

Organs that in segmentation are repeated in each segment are shells - the exoskeleton, extremities as some kind of legs, a pair in each segment, nervous ganglia and muscles. That is organs mainly created from outside inwards – the characteristic direction of Protostomia. ("Superficial" egg cleavage - on the surface, d-degree 2 - in the class arthropods seems as a connected feature.) The alimentary canal however from vegetative 0-pole runs through the length of the body unsegmented. Cf. threadlike colonies of unicellular organisms that sometimes have a canal straight along the thread with shared protoplasm.

It has been said somewhere in a physical context (no reference here available) that convergent vectors (Vconv) don’t reach the center. If so, the segmentation of worms could be one example!
   Why the multitudes of Masses in macrocosm – and the unity of expanding Space? Perhaps the density of divergent vectors from a center (not identified or defined) hampers and overcomes the convergent vectors "half ways" ? Compare Vdiv, the FA-force, with what is called "dark matter" and estimations of its relative strength or overweight in relation to mass and the FG-force, (70 % to 30 % or other similar figures).

Some features of segmentation appear also among Deuterostomia as chordates, e.g. in muscles of the alimentary canal, in the straight abdominal muscle of humans, in our notochord and in bladders on the neural tube in its embryological development.

Segmentation on the level of organs corresponds to a similar feature on the level of tissues: cells in the outer tissue (ectoderm, the skin) are more or less equal, while cells in endoderm have a high potential for differentiation. Compare about exogastrulation in file Embryology concerning Deuterostomia: isolated tissue material at animal pole doesn’t differentiate while that from invaginated ventral pole can develop rather much of organs.

Hence, there is a quantification and multitude as repetition from outside (the quantifying 00-pole), a continuum and unity from inside (the quantified 0-pole ). In the dimension model the force from 00-pole is the primary polarizing one, the pole from 0-pole the primary integrating one; a relation quantification – continuum related to the one between particles and waves.

Standing waves, longitudinal and/or transversal, could to a certain degree illustrate segmentation - as it seems expressed in motions of worms!

   Fig Ev-11

A wave and its reflection between two borders correspond to opposite directions in the dimension chain, the reflection implying a pole exchange and a quantification. If the illustration is more than a metaphor, why should the double direction be typical for Schizocoela Protostomia? The reflected wave as "inwards" along the F – B axis in this group eventually stronger than in the other?

It’s said that stationary waves appear when "the force is not in phase with the velocity".
   However to interpret and apply that statement, one could speculate about phase displacements of waves as a differentiating factor between animal groups.
   Such phase displacements may correspond to half steps in the dimension model here, a displacement between a d-degree and the jump or d-degree step: border – interval. Cf. first figure in this file, differentiation through primary steps.
   Examples are the relation between vertebrae and spinal cord in humans, also development of extremities from the web between fin bones in fishes.

The opposition plants – animals, if expressed in terms of standing waves could be:
- Plants: longitudinal waves, open ends.
- Animals: transversal waves, closed ends. (Source Wikipedia, Standing wave.)

Segments can during embryological development and historical evolution get differentiated functions and be packed together. In the Protostomia class crawfish (crayfish) the 5 front segments develop to the head. Note once again number 5! It’s about the same when it concerns the human brain. It seems again to be a question of positions, the location of cells along coordinate axes as vectors, here the back - front vector.

Protostomia includes a manifold of groups, Deuterostomia only a few with Chordate as a big one. Thus, the very number of groups could be seen as an expression for the polarity manifolds– unity, the 00- versus 0-pole, and the dominating opposition of directions between Proto- and Deuterostomia.

7. Directions of evolution?

Since all animal phyla of multicellular organisms seem to have been differentiated already in Precambrian times (Ez), it make it difficult to establish a direction in time between them.

Among the bases for classifications above only the one between Protostomia and Deuterostomia represent more clearly a complementary polarity of poles of the same d-degree in the dimension model. The other divisions are more as d-degree steps, to or from more complex forms. To or from? It's easy to be cheated by the clear evolution within phyla, as the steps from water to land living animals and more built-in embryos. We can remember that some scientists regard 1-layer animals as originating from 2-layer ones.

Direction from simple to complex forms is not unambiguous.
In a dimension chain the direction outwards lower degrees implies increasing differentiation as increasing numbers of polarizations. This could be translated to steps towards more simple forms. However, with the differentiations built-in the evolution goes towards more complex forms.   

Fig Ev-12-182-2

A fundamental tone includes all its overtones. A highdimensional cell or tissue is the one with high potential for development - like stem cells in relation to specialized ones. In the
embryological development first stages are naturally most high dimensional in this sense, most "simple" stages as blastula and the two wall bladder in chordates resembling the 1- to 2-layer organisms.

There are features of these 1- to 2-layer animals that connect them with highest d-degrees in the dimension chain and could point to an origin from these steps 5 - 4 - 3.
   They are often stuck to the sea bed or other solid surface or have such stages in their development. External, "endogenous" locomotion is not developed.
   Density is a factor too (the physical quantity proposed in step 5 → 4 These 1- to 2-layer classes are often building colonies as the unicellular ones. In colonies the external relations are still structural, individualization only partial. (Certain 2-layer hydroids have the individuals on a common "stalk" where they get mutually differentiated in function.). Thus features from the further functional differentiation into organs in 3-layer classes appear already on these elementary tissue levels.
    The symmetry (see below) of 1-2-layer classes are mostly radial or biradial symmetric, like 5- to 4-merous diploblastica among plants.

One aspect on this ambiguity may be illustrated by the loop version of the dimension chain: d-degrees from higher steps debranched and meeting the other way around in synthesizing direction inwards:
      Fig Ev-13

(It's possible to see some structural relationships between corresponding steps outwards - inwards in this the loop version of a dimension chain:
- Naturally the egg with the whole organism (as between d-degree 5 and 0/00).
- The above mentioned inversion of the blastula in a species of 1-layer sponges and differentiations of directions in Proto-/Deuterostomia. (4 - 1)
- The septa (developed inwards) in many 2-layer organisms and of coelom (outward from archenteron) in 3-layer animals:


Fig Ev-14-179-3x )

In the history of evolution the direction may appear to be mainly towards "invelopment".Yet, in other respects it seems possible to suppose simpler animal forms as debranched or reduced variants of species from more complex levels in the evolution.
   5 →> 4 →>3 →> .......
       \ 1   \ 2   \ 2,5

A more acceptable interpretation in accordance with the figure is probably to see the high dimensional potential of an embryo going stepwise further, implying stepwise more involvement towards more complex forms the other way around: Hence, an evolution that stops of some reason (e.g. complexity of DNA?), earlier or later:

   5 →>4 →>3 →>2.......3 ← 2 ← 1 ← 0
   5 →>4 →>3......................2 ← 1 ← 0
   5 →>4.................................... 1 ← 0

Another aspect concerns the fundamental polarity between plants and animals. They represent opposite directions in the dimension chain.

Fig Ev-15

   Fig Ev-16

In essential functions the animals in their involvement represent the "a-poles" versus plants as "b-poles" in a dimension chain, "motions to each other", outside in eating, circular versus radial structure and inward direction embryologically as in their dependence.

In symmetries the opposition in directions seems clear:
While symmetry in flowers of plants generally decreases from 2-cotyledons to 1-cotyledons, from 5- and 4-merous plans to 3-merous plans, the polarizations in animals into coordinate axes become coupled with increasing "invelopment" inwards, with increasing number of tissue layers:
   - 1-layer animals are radial-symmetrical.
   - 2-layer animals are radial-symmetrical - or bilateral symmetrical in certain cases.
   - 3-layer animals are bilateral symmetric.

In the embryological development the steps from the "vertical" axis Animal-Vegetative poles to the axis Front – Back to the Right-Left axes has been interpreted as a process through 4th to 3rd to 2nd d-degree. In numbers of axes corresponding to 1 - 2 - 3.
   With only the V-A-axis defined, the symmetry becomes principally radial, and 1- to 2-layer animals retain thus this main axis of the invagination gastrula.
   With development of the F-B-axis in 3-layer animals the symmetry becomes bilateral. The symmetry axes can be described as stepwise "crystallized" and decreasing towards more "inveloped" animals:

Fig Ev-17-183-1


Fig Ev-18-183-2

(The fundamental reason why the symmetry decreases with new coordinate axes is naturally that the axes are defined by complementary poles).

The step from radial to bilateral symmetries resembles the one in atoms from s- to p-orbitals: s-orbitals circularly "all-directed" versus electrons divided on perpendicular coordinate axes in p-orbitals.(Orbital numbers given from intervals in steps ← 2 ← 1 ← 0 in the 2x2-chain. It's surely not a coincidence, cf. numbers of septa in corals above.. .

It can be observed that radial-symmetric 2-(2.5-)layer species as cnidarians often have a 4-numbered symmetry (or n x 4) in outer and inner structure as related to 4-merous plants.
   A primary group of 3-layer Deuterostomia as the echinoderms is said to have had a 3-numbered symmetrical form in Precambrian (Fc p. 121), hence a decreased symmetry towards more "inveloped" forms in agreement with the views above. Their radial, 5-numbered symmetry of today is a secondary feature, superposed their bilateral symmetry on their larval stage (Ez). Compare number 5 in the superposed chain 9 – 7 – 5 – 3 – 1, halved orbital numbers as intervals in the 2x2-chain behind the periodic system:


Fig Ev-19-192x

The 3rd coordinate axis right – left (R-L), the bilateral one, shows many signs to also develop towards asymmetry, as out of complementary poles:
   Examples are for instance the dominance of the claw on one side in some species of crayfishes and their lateral motional direction, the increased growth of one tooth on one side in toothed whales, the differences between right and left cerebral hemispheres in humans as well as the asymmetric positions of inner organs and different division of lung lobes. Etceteras. Birds retain the right arc of aorta, mammals the left one (Kz p. 210).

In mollusks, believed to have developed from an early bilateral symmetric species, the intestines undergo a rotation half a turn that leads to a radical asymmetry bilaterally (Ez p. 259).
   In the dimension model rotation and spiral motions are assumed as the external motions in structures of d-degree 3 and 2 respectively. We could wonder if perhaps a slow process of such motions goes on during history of evolution, which have proceeded unequal length of time within different species?

About fundamental tones and overtones, an association:
How overtones on a cello string may appear? Development of tissue layers and specializations (figure from end of file Embryology).

A"Medusa" figure :


Fig Ev-20-188

8. Miscellaneous notes:
About secondary polarizations of directions, numbers and design with associations to the dimension chain:

a) Directions, some examples:
A secondary polarization within the 2-layer phyla cnidarians:
- in hydromedusae do the sex cells and statocysts derive from ectoderm, the outer layer,
- in jellyfishes the same cells from endoderm, the inner layer.

3-layer species with 2 shells or valves, polarized in orthogonal directions:
- Brachiopods, among the oldest, have the shells along the ventral – distal axis V-D.
- Bivalves have the two shells divided right-left, the R-L-axis.
Both examples illustrate complementary poles or d-degree steps according to the interpretations here.

Shells of different mollusks illustrating motional degrees of growth:
- Some have a plane-spiraled shell that illustrates rotation + pathway motions, a 3-dimensional motion.
- Others have a conical-spiraled shell that illustrates an extra factor of growth with increasing radius as a 4-dimensional motion.

- Flagellates: Some has 2 flagella at one end, others one at each end: a step from 2-way-direction to one-way direction or the opposite, a depolarization.

b) Numbers of things:
From the aspect of a dimension chain: 5 – 4 – 3 – 2- - 1 – 0/00, sum 15;
sums of pole values: 10 – 8 – 6 – 4 – 2.

  • Trilobites, old extinct group: 5 pair of legs, the front pair of which becomes antennae.
  • Cuttlefishes (Cephalopods): 10 arms → 8 arms – or a multitude of arms (Nautilidae).
  • Legs: Spiders 8, insects 6, tetrapods 4, birds, humans 2, mollusks "1-footed" or "0-footed" (stuck to some solid surface).
  • Body divided: Spiders in 2 parts, insects in 3 parts.
  • Wings of insects: 3 pairs first, became 2 pairs. Mosquitoes and flies: 2 pairs became 1 pair.
  • Myriapods (among Arthropods): Diplopods 2 pair of legs in each segment, Chilopods 1 pair.

Examples of reductions?

  • Sea urchins: early species had 15 → 100 shell plates, today 20 ( 5 x 4).
  • Fishes: fen rays a multitude → tetrapods 5 "rays" of toes and fingers.
  • Hominids: 5 knobs on the masticating surface of molars, East apes 4.

An insect as Rhodnius prolixus undergoes 5 stages as a larva with changes of skin before metamorphosis (JB).


c) Structural design:
Biological shapes on a macro-scale reflect in many cases structures on underlying levels. According to the presumptions in the dimension model this depends on the same dimensional processes reappearing on all levels (cf. fractals).

We could imagine that the structures of light and of water have "induced" early organic molecules and these further biological forms: for instance
- the 6-rings of H2O-molecules in water → rings of carbohydrates → leave forms, or
- polarized light with appearing rotation → spiraled DNA and proteins.

A fatty acid looks like a simplified form of a Myriapod.

A ganglioside as a larva that eats leaves, (sketch after P. Karlson 1976, p. 169):

Fig Ev-21

(Content of this ganglioside, a glycolipid with aminated carbohydrates: Glucose, Galactose, N-acetylgalactosamine, N-acetylneuraminate + sphingosine + fatty acids. Mass sum: 1836, the p/e-quotient, if with uncharged COOH-groups.)
   Cf. also the figure here.

- A segmented worm has a macro-form as a circular magnetic field around an electric cable. Plants as trees have forms like the magnetic fields around a staff.

- A fish with its fens can resemble the illustration of a light beam with the fens like the electric and magnetic oscillation planes perpendicular to the pathway direction.
- A backbone of vertebrae in relation to the spinal cord illustrates the quantified light beam with phase displacements.
- All pacing motions have a parallel in this phase displacement of light propagation.
- Bird wings get a form that resembles motions of the cell material towards the primitive groove at their embryological development.

Body forms in the broad outline may become nearly anything but tend to develop towards the basic dimensional geometries of d-degree 1 – 2 – 3 – 4, especially among organisms living in water: needle- or worm-like – plane – oval – spherical – star-like and flower-like or tree-like as divergent vector fields.

It’s pointed out by scientists that worm-like forms exist in many different phyla, e.g. in Mesozoa, Acoela, Pseudocoela, Schizocoela (annelids - arthropods – mollusks), Lophophorata and Deuterostomia.
   Regarded as 1-dimensional forms, they can be seen from the aspect of one debranched degree in each step of the dimension chain outwards. Only in the 1- to 2-layer animals these forms are lacking, i.e. where the lengthwise coordinate axes front-back isn’t yet defined.

There are many examples showing that elementary geometrical forms are more obvious and distinct the closer a structure is studied in details. (Cf. liver lobes for instance.) In viruses all round forms are actually polyhedrons.

9. Some general questions:

Characteristic features of classes and species of organisms are coded in DNA. If now the mutations in genes were totally haphazard, there wouldn’t exist any system in the classifications of animal groups, no "tree of evolution", no connected properties, would it?
   In that case a multitude of variations and combinations should exist without any mutual stepwise development between or within families and classes and it would be without reason to look for "missing links" ?
   If certain genes code for a notochord, other genes for segmentation, some others for 6 arms, then one thinks that a segmented chordate with 6 arms should be possible, perhaps even practical? Flying hominids or trees with brains?
   Presumably it shows on the hierarchy of genes, more general and more specialized, a system that groups properties.
   How about complementary genes? Do they exist – in the sense of "complementarity" used in the dimension model? Since there are proteins with opposite, complementary functions, it seems reasonable to assume.
   And how could such hierarchies develop through only random mutations and adaptation to an environment?
   A fundamental scheme of some kind – as the dimensional model here proposed or the like - seems necessary to presume.

The evolution on different levels, system level, organ level, tissue level… could also be presumed to follow different time-scales, which could cause many obscurities in systematics and relationships.
   The big jumps in evolution, which some biologists talk about, could be imagined as the end of a dimension or level chain and start of a new? (Or, which becomes the same, at a certain d-degree in the most fundamental dimension chain.)

A speculation about the time aspect: If the time-scale on two different levels correspond to frequencies, and it takes a certain time for nodes of the two "waves" to coincide, this conjunction could perhaps imply a decisive mutation? "A certain time" eventually interpreted as a certain number of generations when the chain of propagation is regarded as a wave?

Adaptation to the environment is the other component in the Darwinian view on evolution. However, the coelacanth, the fish with 4 legs, is said to be "pre-adapted" (Ez p. 131) to a life on land, i.e. the 4 legs developed first in the sea before it became practical for a life on land. Such "pre-adaptation" points sooner to an "endogenous" factor in the steps of evolution of the here presumed kind, an underlying dimensional scheme in evolution of genes. (Not "pre-adaptation" as an anticipation of future life on land.)
   (We should perhaps count on two different kinds of mutations, on one hand the haphazard that mostly seem to be negative mistakes, on the other such mutations which represent a given evolutionary scheme of the type dimension chains – in accordance with the model "trial and error"?)

Similarities versus relationships:
In the dimension model a step of polarization leads to complementary poles as partial structures). This should imply that animals with complementary characteristics have a closer relationship than animals showing similarities but on different levels:

      Fig Ev-22-197-1

Protostomia and Deuterostomia could be one example, prokaryotic and eukaryotic cells another if both derive from Archae bacteria (now Archae regarded as perhaps a special phyla).
- Relationship could correspond to what biologists call ramifications in the tree of evolution.
- Similarities appear in what biologists call convergence, similar features developed in different phyla as on different levels, "without relationship".
   The idea of polarizations seems inherent in biologists’ descriptions without being expressly pronounced.

Just a figure of primary and secondary polarizations:

Fig Ev-23-197-2

An outline of classifications (freely after Ez and Sb):

Fig Ev-24-198-2

Log-scale for times of the Earth; middle ~ 9,65 / 2 = circa 67000 years ago, a time for Neanderthals and Homo Sapiens:

9,65 Born of the Earth
9,5 Prokaryotic cells ?
9,3 Eukaryotic cells ?
8,7 The Cambrian – most phyla of today assumed differentiated
8,4 Reptiles. – Supercontinent Pangaea – gets split later
7,6 India collides with Asia
6,6 Hominids
5,1 Homo Sapiens
4,0 End of last ice age
3,7 Egypt’s pyramids
2,2 The industrialism
1,0 ~ New millennium 2000.


To Evolution - Addition
- Subcellular level


© Åsa Wohlin
Free to distribute if the source is mentioned.
Texts are mostly extractions from a booklet series, made publicly available in year 2000

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