The Barnsley Fern

This post will briefly describe the Barnsley Fern fractal and will suggest that something akin to this is actually suitable for genetic inheritance with the fern template being analogous to the genetic information and the generation mechanism being a solution for the general stability problem in biological systems.

British mathematician Michael Barnsley first described the fractal in 1988 in his book Fractals Everywhere. The shape is generated via purely mathematical means but with the final shape resembling an organic form. Varying the parameters will result in variations of shape and a method exists for producing cross breeds of two different species.

Barnsley’s paper describes the process in detail but to simplify we start with a template illustrated right by the red and blue triangles and then apply a fixed rule for generating the shape a single pixel at a time. The rule involves moving a determined distance but in a random direction for each pixel. See Numberphile

Variations in the original template will give variations in phenotype (right) but the dots are drawn in a completely random order and this order has no impact on the final outcome; the same shape is always generated. We have a morphogenesis that is short term random but long term deterministic; a pattern found throughout biological systems.

Barnsley found a way of combining two templates to create hybrid offspring that displayed traits of both parents. Here a fern and a lettuce are paired to produce four different looking offspring.

The stability problem in biology is just that the interior of a cell is a highly volatile environment with proteins in continual state of rearrangement and with DNA caught in an endless cycle of damage and repair, with actually millions of errors every time it is replicated which are actually repaired somehow by the rest of the cell.

So how, in this environment, does the cell maintain order, both physical and regulatory? Using DNA as an example, we have short term instability with long term predictability. This pattern of behaviour is apparent from the molecular level up to the cellular and all the way up to the scale of an entire organism.

‘Goal oriented behaviour‘ or teleonomy is one common way of describing this. The cell wants to reproduce and will do so despite being subjected to all sorts of vibrations and other energetic perturbations at the molecular level. The exact molecular movements involved in any transaction at this level are unpredictable but the end result is inevitably the same.

So many different paths will converge upon same result despite persistent random disturbances. This is an inversion somewhat of the usual cause and effect process whereby an initial impulse could lead to many different outcomes dependent upon tiny accidents along the way (the sliding doors effect). The end effect here seems not so much to depend upon the precise events that precede it but rather in spite of them to some degree. The important consideration appears to be the end goal instead of the initial causes.

We appear to have short term chaos but long term order.

The Barnsley Fern (here with extra colours to create a third dimension) exhibits similar behaviour in that the end pattern is always the same despite a considerable input of noise into the system. The fern forms a stable attractor in mathematical terms and will always reproduce precisely as long as the template remains intact.

To recap then, the system consists of several components:

  • A stable template whose variations will result in a variety of shapes
  • A rule for generation i.e. a fixed law of nature
  • Morphological expression via a noisy (random) system, variations in which have no effect upon the outcome whatsoever
  • A method for combining two templates to create hybrid outcomes
  • The illusion of goal directed behaviour
  • Creation of a natural form from a very simple mathematical template

Note that the original template is not obviously identifiable from the final shape and very likely it is not derivable from it.

The parallels with nature are so striking that it is appropriate to suggest that something of this character is in place as a guiding principle for organising biological forms.

  • The template here is the gene (not necessarily DNA), the compressed information field that somehow is passed down through generations and whose combinations with others at reproduction is responsible for individual variations.
  • The random noise is the Brownian motion of molecules within the cell which is seen now to have no compromising effect upon the final shape
  • The template is able to generate shapes outside of its own boundaries and so to organise on a global scale rather than just a single cell at a time
  • Long term evolution has the character of attractor evolution rather than evolution by increment, with the fossil record showing stability of species for millennia followed by a sudden emergence of a new stable phenotype
  • The stability problem is solved
  • Goal oriented behaviour is seen here as an illusion, an inevitable outcome of the algorithm. The outcome is always the same so it can’t really be described as a goal as there is no foresight or intention involved.

Inheritance

We now have a clue as to what it is that is actually inherited from one generation to another. Currently we are stuck wondering as to how a complex shape is passed down through the generations and how such a shape may be implemented but the Barnsley Fern gives us something to think about at least.

The pattern of goal oriented behaviour (and its inheritance) is so common to biology that it has been described as the central problem in biology (Jacques Monod) but we now have some idea as to what mechanisms to look for and how they might be encoded in a transmissible ‘gene’ .

How would this be implemented? Admittedly this is highly unclear but consider the following:

“The Holy Grail in this area, as outlined some years ago by Orgel, remains unchanged- the establishment of a general biology, a biology that is not necessarily linked to the specific biomolecules from which biological complexity is structured (nucleic acids, proteins, lipids, etc.), but one that will be based on the chemistry of the class of molecular replicators.” – Pross

So it is thought by an expert in the field that the physical structure of bio-molecules is insufficient or even irrelevant in explaining biological organisation.

Global organisation indicates a global morphogenetic field – discuss!

The Implicate Order of David Bohm is a generalised version of this idea. Here the template resides in the Implicate Order, hidden from our perceptions whilst the generator mechanism is almost precisely how Bohm describes the Unfolding of the Wholeness. All we are presented with in the Explicate Order (the physical world) is the final fern shape.

The way this would work is that some super fast process continually operates to maintain the final form and as the template evolves over a lifetime, the generation process reflects that and expresses the shape as best it can given the noise and imperfection of the physical world. The impression given then, in the timescale of our perceptions is that of a stable, self-healing morphology that slowly ages over time.

Scientists, according to this model, are studying an epiphenomenon of an altogether different reality to the one which they are presented with. They are happy when they find patterns here and there that can be declared as laws but the organisation of the whole will remain elusive as long as they look to the physical properties of matter to somehow organise the physical matter itself via its own properties.

Peter Gariaev photographed leaves under a strong magnetic field and found that even after removing part of the leaf, a ghost image remained for several minutes before disappearing. Here a piece of leaf at the top right has been removed but still appears in the photograph.

Other parts of the leaf were removed earlier and their ghost images have disappeared. So for a short while then it appears that some sort of bio-field is still present and is managing to manifest in energetic form via amplification from the applied magnetic field.

Relationship to Naturopathy. The existence of such a scheme clearly supports a naturopathic approach to health. The body here is not so much self-healing but continually regenerating according to some hidden mechanism drawing on some unseen information contained in a possibly read-only script.

The job of the healer then is to maintain the physical condition in as receptive state as possible to allow the regeneration to take place.

The difference here is that whereas naturopaths view the body as a self-healing system, we now have a separation between the body itself and the guiding information that is responsible for maintaining that body.

Any interventions in this system can only affect the physical body. In the above example, causality is clearly a one way process as no possible alteration of the outcome can alter the original template. The process represents a sort of causal event horizon, a new Central Dogma where information can only flow one way.

Related pages:

References:

V -variable fractals and superfractals
Authors: Michael Barnsley, John E Hutchinson, Örjan Stenflo
https://maths-people.anu.edu.au/~barnsley/pdfs/V-var_super_fractals.pdf

Theory and Applications of Fractal Tops – Michael F Barnsley
https://www.researchgate.net/publication/225263258_Theory_and_Applications_of_Fractal_Tops

A Barnsley fern generator (create your own ferns)
https://www.chradams.co.uk/fern/maker.html

The Chaos Game – Numberphile
https://youtu.be/kbKtFN71Lfs?t=336

Barnsley ferns, but actually explaining each function – Leios Labs
https://youtu.be/xoXe0AljUMA

Wikipedia on the Chaos Game
https://en.wikipedia.org/wiki/Chaos_game

Stability in chemistry and biology: Life as a kinetic state of matter
Author: Addy Pross
https://www.degruyter.com/document/doi/10.1351/pac200577111905/pdf

The Illusions of the Modern Synthesis
Author: Denis Noble
https://link.springer.com/article/10.1007/s12304-021-09405-3

Principles of Linguistic-Wave Genetics – Peter Petrovich Gariaev
https://www.researchgate.net/publication/228926241_Principles_of_Linguistic-Wave_Genetics