William Blake

(1757 – 1827)

To see a World in

a Grain of Sand

And a Heaven in a Wild Flower

Hold Infinity in the palm of your hand

And Eternity in an hour…

-‘Auguries of Innocence’

My Fractal Journey

It all started, I suppose  now when I look back, when I was about seven years of age and standing in-line with the rest of my class (funny word to use – we don’t do grades in Ireland). It was the height of the ‘troubles’ and it was a convent school for girls right in the centre of it all. (I have given away my age and background here; perhaps a little too much). Anyway, what has this to do with my first fractal revelation?

Well, the setting is quite relevant, but the time may not be so. The room that we were all standing in was filled with lines of girls according to their age and class, and this room had a special name – ‘The Red-Room’  on account of it having a checkered patterned floor of red and pale coloured tiles. This room  should however, have been called the mirrored room as the two main opposing walls were adorned with massive gilded framed mirrors and of course, there was the expected statuette and other religious paraphernalia, along with obligatory singing high-pitched Nun vamping on the slightly discordant ivory keys (she was one of those rare fun-loving Nuns and her name was Sister Anne).

Some of you may have already guessed from the set-up, what I suddenly and profoundly came to understand – in what could only be described as a silent eureka moment;  remember, I’m only seven – so I don’t know any of this intellectually, nor do I have a language for it – at least not  until I was a very mature adult – my early education was really woefully poor – but you could say that all that I learned in that moment was good enough for anyone’s life-time and I was in school at the time.

Suddenly, everything came into sharp focus. The long sequential rows, on every scale, reflected in the opposing panes, ran to what seemed to be infinity in ever-decreasing lines of green-clad girls. Anyone who has ever studied fractals knows that self-similar patterns are reflected at scales within scales and the more you zoom into the deeper and smaller levels, you find the same type of pattern  reflected repeatedly. It’s just a matter of scale.


Mandelbrot Set Zoom CC. licence Wikimedia

I cannot put into words how profound this deep understanding of things, the world (before the usual indoctrination of how the world works according to the cultural dictates of our time) that came like an instant download in that moment. But it probably did guide me at some deeper level (after several years of research) towards an understanding of the inherent fractal nature of Nature. This brings us to one of the main players in a very distinct (non-Darwinian) but, dynamically  much more explicable model of evolutionary thinking (click on the title below for a short bio and a summary of his ideas along with some other forgotten theories of evolution in a related article on this site and/or read on for the summary of a more universal scaling fractal explanation of his concepts and how they relate to an over-arching alternative evolutionary model, which dovetails (fractally at every scale) with several other alternatives:

D’Arcy Wentworth Thompson

(1860 – 1948)



D’Arcy Thompson was born the year after Darwin published his famous book On the Origin of Species. And in 1917 D’Arcy published his own quite famous book: On the Growth and Form and much extended and updated version in the 1940s (See Thompson 1917 [link] and Thompson 1945 [Link]). In his major treatise, he uses the known processes and principles of the physical sciences of chemistry, physics etc in conjunction with biology and by applying the language and tools of mathematics, he essentially quantified and established inherent patterns of growth and form that Nature appeared to consistently use in the production of even the most complex organisms.

He essentially, came to a distinctively different way of understanding evolutionary development and ultimate complexity by applying the main principles of the processes known to operate within non-biological complex systems, to biological systems and discovering that there was something universal underpinning them all, as indicated in his 1945 edition On Growth and Form.

There is something, an essential and indispensable something, which is common to them all, something which is the subject of all our transformations, and remains invariant (as the mathematicians say) under them all.

D’Arcy Thompson (1945, 1085)

About D’Arcy
… the dynamic influence of starting conditions lies in the morphology of shells and horns. These are the permanent, non-living, three-dimensional record of a temporary, two-dimensional living state – the base of the horn, or the mantle of the shellfish. D’Arcy Thompson showed that all horn and shell morphologies could be described in simple mathematical terms readily derived from the incremental nature of growth… For instance, geometrical rules of packing determine cell arrangements. These need not be specified, but can arise spontaneously. Yet the packing arrangement may be “useful” in minimising the space occupied by the cells, by maximizing cell-cell contacts, by establishing different categories of cells (“inside” versus “outside”), and so on…

― University of Dundee and the University of St Andrews (2010)

In other words, D’Arcy Thompson’s general alternative views on evolutionary processes were in principle based upon scaling laws of fractal patterning and these universals embedded in Nature,  led him to propose and apply a very different kind of descent with modification: as summarised in the excerpts given below taken from his epilogue of the 1945 edition On Growth and Form.

Note that protozoa refer to the entire kingdom of animals, and are four-limbed animals and that this is a later edition of D’Arcy’s book (1945).

On Growth & Form
…for eighty years’ study of Darwinian evolution has not taught us how birds descend from reptiles, mammals from earlier quadrupeds, quadrupeds from fishes, nor vertebrates from the invertebrate stock…
… Our geometrical analogies weigh heavily against Darwin’s conception of endless small continuous variations: they help to show that discontinuous variations are a natural thing, that “mutations” -or sudden changes, greater or less-are bound to have taken place, and new “types” to have arisen, now and then. Our argument indicates, if it does not prove, that such mutations, occurring on a comparatively few definite lines, or plain alternatives, of physico-mathematical possibility, are likely to repeat themselves: that the “higher” protozoa, for instance, may have sprung not from or through one another, but severally from the simpler forms; or that the worm-type, to take another example, may have come into being again and again.

Thompson (1945, 1093-95, Epilogue)


In summary, D’Arcy Thompson seems to propose throughout all his chapters in relation to the main topics addressed, that biological life has followed the same fundamental processes inherent in other complex and naturally organising phenomenon and he believed that by tracing the intrinsic patterning and scale employed by Nature, we can begin to see the methods of producing continuous novel and increasingly diverse formations of shape and form that could be implicated in the great variety of species; and yet may be commonly connected by sharing underlying patterns of fundamentally similar properties and employing the same means of development to reach their own particular level of intrinsic complexity. How he arrived at his conclusions are best summarised below:

Transformations: The Visual Influence of D’Arcy Thompson
D’Arcy passionately believed in giving students as great a breadth of knowledge as possible, telling them if you dream, as some of you, I doubt not, have a right to dream, of future discoveries and inventions, let me tell you that the fertile field of discovery lies for the most part on those borderlands where one science meets another. There is a cry in the land for specialization. . . but depend upon it, that the specialist who is not reinforced by a breadth of knowledge beyond his own specialty is apt very soon to find himself only the highly trained assistant to some other man. . .
Try also to understand that though the sciences are defined from one another in books, there runs through them all what philosophers used to call the commune vinculum, a golden interweaving link, to their mutual support and interpretation.

Jarron (2013, 83-4).
Google books

For instance, as a result of more recent collaborations between physicists and biologists attempting to identify the obvious universals embedded in life (in the spirit of what D’Arcy Thompson had proposed as the most productive way to pursue scientific investigations), Geoffrey West and others have been led to discover consistent power laws, universals and invariants in Nature, along with fractal networks common at all scales and all of this phenomenon having predictable properties, led West to declare the following in the excerpts from an interview in the New York Times below:

”Everything around us is scale dependent…
It’s woven into the fabric of the universe…

It is truly amazing because life is easily the most complex of complex systems,’ …. ‘But in spite of this, it has this absurdly simple scaling law. Something universal is going on …”

Johnson (1999, ‘New York Times’ 12th January)

The video below should give you a general insight into the type of research and the findings which point to the fact that underlying all of biological complexity are, surprisingly simple universal scaling laws governing fractal-like networks.

Perhaps, I should point out at this stage, that the more recent collaborations between physicists and biologists have led to the re-evaluation and acknowledgment of D’Arcy Thompson’s work and contributions to revealing the deeper complexity of biology, but they seem to take the cautionary approach to stating these laws and invariants of scale and form in more general and less emphatic terms than D’Arcy Thompson did as indicated below. They also do not go as far as re-assessing our current model of evolutionary complexity in the light of these findings as D’Arcy Thompson was prompted to do (but perhaps they don’t know about the fundamental flaws in our existing paradigm in the first place). If you are in any doubt, I would suggest reading the link to the readable quote book here , it’s a flip book and reads just like the real thing and see what an increasing number of well-respected scientists thing about the science behind the Darwinian form of evolution, particularly our modern synthesis):

Life’s Universal Scaling Laws
Although few today would articulate Thompson’s position so provocatively, the spirit of his characterization remains to a large extent valid, despite the extraordinary progress during the intervening century. The basic question implicit in his discussion remains unanswered: Do biological phenomena obey underlying universal laws of life that can be mathematized so that biology can be formulated as a predictive, quantitative science?
Most would regard it as unlikely that scientists will ever discover “Newton’s laws of biology” that could lead to precise calculations of detailed biological phenomena. Indeed, one could convincingly argue that the extraordinary complexity of most biological systems precludes such a possibility. Nevertheless, it is reasonable to conjecture that the coarse-grained behavior of living systems might obey quantifiable universal laws that capture the systems’ essential features. This more modest view presumes that, at every organizational level, one can construct idealized biological systems whose average properties are calculable.

West and Brown (2004, 36)

Therefore, we could go one of two ways here and either acknowledge that even in the light of our more modern approaches to trying to understand biological complexity, D’Arcy Thompson’s model is still highly relevant; albeit as an idealised concept and useful for grasping the essence of biological systems and how they appear to work. On the other hand, we could take the position that D’Arcy Thompson took, based upon a much broader approach to the most up-to-date evidence from many fields of study and apply them to biological systems and restate his universal patterning of life in terms of biological laws that hold at every level in Nature – This is the underlying principle – The fractal Nature of Nature; the apparent driver and the underpinnings of the non-random directional change and the principle that makes meaningful and highly efficient forms, shapes and patterns in nature. See first five chapters of forthcoming book (needs final edit) to get an idea of how all this fractal scaling applies to biological complexity and by inference evolution.

final backcover info for evolution book one

In the aforementioned book, the scientific evidence has led to me taking the latter approach and as you will see: his dream of placing biology en par with the mathematical descriptions afforded physics and chemistry really do appear to hold across the entire spectrum and at every scale of life. They are quantifiable and predictable, not just descriptions of life, but a key to unlocking many of its mysteries embedded in the evolutionary process itself. In my mind, this is good science, as these laws and equations of scale can be tested and applied to find out things we could not otherwise directly measure.

For instance, by looking to fundamental properties within naturally developing non-biological complex systems, where, under certain conditions, rapid and profound changes can be brought about via catalysing the whole system in a single leap, D’Arcy Thompson applied this principle to evolutionary development and suggested that whole functional cellular life and its highly organised organisms could have come into being likewise a number of times under broadly similar catalytic conditions. He also understood that a very small change (initial starting conditions) at the beginning; can make a huge difference further down the line, reflecting yet another fundamental characteristic of natural complex systems. This led him to suggest that primitive life could have been triggered into taking fundamentally divergent paths by simply having different starting conditions.

D’Arcy Thompson suggested that as several distinct groups were discernible, based upon his study and observations of quite distinct body-plans and internal systems of a broad range of species, that these may represent the resultant divergent forms alluded to above. He referred to these distinct groupings of organisms as “discontinuous types”, meaning that Nature drew upon their underlying form and inherent systems to produce many scaled-up and increasingly complex versions of the same fundamental forms “Types” with degrees of modification according to the group’s intrinsic evolutionary potential.

Who Wrote The Book of Life?
Picking Up Where D’Arcy Thompson Left Off
NASA scientists are using Thompson’s biomathematical studies of life forms on Earth to postulate about life forms throughout the universe. There are certain universal conditions that will always affect the shape of a life form, wherever that life may be.”Everywhere Nature works true to scale, and everything has a proper size accordingly,” wrote Thompson. “Cell and tissue, shell and bone, leaf and flower are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed.” … Gravity, for instance, acts on all particles and affects matter cohesion, chemical affinity and body volume. Other influences that are consistent throughout the universe are temperature, pressure, electrical charge and chemistry.

NASA (1999, May 28th)

  Is the nature of Nature, simply a matter of fractal scales…?

DNA nebula molecule compared

What do you think?

Please comment, share and otherwise join me in this conversation if you are as fascinated,  as I am, by our Natural world and seek some serious answers, email me at: diggingupthefuture@gmail.com


Cheers Maria Brigit






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