Continuing on from last week’s focus on D’Arcy Thompson’s alternative evolutionary understanding – universal scaling laws, which are finding strong scientific support in more recent times, particularly within the field of physics and its application to biological systems, as noted previously: D’Arcy Thompson’s belief that a broader approach to the bigger questions would produce  exciting, predictable patterning in Nature that can be summarised in powerful, yet simple equations, is coming to pass.

”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 (New York Times Jan 12th 1999)

   This brings us to the other dimension of Growth & Form which D’Arcy Thompson explored: Time in relation to growth. Indeed, D’Arcy discussed such patterns of non-linear and predictable patterns of natural growth and development in his extended (1942) edition of his book: On Growth & Form Link

Once, again these universal predicable scales of magnitude continue to be in evidence everywhere we look. Perhaps D’Arcy was actually on to something fundamental about biology.

 The Universal Growth/Evolution Curve

For example, West draws attention to yet another fundamental invariant – the universal (or almost universal) growth curve in the following:

…Still largely missing, however, is a theoretical framework for understanding the mechanisms that affect whole-organism growth trajectories. So questions such as why growth curves are almost universally sigmoidal, what controls the final or mature body size, and what affects the allocation of energy and materials to growth and development remain largely unanswered.

 sigmoidal simplified growth curve.jpg

Fig. 1:  Idealised chart showing the Sigmoidal growth/evolution over time curve. Left-hand section shows typical ‘Lag Phase’, the centre section where ‘S’ (Sigmoidal curve) rises rapidly and is sometimes referred to as the ‘Exponential Phase’ and finally, the third section (right) is the slow growth/evolutionary stage at the top of the ‘S-Curve’ which could be described as the ‘Stablisation Phase’.

 This is seemingly the pattern followed by everything that we have observed and recorded growing or developing, colonising or multiplying in the natural world or systems created by us, follow the Sigmoidal growth pattern as seen in the evolution of internet, computer technology and even social media platform such as Facebook as presented in simplified (idealised) form in the diagrams below.  Therefore, it is seemingly – universal and we should therefore pay attention to this pattern to see if we can establish how evolution itself evolved (growth and development at another scale and level of complexity. For instance, the things that we can observe and study in real time where, the Signoidal growth pattern is clearly the pattern commencing with a lag phase (fairly slow coalescence of all the necessary aspects needed to create the next level of the whole system), followed by an explosion of growth (an exponential phase) and finally a leveling out of growth and stability with further refinements of the existing system (or organism) – filling out the detail.

This Sigmoidal growth pattern can be seen in the slow development of a woodland, to later (exponential) colonisation of an area and finally, stability. See example below in Figure 2.

scaling seedling

Fig. 2: Sigmoidal growth/evolution over time curve applies to the growth of a seed to a sapling to maturity. Plant biomass is Sigmoidal (See:

.scaling trees

Fig.3:  Sigmoidal growth/evolution over time curve applies to the growth of an entire woodland. (See figure 11.5  Exponentlal growth of a colonizing population of Scots pine. Pinus sylvestris. Data from Bennen 1983.

From the seed to the sapling and from the mature plant to the entire colonisation and biomass, follow the same Sigmoidal growth pattern from a lag, exponential and stabilisation stage. The growth of yeast populations and their colonisation are no different, their growth rate is much faster, but this is relative to and proportional to their scale (size).

scaling yeast

Fig.4:  Sigmoidal growth/evolution over time curve applies to the growth of colonies of yeast. as seen in: figure 11.8 Sigmoidal growth by a population of the yeast Saccharomyoces cercvisiac . Data from Gause 1934 Or bacterial colonies is also Sigmoidal, but will rapidly decline in closed system such as a Petrie Dish (See:

Embryological development has a lag phase, quickly followed by an exponential growth phase and stabilisation – it is one again Sigmoidal. And I could fill a page with all the other instances, in other words, biological growth and development and even evolution is predictably Sigmoidal. Basically, as Thompson proposed: at every scale and level, things grow proportionally and in a predictable manner and can therefore be used to calculate otherwise unknown things in nature. This is the beauty of the mathematical formula.

scaling embryo

Fig.5:  Sigmoidal growth/evolution over time curve applies to the development and growth of an embryo. Then the embryos grew up to develop social media and the internet that also followed a lag phase, an exponential phase and finally reached maturity and stablisation.
scaling facebood
Fig.6:  Sigmoidal growth/evolution over time curve applies to the growth of social media: facebook being one well-known example.

What makes the Sigmoidal curve, not only important as a model for seemingly all natural systems that grow and evolve, from the internet (see above illustration of facebook, a successful mega-social media forum on the internet, which may be obsolete by the time you read this, if it hasn’t ground to a slow stabilised trickle by the time you are reading this. From cities to rail networks, as long as they continue to be required and fashions don’t change too much and from broccoli to bacteria, the Sigmoidal curve applies. However, biological life has apparently found a way of not becoming easily extinct, unlike our ‘manmade’ systems, and indeed, life has seemingly learned to beat the law of entropy (tends to disorder and decay) or become both an ‘open’ and ‘closed ‘system at the same time (look up the second law of thermodynamics). But essentially, our bodies may wear out and die, but life itself seems to go on and manage to keep doing against all the odds.

The overarching principle or growth/evolution and species formation according to natural forces that can be predicted, quantified and explained mathematically (or at least a lot of it, but certainly not all – such as the very beginning of it all – but it has tried and that is the topic of another book in the making), forms a major part of the larger evolutionary puzzle which will only probably make more sense when we have applied all that follows and much of what has been said already to the fossil record in Part Two of the forthcoming book: Evolution: If it didn’t Happen by Darwinian means, how did it occur?

D’Arcy Thompson’s model of evolutionary change (which forms Chapter One of the forthcomming book and initiates Part One, the alternative scientific explanations of evolutionary processess, which collectively form a natural distinct synthesis),  proposed that the eloquent underlying cohesion of Nature and biological life is discoverable and predictable at all magnitudes (scales) and therefore can be applied to trying to figure out the processes of evolution itself.

As the Sigmoid growth/evolutionary curve appears to be a universal (or almost universal as noted above by West), it is therefore not perhaps that surprising that it could possibly reveal things that we couldn’t otherwise know about Nature, such as the tempo of evolution itself.  For instance, the Sigmoidal universal gives us a clue the tempo and pattern of life and therefore an insight into evolution itself may have occurred, if it wasn’t by Darwinian means as for example, the ‘S’ Curve also clearly shows us the tempo of life in the Pre-Cambrian period (the lag phase), the explosion of life itself (the exponential phase) and finally, the stabilisation of the species as a whole (stabilisation).

It even looks like the Sigmoidal growth equation could be applied to the embryological development and reflect on a smaller scale, the greater Sigmoidal tempo of the development (evolutionary) of the species itself, as indicated within the alternative evolutionary model presented in Chapter Three of the forthcoming book and the fourth part of this present series within this blog.

The possible reason for all Natural things (and systems created by Natural things) being Sigmoidal, with an initial lag phase, followed by an exponential phase and then stabalisation, may never be understood, but seemingly it is the most efficient, time and space-saving way. As, seemingly, Nature abides by natural laws of area/space filling rules in relation to energy concerns and available resources (which is fractally definable/measurable), as the work of West and others suggests.

Follow this link to view a video [You Tube] where Geoffrey West gives a very good lecture on the topic, see Scaling Laws In Biology And Other Complex Systems (2012) Link

 The seemngly universal pattern of growth and development according to the Sigmoidal growth (evolutionary?) curve and a very simple equations, that being universal, should therefore, apply at every scale of life, as we have observed it for all other scales that can be quantified.  In principle, therefore, the pattern that evolution itself and the formation of the species should be knowable, eventhough, we do not a have an actual time machine to go back and observe the dramatic eruption and incredible adaptive moulding and shaping of life, we can at last begin to make more than a guess based upon assumptions, that are firmly grounded in true scientific understanding and application.

Everything seems to be a tiny whole system from its origin and reflected at every scale and magnitude and embedded within a larger whole system. This is a fractal system of self-similar repeating (reiterating) patterns at every scale. One pattern reflects the whole.

Here is a little thought experiment. Forget everything you have ever been told about evolution – Try, I know it is very difficult….Think of the principle of Russian Nested Dolls and how the very small (unformed and less defined) origin doll is made from the same materials and directs the scaled up form of the next  leap of complexity. Nature will seemingly continue to grow more complex versions of the same fundamental form according to available resources. She would appear to cut her cloth to her measure (the old axim used to refer to tailors) and like an expert nested doll maker, each form is sculpted (shaped and formed) according to well established, tried and tested rules. Each increasing scale fits snuggly over the previous form, allowing more detail and complexity to be filled in and expanded upon. This is the overarching principle of the emerging concept of The Nested Doll Evolutionary Model.

nested dolls


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