The Damp Mathematics of Evolution. Volume 0.

By on 20/07/2017

“The Damp Mathematics of Evolution” is a historical column in the Italian site MaddMaths! (here the Italian version), which is devoted to a mathematical approach to natural evolution. The author is Davide Palmigiani, a PhD student in Mathematical Biology at Sapienza Università di Roma. This is the English translation of the first episode.

When we observe Nature (Nature as “all that is alive”), we remain amazed, marveled, sometimes frightened by its incredible variety of forms and behaviors. It is pratically certain that a little-trained eye will conclude that, beyond every reasonable doubt, a miracle must have happened, some divine, extraterrestrial, mystical, esoteric, certainly intelligent entities must be intervened, because without “external” help, to create something so stunning is impossible. While observing the complicated patterns on the wings of a butterfly, the impressive camouflage of a stick insect, the surprising resemblance of behaviors between a child and a chimpanzee, our intelligence, capable of even thinking about ourselves, we can only repeat that something so complicated could not have been generated alone, without a “maker”. After all, without us, a palace would not stand up.

The Theory of Evolution explains that the Intelligent Design is not the only way of thinking. We must learn that the world we live in is the visible expression of a process of creation still in progress, that does not want to shape the Reality in the way we see it, based on some pre-printed project, but rather considering rules that select, for one reason or another, single forms between a plurality of beings. We observe this singularity without thinking about how and why it came to be, and so we need to find an external director, a demiurge.

If we learn about the rules of the Big Game of Evolution, we will end up to appreciate even more the damp swarm of life around us. We will find that these rules are written in the language of the branch of knowledge which, according to many, can not absolutely be defined “damp”: Mathematics. The image I want to recall to mind with the adjective “damp” is the one of Nature and of those who study it: biologists and ethologists, with boots immersed in mud to contemplate this or that species.


The portrait of the typical mathematician is quite different, much less tied to Nature, lost in the search for beauty in abstract reasoning.


In this article, we will try to understand the meaning of Darwin’s theory to be ready to look for events in Nature that can only be explained by collaboration of Evolution and Mathematics, where both disciplines may get dirty in looking for insects under this or that stone.

What is Evolution, in short?
Evolution is a phenomenon that acts on the fundamental entity of life: information. It is a recipe that needs three ingredients:

First ingredient: the information and its bearer.
Living beings are vehicles for information, both the one they use to live, both the one needed for their creation: the latter is preserved in the genes, DNA fragments, a almost two meters long molecule enclosed in our cells.

Second ingredient: a set of events that bring information to change.
In Nature there are many ways to generate variety, from random mutations in DNA, to genetic drift, to epigenetic modifications … etcetera ad infinitum

Third ingredient: an environment in which information lives, which poses challenges and selects only a few forms between the generated variety, the ones fitted to the environment.
This is the role that Nature itself plays, the theater of Evolution. Thanks to Paul Andersen’s video “Five Fingers of Evolution,” which brilliantly synthesize without trivializing, we can summarize the second and third ingredient … in one hand:

YouTube Preview Image

The little finger, small, represents variations in genetic information due to casual changes occurring when the population size falls drastically.

The ring finger, represents variations due to non-random mating between members of a population.

The middle finger, first letter M, is the main factor of variability, the Mutation of the genes.

The forefinger reminds us of the movement, the flow of individuals moving from one area to another, bringing with them new features, thus increasing the variability.

With the thumb we explain the adaptation, the environment that selects: changes that receive a “thumb down” because inessential or harmful are not selected because those who get them live worse; As a result, the “thumbs up” organisms have greater probability to pass their features to the offspring.

But what we care most is the first ingredient. The construction project of a living creature is engraved in the DNA, but it is not written as the instruction on an Ikea manual where every step is perfectly explained.


The approach is much simpler, incredibly less expensive from information point of view and very elegant: it works just like an axiom scheme (a purely mathematical concept), like Nonograms.


The game consists in blacken some of the boxes in the scheme, to reveal the hidden figure. The numbers next to each row and column indicate the groups of boxes that have to be blacken in the respective row or column: each number corresponds to a group of boxes and its value indicates how many boxes are composed. Between a group and another there is always at least one white box.

Guided example:


The fifth column has no group of black boxes, so it is made up of white boxes alone. In the fourth column there is a group of 1 and a group of 3, plus at least one white box in the middle; In total 5 boxes (at least) and therefore – since the total boxes are 5 – we can fill the column from start to finish: first a black, then a white and then the other 3 black boxes. In the first line there is only one black box, one that is already in the scheme. Then … the complete figure can be easily obtained.


Try to solve the above scheme before continuing to read will help you to understand the next part. The rules of Nonograms are few and do not help at all to understand what the final image will be, they give only explanations about how to achieve the goal but not about the image itself; likewise, the final form of the body is not written in the DNA, is not written that a hand should have five fingers, but rather how to create substances that, working together, will give birth to a little finger, a ring finger, a middle finger, a forefinger and a thumb. In Nonograms the lines and columns with the numbers are the equivalent of the genes and as such they contain everything that is needed for the image, encoded in a small space (a bunch of numbers). While solving the puzzle, we soon realize that the image that is forming is not created smoothly, from top to bottom, with some kind of linear logic. The final body is the result of the process of creation, an emerging entity, which seems more than the sum of parts that form it.

Finally, Nonograms become enlightening if we want to understand the relationship between mutations and selection, at the basis of evolution theory: mutations occur at the level of genes, selection at the higher level of the body size. Let’s take the scheme above (The one that produces a camel… ops! Hint!) and imagine that numbers changed randomly, that the first line has become 0 (instead of 1), the fifth column 5-7 (instead of 6- 7). The mutation is random and does not occur “in order to”, or “to avoid that”, are just the numbers that vary – the level of the genes. The new numbers will however give rise to a new image, a camel without ears (try to believe). At this level – the level of the organism – the selection occurs: a deaf camel can not hear the predators, “thumb down”. Probably the unlucky bearers of this character will fail to reproduce and will not transfer their genes over time. So a species slowly changes gaining benefits and passing them to the offspring.

Multiple levels, complex dynamics, axioms, emerging characters … math is hidden but lurking. In addition, in many cases the events have led the species to behave as true mathematicians: bees build almost perfectly hexagonal cells; some cicadas can count prime numbers; fractals, complex abstract constructions, appear on multiple levels on many occasions in the shape of plants, organs of animals; some particular numbers are recurrent in botany. In each of these cases Evolution has found mathematical curiosities usable by organisms. If we look at mathematical forms in Nature, we should not be surprised at all.

Over time we move towards forms more and more perfectly fitted to their enviroment, and it is “normal” that Mathematics is part of this perfection, because from Euclide’s “The Elements” is known as the subject that knows how to deal with “Perfect” forms.

Davide Palmigiani

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