FunNature & Animal3 mathematical constants in nature

3 mathematical constants in nature

Mathematics has proven to be the most useful tool for the study of nature and for the rest of the sciences. Nature is full of patterns and proportions. From the arrangement of the petals in a flower to the growth of the population of a colony of bacteria, and from the trajectory that a hawk takes when pouncing on its prey to the shape that an octopus’s cloud of ink generates, geometric shapes appear and equations that can be modeled mathematically . And in those models certain values arise that are mathematical constants.

Today is March 14, International Mathematics Day, and Pi Day. It is only fair that we pay tribute to them. We will deal with three mathematical constants that appear repeatedly in nature.

The number e and the spiral of the mollusk

There is a type of spirals in mathematics that are called logarithmic .
They start from an origin and are separated in such a way that one lap maintains a geometric proportion with respect to the next. One property of this type of spiral is that if you draw a circle around its center, the angle it makes with the spiral at the point of intersection is always the same , regardless of the size of the circle. And that angle is defined by a logarithm , hence its name.

The shells of molluscs —and of many other animals— respond to approximations of natural logarithms. These are the ones that use the number e , or Euler’s number, as a base. It is an irrational number, that is, whose infinite decimals are never repeated, and its value is approximately 2.718 281 828…

This is due to the way the shell grows. At a given time of mollusk growth, its shell grows approximately to the same extent as the mollusk itself, with a constant tilt to the axis of symmetry, genetically determined. In this way, the angle of the spiral with respect to the circumference remains constant throughout the life of the animal. The variation that is observed between the real proportions of the mollusc shells and the mathematical model that defines them is sometimes so small that extremely precise and exact measurements are necessary to find it.

The golden ratio and the Romanesco

1, 1, 2, 3, 5, 8, 13, 21… ; This sequence has a peculiarity. Starting with the number 1, each subsequent number is the result of adding the two previous numbers . It is the Fibonacci sequence , and the relationship between each number and its previous one defines a proportion. It changes in each link of the chain. If we start at the beginning, that proportion is equal to 1; then 2; then 1.5; 1,666…; 1.6; 1,625; but each time it is bounded more and more, until, when the sequence becomes infinite, it ends up defining a number, which we call the golden number or number φ ( phi ). It is also an irrational number and its value is approximately 1,618,033,988…

Let’s go to the plants. At the apex, the end where they grow, some tissues called meristems are distributed, which produce branches, leaves or floral pieces. We call the arrangement of these structures phyllotaxis, and they always follow regular spiral patterns.

In each species, the organs are distributed in a specific way, but in all cases a general pattern is preserved, which is defined by two values: the number of turns to the stem in each spiral , starting from one piece and ending in the next that is in the same position (m), and the number of pieces in each spiral (n). In this way, with a division, the fraction of the turn between one piece and the next is obtained: m/n.

One peculiarity is that the number of laps is not random. Each spiral can go around the stem once, two, three, five, eight… but never four, six or seven. The value “m” follows the Fibonacci sequence. Similarly, the value “n” also follows the same sequence. However, the most amazing thing is that they do not do it independently. We do not find in nature any plant that has, for example, an angle of 5/21, although both values correspond to Fibonacci numbers. The number “n” is always two numbers after “m” in the Fibonacci sequence.

In this way, we find leaves that are distributed in 1/2; each spiral occupies one turn and carries two sheets. Or, 1/3 ; each spiral is one turn and has three leaves, as in beech. Or 2/5 ; each spiral occupies two turns and carries five leaves, like the oak. Or 3/8 like the sunflower. Or 5/13 like the almond tree…

But the most characteristic thing about this type of pattern is that, when the pieces are close together, they generate clearly visible spirals. That’s what happens in romanesco , that broccoli that looks like a fractal, and it does so with good reason; it really is a natural approximation to a mathematical fractal . A fractal in whose background, thanks to the Fibonacci sequence, the number φ is hidden in plain sight.


The ubiquitous number pi


Its approximate value is known to all: 3.141 592 653… Like the previous ones, it is an irrational number, and its definition is also clear: it is the relationship between the length of a circumference and its diameter. Given this fact, the number π is present everywhere there is a circle, fragments of one, or whatever is obtained from a circle.

For example, a sphere.

The mathematical formula for the volume of the sphere is 4⁄3 π r3 ; where “r” is the radius

So wherever a sphere exists in nature, the number π is hidden. From cell nuclei to pollen grains. And the shapes that approximate the sphere, like the eggs of birds, although they require more complex formulas to be modeled, also have the number π in their composition.

And why the sphere? Because a sphere, by definition, is that volume that presents all the points of its surface equidistant from a central point, and also, it is the way in which a given volume can be enclosed using the smallest surface, or using a given surface, contain the largest volume. It is the optimal shape to maintain hydrostatic balance and this makes it the most stable, whether to speak of the tiny nucleus of a cell or the largest of stars.


Cortie, M. 1992. The form, function, and synthesis of the molluscan shell. En I. Hargittai & C. A. Pickover, Spiral Symmetry (pp. 369-387). World Scientific. DOI: 10.1142/9789814343084_0019

Jean, R. V. 1992. On the origins of spiral symmetry in plants. En I. Hargittai & C. A. Pickover, Spiral Symmetry (pp. 323-351). World Scientific. DOI: 10.1142/9789814343084_0017

Pérez Morales, C. 1999. Morfología de espermatófitos. Ed. Celarayn.

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