Since the surface area of the skin through which heat can be expelled grows in size by a power of two, and the amount of energy produced is proportional to the volume, so size by a power of three, one would rather expect the elephant and any other animal to boil and not cool down - they should maintain metabolism proportional to weight by a power of two-thirds.

**There can be no mistake, however: it is three-quarters rule that stubbornly appears in the laws of nature noticed by scientists.** In all forest ecosystems, from the Amazon River to the tundra, the relationship between tree species density is inversely proportional to a tree's mass by about three-quarters. The rate of growth of plants also depends on the mass of three-quarters. The same relationship relates the thickness of the tree trunk to the weight of the plant. The territory an animal needs to survive also increases with its weight. In what ratio? Three-quarters of the way, of course.

### Why does biology have such a penchant for three-quarters?

According to American ecologists James Brown and Brian Enquist, and Geoffrey West's Los Alamos physicist collaborating with them, **the key to solving the mystery is the observation that the size of the organism depends on the efficiency of the internal transport system - water, oxygen, and nutrients. Therefore, cockroaches cannot grow too large, because they are not allowed to do so by the constraints of the respiratory system.**

The efficiency of the internal supply system also seems to explain the mystery of why whales grow so big. Indeed, marine mammals are impressive in size, and fish are not as big as they are. What is the cause of these disproportions?

**First of all, mammals find it easier to breathe than fish, because while a liter of seawater at a temperature of 10 degrees C contains only 3.07 milliliters of oxygen, the same volume of air contains 50 times more oxygen.**

Both groups of animals also solved the architecture of the circulatory system differently. Fish has kept it simple: they have one universal bloodstream. Mammals are equipped with two circuits: the pulmonary circuit, which is under low pressure so as not to damage the crown of the vessels surrounding the gas exchange vesicles, and the other one - forcing blood into all corners of the body. The latter has a much higher pressure that allows blood to be pushed to the ends of the body.

### Life is a fractal

**The branches of trees, a tangle of veins and arteries, a network of nerves and branches of the respiratory system are amazingly similar**. Even more than similar - all of these structures have what mathematicians call self-similarity. What does it mean?

The larger cauliflower rose looks like a whole plant, a branch of a feathery fern leaf - like a whole leaf. **Figures whose parts have a structure identical to the whole and which, when enlarged or reduced, do not seem to change, are called fractals by mathematicians. The world is full of such creations.** Geometry discovered them relatively late, previously it was limited to the study of classical figures - circles, triangles, or rotational ellipsoids. You can see how difficult it is to put together real pictures from them by looking at the paintings of cubists.

Figures of traditional geometry are not self-similar: a strongly reduced circle looks like a point, while the enlarged one - deceptively resembles a straight line. In the real world, objects do not simplify when they zoom in, but - like fractals - they reveal successive levels of their complex structure.

Traditional figures always have overall dimensions. So we have one-dimensional lines, two-dimensional objects on surfaces (triangles, circles, polygons), and three-dimensional spheres and polyhedra. Fractals complete this collection with dimensions expressed by all possible fractions that determine their complexity. Fractals drawn on a sheet of paper have dimensions between one and two, and spatial fractals, like a cauliflower rose, have dimensions between two and three, for example, the hemoglobin protein has a dimension of 2.4.

### Everything that lives, from trees to animals to single-celled organisms, nourishes itself through a "tree-like" structure.

**Everything that lives - argues the three researchers - from trees to animals to single-celled organisms - feeds through the "tree-like" structure.** The network of blood vessels, respiratory vessels, tree xylems, and even mitochondria, or "power stations" of cells, all have a similar, fractal shape. **Clearly, the general equation of life will not contain two-thirds - the surface-to-volume ratio, as would be required by traditional geometry - but a fraction depending on the dimension of the fractal lattice. According to the calculations of the West trio, Brown, and Enquist will be three-quarters of it.**

### WBE theory

The theory was published in 1997 in Science and named after its authors __The WBE Theory__ caused a storm in the biological community. Almost simultaneously with its publication in the pages of American Naturalist, a work by Polish scientists from the Jagiellonian University, Jan Kozlowski, and January Weber, appeared, presenting a competitive approach ...

To be continued in the next episode.

*The author of the article is Irena Cieślińska, and it was originally published in Polish in "Przekrój Nauki" magazine, no. 8/2008*

*Translation from Polish: Empowerment Coaching*

**See also: **

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