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the total volume of blood in the body—must be directly proportional to the volume of the body itself,
It is the mathematical interplay between the cube root scaling law for lengths and the square root scaling law for radii, constrained by the linear scaling of blood volume and the invariance of the terminal units, that leads to quarter-power allometric exponents across organisms.
organisms operate as if they were in four dimensions, rather than the canonical three. In this sense the ubiquitous number four is actually 3 + 1.
It fell to the French mathematician Benoit Mandelbrot to make the crucial insight that, quite to the contrary, crinkliness, discontinuity, roughness, and self-similarity—in a word, fractality—are, in fact, ubiquitous features of the complex world we live in.
the Platonic ideal of smoothness embodied in classical Euclidean geometry was so firmly ingrained in our psyches that it had to wait a very long time for someone to actually check that it was valid with real-life examples. That person was an unusual British polymath named Lewis Fry Richardson, who almost accidentally laid the foundation for inspiring Mandelbrot’s invention of fractals.
power law scaling is the mathematical expression of self-similarity and fractality.
Stimulated by his passionate pacifism, Richardson embarked on an ambitious program to develop a quantitative theory for understanding the origins of war and international conflict in order to devise a strategy for their ultimate prevention. His aim was nothing less than to develop a science of war.
he introduced a general concept, which he called the deadly quarrel, defined as any violent conflict between human beings resulting in death. War is then viewed as a particular case of a deadly quarrel, but so is an individual murder.
Thus, by analogy with the Richter scale, the Richardson scale begins with zero for a single individual murder and ends with a magnitude of almost eight for the two world wars
the frequency distribution of wars follows simple power law scaling indicating that conflicts are approximately self-similar.
they very likely reflect the fractal-like network characteristics of national economies, social behavior, and competitive forces.
he hypothesized that the probability of war between neighboring states was proportional to the length of their common border.
he turned his attention to figuring out how the lengths of borders are measured . . . and in so doing inadvertently discovered fractals.
Implicit in our measurement process, whatever it is, is the assumption that with increasing resolution the result converges to an increasingly accurate fixed number,
he discovered that the finer the resolution, and therefore the greater the expected accuracy, the longer the border got, rather than converging to some specific value!
fractality, is quantified by how steep the slopes of the corresponding straight lines are on Richardson’s logarithmic plots: the steeper the slope, the more crinkly the curve.
For more crinkly coastlines like those of Norway with its magnificent fjords and multiple levels of bays and inlets that successively branch into increasingly smaller bays and inlets, the slope has the whopping value of 0.52. On the other hand, Richardson found that the South African coast is unlike almost any other coastline, with a slope of only 0.02, closely approximating a smooth curve.
Mandelbrot introduced the concept of a fractal dimension, defined by adding 1 to the exponent of the power law (the value of the slopes).
Recall that a smooth line has dimension 1, a smooth surface dimension 2, and a volume dimension 3. Thus the South African coast is very close to being a smooth line because its fractal dimension is 1.02, which is very close to 1, whereas Norway is far from it because its fractal dimension of 1.52 is so much greater than 1.
In the natural world almost nothing is smooth—most things are crinkly, irregular, and crenulated, very often in a self-similar way.
Consequently, most physical objects have no absolute objective length, and it is crucial to quote the resolution when stating the measurement.
In this new world of artifacts we inevitably became conditioned to seeing it through the lens of Euclidian geometry—straight lines, smooth curves, and smooth surfaces—blinding
As Mandelbrot succinctly put it: “Smooth shapes are very rare in the wild but extremely important in the ivory tower and the factory.”
Perhaps of greater importance is that he realized that these ideas are generalizable far beyond considerations of borders and coastlines to almost anything that can be measured, even including times and frequencies.
For instance, it turns out that the pattern of fluctuations in financial markets during an hour of trading is, on average, the same as that for a day, a month, a year, or a decade.
The reason that being healthy and robust equates with greater variance and larger fluctuations, and therefore a larger fractal dimension as in an EKG, is closely related to the resilience of such systems.
In 1982 Mandelbrot published a highly influential and very readable semipopular book titled The Fractal Geometry of Nature.
It is claimed that the fractal dimensions of musical scores can be used to quantify the signature nature and characteristics of different composers, such as between Beethoven, Bach, and Mozart,
Terminal units therefore play a critical role not only because they are invariant but also because they are the interface with the resource environment, whether internal as in the case of capillaries or external as in the case of leaves.
Geometrically, the nested levels of continuous branching and crenulations inherent in fractal-like structures optimize the transport of information, energy, and resources by maximizing the surface areas across which these essential features of life flow.
a crinkly enough line that is space filling can scale as if it’s an area. Its fractality effectively endows it with an additional dimension.
This additional dimension, which arises from optimizing network performance, leads to organisms’ functioning as if they are operating in four dimensions. This is the geometric origin of the quarter power. Thus, instead of scaling with classic ⅓ exponents, as would be the case if they were smooth nonfractal Euclidean objects, they scale with ¼ exponents.
Unlike the genetic code, which has evolved only once in the history of life, fractal-like distribution networks that confer an additional effective fourth dimension have originated many times.
Quarter-power scaling laws are perhaps as universal and as uniquely biological as the biochemical pathways of metabolism, the structure and function of the genetic code, and the process of natural selection.
In marked contrast to this, almost none of our man-made engineered artifacts and systems, whether automobiles, houses, washing machines, or television sets, invoke the power of fractals to optimize performance.
Idealized mathematical fractals continue “forever.”
But in real life there are clear limits.
Unlike most biological networks, mammalian circulatory systems are not single self-similar fractals but an admixture of two different ones, reflecting the change in flow from predominantly pulsatile AC to predominantly nonpulsatile DC as blood flows from the aorta to the capillaries.
all mammals have roughly the same number of branching levels, about fifteen, where the flow is predominantly steady nonpulsatile DC. The distinction among mammals as their size increases is the increasing number of levels where the flow is pulsatile AC. For example, we have about seven to eight, the whale has about sixteen to seventeen, and the shrew just one or two.
This argument shows that only mammals that are large enough for their circulatory systems to support pulsatile waves through at the very least the first couple of branching levels would have evolved, thereby providing a fundamental reason why there is a minimum size.
Capillaries service cells, so the opening up of the network means that there is increasingly more tissue that needs to be serviced situated between adjacent capillaries as size increases.
However, there is a limit to how far this can be pushed. Each capillary, being an invariant unit, can deliver only so much oxygen to the tissue.
there is a limit to how far oxygen can diffuse before there isn’t sufficient left to sustain the cells that are too far away. This distance is known as the maximal Krogh radius, which is the radius of an imaginary cylinder surrounding the length of a capillary, like a sheath, and which contains all of the cells that can be sustained
Based on this, one can calculate how large an animal could be before the separation distance between its capillaries gets so large that significant hypoxia develops. This leads to an estimate of about 100 kilograms for the maximum size, roughly equivalent to the largest blue whales, suggesting that they represent the end of the line for the mammalian family.
You eat, you metabolize, you transport metabolic energy through networks to your cells, where some is allocated to the repair and maintenance of existing cells, some to replace those that have died, and some to create new ones that add to your overall biomass.
Why is it that we reach a relatively stable size and don’t continue to grow by adding more and more tissue the way some organisms do?
the network theory naturally leads to a quantitative derivation of how the weight of an organism changes with its age and, in particular, explains why we stop growing.
All mammals and many other animals share the same kind of growth trajectory that we follow, called determinate growth by biologists to distinguish it from indeterminate growth, typically observed in fish, plants, and trees, where growth continues indefinitely until death.
the rate at which energy is needed for maintenance increases faster than the rate at which metabolic energy can be supplied, forcing the amount of energy available for growth to systematically decrease and eventually go to zero, resulting in the cessation of growth. In other words, you stop growing because of the mismatch between the way maintenance and supply scale as size increases.
the increase in the number of supply units (the capillaries) cannot keep up with the demands from the increase in the number of customers (the cells).
