December 08, 2019

No matter the diversity of animals, cities, and companies, most of the characteristics within these groups scale quantitatively with their size. In a sense, a dog is 90% a scaled up version of a mouse.

The commonalities of complex systems give hope to finding a way to understand why they behave in particular ways.

All processes require energy. The amount of energy that these processes require to continue is their metabolic rate. Although we primarily use this notion for living organism, we can expand it to include cities and companies.

When energy gets used, it also produces a certain amount of entropy -- energy that doesn't do any useful work. This excess energy is the cause of damage and degradation of the systems that produce it.

Most of the relationships related to scale do not relate in a linear fashion. Just because something is twice the size of something else doesn't mean it will need twice the energy.

Scaling relationships come in two varieties:

- Sublinear relationships decrease the need per capita as size increases. -

Superlinear relationships increase the need per capita as size increases.

A complex system is categorized by simple components who's actions add up to

more than the linear sum of their output. Often times, these systems take on a life of their own outside of their individual components. We are great examples of a complex system.

Simplicity enables complexity when combine with self organization. The fractal nature of self organizing systems explains why properties rarely scale linearly.

In biology the number 4 comes up again and again when related to scale. As an organism's size increases, it's energy needs increase by 3/4. While the heart rate decreases by 1/4 of the size.

No system can sustain unbounded growth. This paints a grim future for our economies and population that has grown without pause. The only way we kept it growing for this long is through technological innovation, but to sustain further growth the innovation needs to happen even more rapidly. Once we hit the limit of our system, we will either need a reset to start the growth again, or the entire system will collapse in catastrophe.

The properties of systems increase at different orders of magnitude when scaled. For example: If an object is scaled by one order of magnitude, it's area will increase by two order of magnitude, and it's volume will increase by three orders of magnitude.

The strength of an object is determined by it's cross-section -- it's area. Therefore, if the area -- which determines strength -- increase by 2 while the volume -- which determines weight -- increase by 3, then the object will soon grow too heavy to support itself.

The above strength to weight ration demonstrates why the proportional strength decreases as the size increases. This is why an ant can lift 100x it's weight while we can only lift about 1x.

Logarithmic scales allow plotting order of magnitude increases within a limited space. Without this scale, the representation of exponential growth would require a giant scale.

Although many aspects change with scale, there are certain invariant aspects to every system and stay consistent -- like average heart rate for all people.

Scaling drug dosages from animals to people, or even to kids of different ages remains a challenge. Going from one organism to another rarely has identical interactions, and the scaling remains non linear even in the same organisms.

It remains difficult to determine truly invariant factors from simply slow changing ones.

The BMI index incorrectly because it takes weight increase by a factor of two, while it should use a factor of three -- corresponding to the increase in volume.

If we wish to surpass physical limitations of growth, we must innovate by changing materials, design, or the environment surrounding the system.

Scaling theory enables simulations at a fraction of the cost and effort of building the real thing.

Scale invariance represent relationships within a system that don't change regardless of scale or measuring system. These invariances enable the expression of scale as a mathematical formula.

Despite many systems having accurate mathematical representations, these formulas often because uselessly complex. Instead of calculating the solution outright, solving these problems require modeling or simulations.

It's unlikely that there are "Universal Laws of Biology," but it is likely that there are formulas that provide quantitative understanding of biological systems. For example, we may never have the ability to calculate how long you will live, but we might figure out how to calculate the maximum age you could live.

For much of it's existence biology has been a qualitative science without a unifying theory underlain by mathematics.

All the processes of life, even sleep, require energy. The amount of energy an organism needs to sustain basic operation is governed by it's metabolism.

As the size of an organism increases, it's metabolic rate only increases by a factor of 3/4. This provide every doubling in size with an energy savings of 25%.

Almost all biological scaling laws are multiples of 1/4. This thread leads to a commonality among the diverse aspects of life and provide a strong case for a uniting principle that's far less random than natural selection.

Many systems in biology have networks at their hearts. Animal and plan circulatory systems are perfect examples.

All biological networks follow three guidelines:

- Space filling dictates that effective networks to stretch everywhere within

the system and have terminal units that touch every component. - Invariance of terminal units leads to the edges being the same no matter the scale; like power outlets in a building of any size.
- Optimization pushes every network to reduce the work requires to transmit the blood, electricity, or information.

Area preserving branching allows flows through the network with minimal interference or back feed. In combination with the three biological network guidelines, this effect produces fractal networks. This means that the network looks almost identical at varying scales.

The fractal nature of the networks critical to life add a forth dimension to the three dimensional nature of volume present in everything. It guides and limits the growth of organisms, cities, and even companies.

Popular opinions become entrenched in the tissue of society whether it's valid or not. For people to look at measuring length differently, it took Lewis Fry Richardson noticing discrepancies in the length of borders while investigating the cause for war.

By noticing that as the scale decreased, the size of borders increased, Richardson accidentally discovered a fractal dimension.

The fractal nature of object mean that the rougher an object becomes, the closer it gets to gaining another dimensions. In the example of borders, a single dimension border with a jagged enough coast line gains mathematical properties of a two dimensional object.

Mandelbrot expanded on Richardson's finding by noticing that the fractal state was the norm in the natural world and not the exception -- as many Euclidean idealists would like to believe.

Fractals lead to the unique scaling properties of plants, living organisms, cities, companies, and even markets. The influence of fractals spread to anything that maintains a similar structure at different levels or behaves similarly at different time scales.

Copyright © Artem Chernyak 2020