Why would a spider magnified 100 times be crushed by itself?

Today we are going to talk about a topic that is both scientific and interesting - "scaling law". Don't be scared by this high-sounding name, in fact, it is like a "magnifying glass" in our daily life, except that it does not magnify the size of objects, but the subtle and interesting relationships between objects.

First, let me tell you a story. Why would a spider that was magnified a hundred times be crushed by itself?

1. The tragedy of spider magnification

In science fiction movies and TV shows, we often see the land of Lilliputians and giants, or various monsters of gigantic size, such as a spider magnified 100 times (Figure 1). When you see these, you might say, "Wow, a giant spider can easily prey on small birds and mice!"

Figure 1 A spider magnified 100 times

But is this really the case?

Let's use the "scaling law" as a "magnifying glass" to take a closer look. First, from a geometric point of view, when the spider is magnified 100 times, its volume will become 1003 times, or 1,000,000 times, and its leg thickness (cross-sectional area) will become 1002 times, or 10,000 times, which means that the shape has "geometric similarity". From the perspective of force analysis, the body weight pressure (i.e., compressive stress) per unit cross-sectional area of ​​the legs has increased by 1003÷1002=100 times. This means that although the spider's body size has increased, the load-bearing capacity of its legs and body has not increased in the same proportion.

In other words, the spider's weight increased faster than its legs could support it! It's like if you suddenly gained a lot of weight but your legs remained the same thickness, the result is predictable - your legs are likely to break.

So, a spider that is magnified 100 times may eventually collapse because it cannot support its own huge weight. Doesn't this sound unexpected and interesting?

Interested students can refer to my personal WeChat public account "Medical Biomechanics" article: "Putting onions in a pig's nose - pretending to be an elephant". It mentions the principle of similarity, including geometric similarity and mechanical similarity. Mechanical similarity focuses on the proportional relationship between the mechanical properties of the model and the real object. It and the scaling law both involve the proportional relationship between physical quantities in the system. The principle of mechanical similarity can be applied to verify the scaling law.

Next, let’s look at what scaling laws are.

2. What is the scaling law?

Simply put, a scaling law is a law that describes how other related quantities change proportionally when one quantity in a system (such as length, mass, time, etc.) changes.

In nature, many phenomena follow scaling laws. They are like nature’s “codes” waiting for us to decipher.

1. Mathematical expression of scaling law

Scaling laws are usually expressed mathematically in the form of Y=cXk, where Y and X are related physical quantities, c is a constant, and k is the scaling exponent. The value of k is very critical, as it determines the rate at which Y changes with X.

When k=1, we call it a linear relationship. This means that Y and X grow at the same rate, for example, the more money you make, the more things you can buy, which is a linear relationship.

When k>1, we call it a superlinear relationship. This means that Y grows faster than X, just like compound interest investment, the longer the time, the faster the return grows.

When k < 1, we call this a sublinear relationship. This means that Y is growing more slowly than X, just like when you run, your speed may gradually decrease as the distance increases.

Scaling laws are ubiquitous in nature, from the relationship between earthquake frequency and magnitude, to the relationship between urban population distribution and size, to the relationship between Internet traffic distribution and user behavior... They are almost everywhere. They are like a "master key" in nature, revealing the secrets behind many seemingly unrelated but actually closely connected phenomena.

This universality is not accidental, but stems from the inherent characteristics of complex systems described by power-law relationships. In complex systems, there are often complex interactions and dependencies between the various parts, which make the overall behavior of the system show the characteristics of power-law distribution.

2. Manifestations of Scaling Laws

Isokinetic and allometric scaling laws are two manifestations of scaling laws.

Isokinetic scaling refers to the constant proportional relationship between the parts of an organism and the whole during its growth. This growth follows geometric similarity, that is, the size of an organism changes in proportion during growth or evolution. In the mathematical expression of the scaling law above, k=1 represents the isometric scaling law. A typical example is a frog, whose legs always change at a proportional rate to its body in the short period after metamorphosis. Isokinetic growth is determined by Galileo's square-cube law, that is, if an organism's length grows exponentially at an isokinetic rate, its surface area will increase by 4 times, while its volume and weight will increase by 8 times. However, this growth method is not common in real organisms because it may cause the organism to encounter physiological and mechanical limitations during growth, such as the "tragedy" of the spider enlargement mentioned above.

Allometric scaling refers to the change in the proportional relationship between the various parts and the whole of an organism during its growth. This growth mode does not follow geometric similarity, but rather exhibits a nonlinear proportional relationship. In the mathematical expression of the above scaling law, k≠1 represents the allometric scaling law. In biology, the allometric scaling law has a wide range of applications and important biomechanical significance. It reveals how the various parts and the whole of an organism coordinate changes during its growth to adapt to the needs of the environment and function. For example, as the weight of an organism increases, its metabolic rate does not increase linearly, but follows a power law relationship of about 3/4 (Kleiber's law), which reflects the optimization strategy of the organism in energy utilization and distribution. In layman's terms, scaling laws are the embodiment of structural optimization, energy saving and high efficiency. For example, compared with his childhood, the volume and weight of an adult has increased by more than ten or even dozens of times, but the growth of the heart is not that large, and his appetite has not increased that much.

Since allometric scaling laws are more common, the scaling laws generally referred to refer to allometric scaling laws.

3. Scale invariance: the “self-similarity” of nature

The power law relationship also has a very interesting property, which is scale invariance. This means that whether you observe the system as a whole or locally, the power law relationship they follow is the same. In other words, the system shows self-similarities at different scales, demonstrating the "geometric miracle" of nature.

This self-similarity is very common in nature, such as lightning, water droplets, snowflakes, trees, flowers, conchs, rivers, coastlines, mountains, clouds, brains, vascular networks, etc. It is also common in sociology, such as stock trends, etc. Their shapes and distributions show similar characteristics at different scales (Figure 2).

Figure 2 Plant shapes and distributions show similar characteristics at different scales

Fractal is a kind of pattern or structure with self-similar properties. Fractal is usually defined as "a rough or fragmentary geometric shape that can be divided into several parts, and each part is (at least approximately) a reduced version of the whole" (Figure 3).

Figure 3 When the Koch curve is infinitely enlarged, it will show infinitely repeated self-similarity

Fractal theory not only reveals the intrinsic properties of complex geometric shapes and structures in nature, but also provides us with a concise and effective way to describe complex systems. By ignoring the specific details of the system and focusing only on the fractal laws followed by its overall behavior, we can use fractal theory to better understand and predict the behavior of the system.

Seeing this, I wondered: What is my own self-similarity? Am I made up of countless different "small selves"? Do I have countless "same-self" individuals? Do I and they together form a "big self"?

3. Scaling laws in biomechanics

Next, let us turn the scaling law's "magnifying glass" to the interesting field of biomechanics.

Biomechanics is a science that applies the principles and methods of mechanics to quantitatively study mechanical problems in organisms. In this field, scaling laws also play an important role. The following example illustrates this.

1. Vascular networks and fractals

First, let's look at the vascular network in an organism. The vascular network is a typical fractal structure, which means that each of its parts is self-similar to the whole. In other words, whether you are observing a tiny capillary or a large blood vessel in the heart, or a bronchus or blood vessel in the lungs, their shapes and distributions follow the same rules (Figure 4).

Figure 4 Distribution of bronchi or blood vessels in the lungs

This fractal structure of the vascular network enables organisms to efficiently transport oxygen and nutrients to every corner of the body. This efficiency is due to the scaling law followed by the vascular network.

For example, elephants weigh thousands of times more than humans, but their vascular networks are able to supply their large bodies with sufficient oxygen and nutrients with amazing efficiency. This is because their vascular networks follow a specific scaling law, which allows the radius of the vessels and the blood flow to adjust proportionally with the increase in body weight.

Animal experiments have shown that the relationship between myocardial mass and coronary blood flow satisfies the allometric law, i.e., Qcor=kMmyo0.75, where Qcor is coronary blood flow, Mmyo is myocardial mass, and k is a proportionality coefficient that varies from individual to individual. According to this model, the myocardial mass supplied downstream can be calculated from the blood flow of a certain coronary artery (Figure 5), just like knowing how much water flow a pipe uses to irrigate farmland can tell you how much farmland can be irrigated.

Figure 5 Coronary branch blood flow perfusing the downstream myocardium

2. Basal metabolic rate and body weight

Basal metabolic rate (BMR) refers to the energy metabolism rate of the human body when it is awake and extremely quiet, and is not affected by muscle activity, ambient temperature, food, mental stress, etc. It determines the amount of energy required by an organism to maintain life activities.

The relationship between basal metabolic rate and body weight is not a simple linear relationship, but a power law relationship, that is, the metabolic rate is proportional to a certain power of body weight. This relationship is called the famous metabolic scaling law or Kleiber's law.

Kleiber's law states that for most organisms, their metabolic rate is proportional to the 3/4 power of their body weight, that is, the mathematical expression of the scaling law mentioned above becomes q=M34, where q is the metabolic rate and M is the body weight (Figure 5). This means that if the body weight of an organism doubles, its metabolic rate will not double, but only increase by about 80% (because 234≈1.8). If the body weight increases 10 times, the metabolic rate will only increase by about 5.6 times.

Figure 5 The “rat-elephant line” has become one of the most important and famous generalizations in bioenergetics. In animals of all sizes, metabolic rate is proportional to body weight to the 0.75th power.

This law may sound a bit abstract, but we can understand it through a simple example. Compared with a mouse, an elephant weighs tens of thousands of times less, but their metabolic rates are not tens of thousands of times different. On the contrary, according to Kleiber's law, the metabolic rate of an elephant is only a few thousand times higher than that of a mouse. This is because although the number of cells in an elephant is much larger than that of a mouse, the energy required for each cell is relatively small. This is because the body structure of an elephant is more optimized and can better utilize energy for life activities.

Kleiber's law not only reveals the intrinsic connection between the metabolic rate and body weight of an organism, but also provides important clues for us to understand the energy utilization efficiency and growth strategies of organisms.

3. Biomechanical properties

The strength of bones follows a specific scaling law. As organisms gain weight, their bones become thicker and stronger to support the greater weight. This change is not random, but follows a specific power law relationship.

The story of the spider that was magnified 100 times was told earlier. The scaling law is at work behind it. Because volume and weight are three-dimensional quantities, and cross-sectional area is a two-dimensional quantity, as the volume of an organism increases, its supporting structure (such as legs) needs to be thickened according to the scaling law to maintain structural stability. This is just like building a tall building requires a stronger foundation.

Likewise, the contractile force of a muscle follows specific scaling laws that depend on factors such as the muscle's cross-sectional area and fiber type. This allows organisms to adjust muscle strength and endurance according to different exercise demands.

4. Scientific significance of scaling laws

Now that we have seen some vivid examples of scaling laws and their applications in biomechanics, what is the scientific significance of scaling laws?

First, scaling laws help us understand the behavioral laws of complex systems in nature. Whether it is the vascular network, metabolic process or skeletal muscle system in organisms, they all follow specific scaling laws. These laws not only reveal the intrinsic connection between the structure and function of organisms, but also provide important clues for us to understand the evolution, adaptation and survival strategies of organisms.

Secondly, scaling laws have a wide range of applications in mathematics and physics. They provide us with a concise and effective way to describe complex systems, allowing us to better predict and control the behavior of the system.

Finally, scaling laws can also help us promote scientific and technological progress and innovation. Through in-depth research on scaling laws, we can develop more efficient, energy-saving and environmentally friendly technologies and products, and contribute to the sustainable development of mankind.

IV. Application of Scaling Laws in Scientific Research

Scaling laws and their related power law relations, scaling invariance, and fractal theory have wide applications in both natural and social science research.

1. Ecology and biodiversity

In ecology, scaling laws are used to study the relationship between species distribution, biodiversity, and ecosystem function. For example, the relationship between the number of species and the area of ​​an island follows a power law distribution, which is called the "island biogeography theory." Studies have found that the number of species on an island is proportional to a certain power of the island area, which is usually between 0.2 and 0.35. This law applies not only to natural islands, but also to artificial ecosystems such as urban green spaces and nature reserves. By understanding this scaling relationship, ecologists can better predict and protect biodiversity.

2. Seismology

In seismology, scaling laws are used to study the relationship between the frequency and magnitude of earthquakes. The famous Gutenberg-Richter law is a power law relationship, which states that the relationship between the frequency and magnitude of earthquakes follows an exponential distribution. Specifically, earthquakes of larger magnitude occur much less frequently than earthquakes of smaller magnitude, and this relationship can be described by a power law. Through statistical analysis of earthquake data, seismologists can better understand the laws of earthquake activity and provide a scientific basis for earthquake prediction and disaster prevention and mitigation.

3. Sociology and Urban Development

In sociology, scaling laws are used to study the relationship between a city's population distribution, economic development, and social structure. For example, Zipf's law states that the relationship between a city's size (usually measured by population) and its ranking follows a power law distribution. Specifically, the population of the second-ranked city is half that of the first-ranked city, the population of the third-ranked city is one-third of the first-ranked city, and so on. This law applies not only to urban population, but also to many other social phenomena, such as company size, language usage frequency, etc. This is the scale effect produced by "intensification". Through the study of these scaling laws, sociologists can better understand the driving mechanism of urban development and the evolution of social structure.

4. Physics and Materials Science

In physics and materials science, scaling laws are used to study the relationship between the mechanical, thermal and electrical properties of materials and their size. For example, at the nanoscale, many physical properties of materials change significantly, and these changes often follow power-law relationships. By studying these scaling relationships, physicists and materials scientists can design new materials with special properties, providing theoretical support for the development of nanotechnology and new materials.

5. Future Prospects of Scaling Laws

As a powerful tool, scaling laws have achieved remarkable results in many fields. However, with the continuous advancement of science and technology and the explosive growth of data, the application prospects of scaling laws are still very broad and promising.

Interdisciplinary integration: Scaling laws, as a universal method for describing complex systems, are expected to establish closer links between multiple disciplines such as physics, chemistry, biology, and sociology, and promote interdisciplinary research and cooperation.

Data-driven scientific discovery: With the development of big data and artificial intelligence technology, we can use more advanced data processing and analysis methods to study scaling laws and reveal deeper laws hidden behind the data.

New materials and technologies: Through in-depth research on scaling laws, we can design new materials and technologies with special properties and provide innovative solutions for energy, medical care, environmental protection and other fields.

Policy making and social management: Scaling laws can also be applied to the fields of policy making and social management, helping us better understand the nature and laws of social phenomena and providing a scientific basis for policy making and implementation.

In short, as an important tool for describing the behavior of complex systems, scaling laws have achieved remarkable results in many fields. With the continuous advancement of science and technology and the explosive growth of data, the application prospects of scaling laws will be even broader. Let us look forward to scaling laws continuing to reveal the mysteries of nature and promote the advancement of science and technology and the development of society in the future.

VI. Summary

Scaling laws may sound mysterious, but they are actually like a "master key" to nature. Simply put, scaling laws reveal the specific relationship between the size of an organism and certain physical quantities (such as weight, strength, etc.). In the field of biomechanics, scaling laws are even more ubiquitous. Whether it is a vascular network, muscles, bones, or metabolism, they all obey the laws of scaling laws. Although elephants are larger and eat more than mice, the energy consumption of each of their cells is lower than that of mice. It is true that "big guys also have little secrets."

Scaling laws are like the "magician of size" in nature. They make ants become "strongmen" but prevent elephants from becoming "jumping champions". They tell us that in the biological world, being big does not necessarily mean being strong. You have to follow the rules (scaling laws)!

So, next time when you see a small ant easily lifting food that is several times heavier than itself, or when you are amazed at the height of a flea that is hundreds of times higher than itself, or when you lament that you gain weight even when drinking water (Figure 6), don’t forget to “thank” the scaling law, which makes this world full of wonders and reason.

Figure 6 “Drinking water makes you gain weight”? I’m not afraid – as long as the fat grows on someone else! It’s not my legs that are broken by the weight.