While researching the laws of ecosystems, I came across the Ben-Ford Law, a law about numbers that states that the first number in a number is often 1. The first numbers do not appear evenly between 1 and 9. This may seem like a strange law, but it seems to apply in many cases in the natural world.
Frequency of appearance of numbers 1 to 9
If they appear evenly, it should be 1/9 = about 11%, but the frequency of 1 is about 30%, and the frequency decreases as the number increases, with the frequency of 9 being about 5%. It can be expressed by the formula log10(1+1/n), which is strongly related to exponential growth. If there is a bacterium that doubles to 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, the number of leading digits that appear among these 10 digits is 3 1s, 2 2s, 1 each of 3, 4, 5, 6, and 8, which seems to roughly satisfy the Ben-Ford law.
Ben-Ford’s Law in Nature
Ben-Ford’s law is said to hold true for things that exist in nature, such as the drainage area of rivers and lakes, the elevation of mountains, the specific heat of substances, the molecular weight of compounds, blackbody radiation that objects emit depending on their temperature, and constants in physics and mathematics. If it is closely related to exponents and logarithms, you’ll probably find other things if you look for them. It also has the property of being scale-invariant, which means it holds true even if you change to a different unit system. It may also be related to the fact that there are many tree shapes and fractal structures in nature.
power distribution
There seem to be many things in nature related to power distributions, including the Ben-Ford law, that are easily linked to exponential logarithms. The size and frequency of earthquakes, the damage area and frequency of wildfires, and animal migration patterns known as Lévy walks all seem to follow the Lévy distribution, which is a type of power distribution. It may seem like statistical information, but there seems to be some law at the root of it.
The distribution of 1 is amazing
The bias in the frequency of appearance of the leading numbers appears to be attributed to exponential logarithmic phenomenon. The increase in microorganisms, as well as the tree shape and fractal structure that branch from the mainstream, are exponential. There seems to be a common rule in nature. With regard to molecular weight, there may be some fractal-like symmetry in the way molecules are generated. When it comes to physical constants, is this an indication that there are further rules that follow the rules? The distribution is amazing. The laws of the earth are amazing.
