Why is it that large important complex systems often have about 10-100 billion (or 1E10 to 1E11 give or take an order of magnitude) objects? This seems to apply to the number of stars in the Milky Way (3E11), the number of galaxies in the observable universe (2E11), the number of cells in the first mammal (1E10), the number of neurons in a human brain (1E11), the number of nuclide pairs in human DNA (3E9), the number of proteins in a typical eukaryotic cell (1E10), and the current population of earth (7E9). Yes, many of these are not well determined yet and various estimates exist.

In a complex system there are hierarchies, e.g., organs, tissue types, and cells in humans. Variation is needed such that certain objects in the system can specialize and support the hierarchy. Variations in objects lead to differences of performance among a set of characteristics, i.e., dimensions. If there are multiple dimensions and the overall performance is proportional to the product of the levels in each dimension, then the distribution of overall performance tends to be lognormally distributed, even if the performance within each dimension is normally distributed.

Lognormal distributions are seen throughout the natural world and in economics. They occur whenever there is a process dependent on many factors which are multiplied together. At the extreme ends a lognormal distribution might be seen such as the pay scale of major league baseball players.

If there are N characteristics (dimensions) which have polar opposites (i.e., the scale goes from 1 in one direction and -1 in the other, although both directions may contribute to a beneficial combination), then the probability of an object with a cumulative distribution probability of greater than 97.5% (a standard 2-standard deviation criteria for normal distributions with 2.5% in the two tails) would be (0.025)^N. To ensure there is a good chance of objects in these (2^N) corners of the hypercube of characteristics (for example, a cube with N=3 has 2^3=8 corners), then the number of objects should be about 1/probability or (40)^N.

Then the question becomes “What is the number of characteristics, N?” Based on Howard Gardner’s multiple intelligence, some have suggested 7 factors for intelligence while others have suggested more. These original seven factors included linguistic, logic, kinaesthetic, spatial, musical, interpersonal, and intrapersonal. For astronomical objects such as galaxies and stars, factors might include size, composition, age, time of formation, surrounding external environment, internal environment (e.g., a star’s planets), and activity such as magnetic fields.

With N=7, the number of objects necessary to have a good chance that the extreme (>95%) corners are occupied is 40^7 or 1.6^11. So although this is not an explanation, it is a hypothesis of what may be going on with this magic number. I hope that if you have different insights or have seen other discussions on this please let me know.

2.5E11 Number of stars in Milky Way https://en.wikipedia.org/wiki/Milky_Way

2E11 Number of galaxies in Universe http://www.physics.org/facts/sand-galaxies.asp

7E9: Population of Earth (peak a bit more)

1E10: Cells in first mammal (20-30 g): https://www.smithsonianmag.com/smart-news/there-are-372-trillion-cells-i...

1E11: 100 billion neurons in brain https://www.verywellmind.com/how-many-neurons-are-in-the-brain-2794889

3E9: 3 billion gene pairs in human DNA http://www.genomenewsnetwork.org/resources/whats_a_genome/Chp1_4_2.shtml

1E10: proteins in cells: https://www.ncbi.nlm.nih.gov/books/NBK21473/

Complex systems 2017 http://rsif.royalsocietypublishing.org/content/14/134/20170391

Log-normal distributions across the sciences: https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf