the migration of schools of fish is a very spectacular scene. A school of fish may cover an area of several kilometers and contain hundreds of millions of fish. They swim in one direction and look like a whole, orderly and consistent from the satellite.

and so on, if you think about it, the field of view of each fish is probably only a few meters. There is nothing else to command, how did these fish synchronize to form such a large school of fish? Could it be controlled by some external force? The answer to this question

is given by mathematicians. It's surprisingly simple, and you can even simulate it yourself if you're familiar with ordinary differential equations. Schools of fish do not need any external factors, nor do they need a specific fish to lead. as long as each fish maintains a few simple principles, a large school of fish can be constructed with a field of view of several meters. different models involve different principles, but basically include the following two items:

1. Don't run into the fish around you. that is, if there is a fish very close to you, swim a little farther.

2. Don't be too far away from the fish around you. that is, if there is a fish in the field of vision that is far away from you, chase it forward a little bit.

these two principles will keep a certain distance between the fish and form the spatial structure of the fish school.

in addition, different models may add some other principles, such as

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-- maintain a certain speed. Relative to the school of fish swimming fast, deceleration, too slow to accelerate

this principle will allow the school of fish to produce a specific movement structure. Depending on the parameters, there may be three states: a school of fish swimming in the same direction, a school of circular fish and a school of round fish.

on this basis, some principles for responding to external conditions can also be added:

-swimming in the direction of higher food density.

this principle will make the group have the wisdom of foraging. Because the school of fish covers a much larger field of view than a single fish, the whole school of fish can "search" for food on a larger scale.

of course, whether these models are correct is a question of sub-fish, but they are indeed concise, reasonable, and can effectively explain the existence and movement of schools of fish. A similar swarming phenomenon occurs in other species, including birds, insects and even humans. According to this principle, some scientists are designing robot clusters to allow them to carry out large-scale activities when the distance of signal communication is limited.

simple interaction patterns on a small scale can show completely different rules on a large scale. This is true of physics, so is biology, and so is society, and the laws of mathematics at the bottom govern it. Maybe, as Pythagoras said, everything counts.

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