From Bacteria to Elephants

Mathematical rules often appear in completely unexpected places. An example of that is the graph in figure below, taken from  https://openi.nlm.nih.gov/imgs/512/85/2751747/PMC2751747_1742-4682-6-17-2.png

mass-metabolism-scale

It turns out that from unicellular organisms to huge mammals, the relationship between mass and metabolic rate are linear on the log-log scale.  Many papers have been published trying to explain this relationship till now.  But there are more interesting connections. Prof. Geoffrey West in his research on cities and corporations has found similar relationships between variety of average parameters of cities (wealth, crime rate, walking speed) against population.  Here is a link to his very interesting Ted talk.

https://www.ted.com/talks/geoffrey_west_the_surprising_math_of_cities_and_corporations.

Such a straight line fit to data is a characteristics of power laws, i.e. relationships of the form y = a xk.  Such relationships are known to exist in diverse fields, ranging from linguistics and sociology to neuroscience and geophysics. Here is a link to the Wikipedia article for the interested reader.

https://en.wikipedia.org/wiki/Power_law

In physics such power law behaviors are also known to have close connections to the theory of phase transitions and renormalization groups.