Magic Numbers: The Feigenbaum Constant

Numbers are weird
February 12th, 2020  by Blaine Garrett

If you have even the minimal curiosity about the world, you are probably familiar with various "magic numbers" in nature. For example, you might be aware that objects falling can't accerlate faster than the constant g (9.8 meters per second squared). You may also be aware the Golden Ratio φ (1.618) is found all over nature. These sorts of numbers popup in math and science quite a bit and I find them fascinating. Perhaps it is my logical computer science driven mind, where everything is constructed from scratch, but it blows my mind that there are these seemingly arbitrary numbers scattered throughout the physical world.

One number I recently ran across was the Feigenbaum Constant, technically the first of two such named constants.  It is the magic number 4.67 (rounded the nearest hundreth).  First described in 1975, this magic numbers pops up in population growth modelling, fractals, chaos theory, and more. It was named after American Physist Mitchell J. Feigenbaum, who I envision as being the inspiration for Ian Malcom in Jurassic Park. According, to Wikipedia, the constant is "the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map".  One such map is the population formula that has a single paramater of growth rate. 

The Veritasium YouTube channel does an excellent job of setting up the number and talking about the growth rate formula and other applications. Give it a watch. It is great.

Also, for fun, here is a video of Ian Malcom doing his thing: