Posted July 26, 2013 by Dr. Henri Montandon in computational neuroscience
 
 

Using tetration for recursion: more movement for brain science

 

 

 

 

 

 

 

 

 

 

 

 

 

The problem of conceptualizing movement is one of the oldest human problems. Conceptual models are first static, and only grudgingly achieve movement. In human history, it took thousands of years to be able to accurately model movement, which is why Isaac Newton is so championed by scientists and philosophers alike.

Tetration by Period, (From tetration.org)

Recursion is a form of movement popular in mathematics and computer science. Nature has used recursion to bring us broccoli. Mandelbrot taught us that broccoli is recursive because it exhibits the same pattern through several different scales. He grew so excited by broccoli that he figured out a method for inventing/discovering self-similarity through infinite scales, aka fractals.

The fractal above is the Julia set of the function f(z)=2^z, (From tetration.org)

Nearly exact self-similar fractal forms do occur in nature, but I’d never seen such a beautiful and perfect example until, sometime after moving to Switzerland, I came across a chou Romanesco like the one [below] in a grocery store. This is so visually stunning an object that on first encounter it’s hard to imagine you’re looking at a garden vegetable rather than an alien artifact created with molecular nanotechnology. But of course, then you realize that vegetables are created with molecular nanotechnology, albeit the product of earthly evolution, not extraterrestrial engineering.

 (From:  http://www.fourmilab.ch/images/Romanesco/)

 

Every self-similar pattern in nature breaks down at some scale—at the level of molecules and atoms if not before. The last photo shows the tiny structures near the top level spiral. As the spirals get smaller and smaller approaching the vertex, the spirals that make them up have less and less lower level detail, with the tiniest being little more than bumpy spheroids.
(From: http://www.fourmilab.ch/images/Romanesco/).

For a long time, recursion in mathematics was confined to three families of operations: addition, multiplication and exponentiation. Tetration – the iteration of exponentiation – was little studied before advances in dynamical systems and easily available computer power made such study feasible. The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of  n, while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function.[i]

I don’t know of any use of tetration in consciousness science or neuroscience. I am presenting it here because it’s cool and because it is another way of forming patterns of great beauty and subtlety.

Some fun for the feeble minded. Start with a simple English sentence: Maybe they will be of use to you.  Make it recursive: Maybe they will be of use to you or maybe they will just be of use to you or maybe they might just be of use to you or maybe they might just possibly be of use to you or maybe they might just possibly somehow be of use to you or maybe they might just possibly somehow someday be of use to you or maybe they might just possibly somehow someday for some reason be of use to you.

Recursive sentences bring up interesting questions. Are all sentences (let’s keep it to English) infinitely recursive? Are there classes of infinitely recursive English sentences? If either of these is true, can they be proved?

And, of course, why has the brain been built to use recursion?

 


[i] Wikipedia contributors, “Tetration,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Tetration&oldid=563835966 (accessed July 24, 2013).


Dr. Henri Montandon