Fourier Series

The Fourier transform

The following excerpt is from The New Yorker magazine, March 13, 2015. It is an important introduction to the ubiquitous nature of the Fourier transform and how we build on good concepts, and how these concepts change the quality of our life.

You would not want to live without its functionality! …nor could you.

We thank Prof. Dr. Steven Strogatz of Cornell University who says the following:

Fourier series

Computing the Fourier series (MIT-David Shirokoff)

Note: Thanks goes to David Shirokoff and Ronald J. Adler, Hansen Experimental Physics Laboratory, Gravity Probe B Mission, Stanford University, Stanford, California 94309

True confessions. We have learned to follow every lead. While getting further and further involved with the study of pi (π),  we would slow down a little to make some guesses and postulations knowing full-well that we would be back to attempt to understand each function more deeply.

Symmetries in motion. The circle-sphere dynamics now transition from basic geometries, to symmetries in motion.

The work of a wide range of people, from Pythagoras (c. 570 – c. 495 BC) to Feynman (1918-1988) to Strogatz (1959-  ) to today’s thinkers and tinkerers within music theory, may find a key function and a new paradigm within the Fourier transform understood from the Planck scale.

File:Circle cos sin.gifThere is an inner transformation (pictured, above right) that I would equate to electromagnetism, Maxwell’s equations and Faraday’s intuitions. Then there are the outer transformations which I would equate with the de facto dialogue between Newton and Einstein and the current wrestlings regarding our understanding of gravity and loop quantum gravity (LQG).

See Sir Martin Rees’ Just Six Numbers.

The internal and external dynamic of spheres. The three dynamic images above are each, in their very special ways, based on our most ubiquitous, never-ending, never-repeating, dimensionless constant, pi.

If in the first emergence there are endless strings of spheres, could it also be a face of Planck Charge, Planck Mass, Planck Length and Planck Time? If part of that emergence is the inner transformation of pi, in what ways is it the face electromagnetism and an expression of Planck Time, Planck Length, Planck  Mass and Planck Charge? If another part of that emergence is the outer transformation, in what manner of speaking is this the face of gravity and yet another expression of Planck Time, Planck Length, Planck Mass and Planck Charge?

History will be a guide. How could so many applications not be tied to the fundamentals of our beginning?

The Fourier transform: Cornell mathematician, Stephen Strogatz, loves the Fourier transform and I thank him for his spirited re-introduction to its place within science and with the functions of pi.  If by starting at the Planck scale, the Fourier transform redefines electromagnetism and gravity, our best living scholars throughout the breadth and depth of all those applications will be encouraged to engage this construct.

Key Fourier applications just may be extended in ways to address age-old questions. I have a hunch that there are calculations between the inner and outer transformations that just might confirm Sir Martin Rees first number within his book, Just Six Numbers. It is the ratio of the strength of the electrical force to the gravitational force. Surely this might add an important twist to our understanding of gravity and charge and mass. More….

Fourier series, transform: The goal here is to bring everyday physics and mathematics to bear to grasp the foundations of our universe so there is nothing esoteric or extra-logical about it. How very satisfying it will be if key mathematicians throughout our history, people like J. Kepler-C.F.Gauss-T.C. Hales (cubic-close packing),  Poincaré-Feigenbam (period doubling bifurcation), and Fourier-Dirac-Strogatz (Fourier transform), are responsible for the concepts that describe and predict the behaviors of our infinitesimal universe.