Fourier Series

The Fourier Transform

by Bruce E. Camber, Initiated: September 2019 Updated: December 14, 2020

March 13, 2015, The New Yorker magazine: An important introduction to: (1) the ubiquitous nature of the Fourier transform, (2) new concepts for us with which we can use to build our model of the universe, and (3) processes by which such concepts change the quality of our life and our understanding of the universe. You would not want to live without the functionality of the Fourier Transform! …nor could you.

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

Fourier series

Continuities and Symmetries:
Keys to the Harmonies of the Fourier Transform

It is a package. We cannot have one without the others.

Scale Independent:

All circles and spheres need to be reconsidered for applications within the Planck scale. Because our little model is scale independent, all such images apply within the finite-infinite transformation and within the Notations 1-to 201. Because Notation-202 is still “in process” or asymmetric, that scaling may well be limited.

Sphere Stacking-Packing, Fourier A to Fourier E

Sphere Stacking-Packing: Discovered online in 2014, this image gave us the visualization of the transformation from spheres to interior functions and line and finally the Euclidean-Platonic geometries. From Wikipedia…
Fourier A. “This wave pattern occurs often in nature, including wind waves, sound waves, and light waves. A cosine wave is said to be sinusoidal, because cos ⁡ ( x ) = sin ⁡ ( x + π / 2 ) , \cos(x) = \sin(x + \pi/2), which is also a sine wave with a phase-shift of π/2 radians. Because of this head start, it is often said that the cosine function leads the sine function or the sine lags the cosine.”
Fourier B. “The sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse).”
Fourier C. “A circularly polarized wave as a sum of two linearly polarized components 90° out of phase.” Also, see the Poincaré sphere.
Fourier D. “Animation showing four different polarization states and two orthogonal projections.” Also, see polarization states and circular polarization.
Fourier E. Celestial mechanics comes home to the Planck scale. Here, the Lagrange points are assumed to describe activity at the Planck scale.

Online Resources:

Video: Computing the Fourier series (MIT 18.03SC Differential Equations, Fall 2011), David Shirokoff, OCW, MIT, 2011. Special thanks goes to David (who has had outstanding careers at MIT, McGill).
Online Course: Differential Equations, OCW MIT Profs. Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis.

Six easy roads to the Planck scale, 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.gif

There is an inner transformation (pictured) 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 within our infinitesimal universe.

Of course, there will be much more to come