I have spent some time apprehending the Fourier story, as well as the Fast Fourier Transform. The historical perspective is helpful.
One source linked the idea to Laplace as Fouriers teacher, but this soorce explains the interactions in more detail.
http://todayinsci.com/F/Fourier_JBJ/FourierPoliticianScientistBio.htm#seriesWe find that Fourier worked on various forms: bar,annulus, sphere and prism . When he tried to use the Fourier transform to describe the temperature gradient Lagrange objected. So his full ideas were written in an unpublished memoir. Of the forms the annulus and the sphere are to me the most fundamental, but the method itself clearly derives from rotationally dynamic systems . The important thing was the notion of period.
I think this term is misleading because the derived ratios are trigonometric and relate to a static form. Thus the superposition is always defined over a spatial region and should be called the bound form. Because of the nature of the description several parameters are used which means that the transform is inherently multidimensional
The introduction of t as a 4 th dimension in 3d case is misleading, if it is considered as time. It is another dimension that is used to coordinate data sets that describe a bound form. But in the 1 dimensional case it is precisely a parametric dimension used to "represent" time.
From Fouriers conception this method was a way of modelling or describing a form.
While it is correct to cite Euler, the true source of these ratios are the ancient Greeks, who derived trigonometry from Pythagorean philosophy and more ancient astrological practices. The works of Ptolomey in his Almagest particularly inspired the Indian astrologers to develop the sine ratio from the chord ratio.. The Arabic and Islamic scholars then initiated a centuries long calculation of the sine ratios for the right triangle in the semicircle. In the course of this the binomial expansion was developed and eventually the Multinomials expansion was studied in the formulation of spherical trigonometry and spherical geometry.
In Europe these things were hardly known, but in the Islamic and Indian traditions Multinomials forms were discussed in "algebra" , that is symbolic arithmetic, as a result of developing methods and algorithms to calculate the sine ratios by difference formulae. This is what is referred to as interpolation. However Newton and his acolytes De Moivre and Cotes, excelled in this area. In fact DeMoivre was accepted into the Royal Academy due to his astonishing papers on Multinomials! Both he and Newton shared a private joke. Newton had instructed De Moivre in this published but generally unread area of mathematics!
Further, in his mathematical papers and Waste book it is clear that Newton developed a deep apprehension of spherical trigonometry of the unit sphere and unit circle, and the imaginary magnitude
)
. It is also clear that Cotes together with DeMoivre took this to the furthest extent in the theorems on the roots of unity.
That Euler, the Bernoulli's and Lagrange were aware of this is undoubted, but it was not in their nature to give credit where it was due! This is something Fourier decried. Thus with this background we can see that Fourier made great use of available mathematical methods to support his own methodical investigation of heat transfer. Clearly inspired by Newtons observations on temperature he developed his geometrical differential forms in an attempt to describe the " form" of heat in certain bound forms. The logical objection to his initial concepts derive specifically from his spatial description. Heat transfer clearly has to be dynamic. He corrected one differential from a spatial into a time varying one, but few noticed or even understood the difference.
In effect he had created the first formulation of " wave" mechanics, but it was not until Helmholtz, Lord Kelvin and Lord Rayleigh that a corpus of able physicists were able to draw out the incredible applicability of what he had done. We might also include Navier and Stokes who struggled to describe fluid motion by differentIal equations which had simplistic solutions in this form.
It is an important aside that Kelvin was a great advocate of Fouriers analysis and his series. He apparently mastered it within a short period and felt it was of fundamental significance. He developed his own kinetic theories on the back of it. Thus his bitter opposition to Hamiltonian mechanics, based as it was on" nonsensical" imaginaries was in part fueled by his abreaction to the general ignorance of the Fourier transform.
When Gibbs, an American acolyte of Kelvin developed the fledgling vector analysis of mechanics which relied not on imaginaries but on good old trig ratios he fully supported Gibbs baudlerised version of Grassmann algebras!
Further he lead a private campaign against the use of Quaternions, forcing Maxwell to recant his erstwhile proclamation of them. The debacle ends when the American academies decline the introduction of Quaternions into the curriculum in favour of Gibbs vector and statistical mechanics, supported by Kelvin.
This is one of many shameful incidents in Academia. However, in Europe the value of Grassmanns analytical and synthetical method was being fully appreciated. Bill Clifford in England was among a number of English academics who were heavily influenced by the Prussian adoption of Grassmanns style. Consequently the imaginaries were never divorced from the original Grassmann lineal Algebra of Strecken.
This had a profound effect on the study of the Barycentric calculus and how it was presented, and the study and development of the Fourier transform, as well as the Laplace transform. These innovations did not pass Lord Rayleigh by and helped him considerably in his notes on Wave Mechanics.
Today the Fourier transform is hard to explain without the imaginaries, although it was and still is a fully trigonometric analysis. Its meaning however is more clearly expressed in Grassmann Ausdehnungs Größe, or what I now call Grassmann Twistors.
Grassmann twistors, in fact the whole Grassmann methods derive from a bound form: the line segment. This line segment is of 2 Types: ordinary and trigonometric. It turns out that circle arc segments can be represented by these trig line segment in a modified product sum AB + BC= AFC where AFC is an arc segment described by 3 points and AB , BC and BF are radii of the arc.
It is to be noted that this product sum encompasses the notion of a rotation being representable as 2 reflections or a single reflection in intermediary line segments or radii. But i am more convinced as time goes on that Grassmann thought of rotation as a projection of a line segment onto another or into another 'position". The trig line segments as vertical projections thus record this rotation. In this sense the cosine laws are projections onto other line segments but clearly elliptical or even hyperbolic.
Of these trig line segments the simplest are those in the unit circ le or sphere, and they represent circular arc dynamics..
In the context of this representation of rotation the "sign" takes on a differemt meaning. In a single line segment the "sign" represents a half rotation about some arbitrary and unspecified point. The principal orientation is preserved but the principal direction is contra in that orientation.
Howeer in a product of 2 line segments the sign" represents a "quarter " turn . To be precise it represents a cyclical rotation of the points by 1 out of the 4 possibilities, that is the next in the cycle. So far from meaning negative in the contra sense it actually means a rotation! Thus when we annihilate AB with BA w are in fact not making physical sense, because the form exists but not in its original orientation.
For example, if i turn a picture from portrait into landscape position the object still exists! However negation is really only about a discounting process, it annihilation! This is why our math often gives the wrong result in physical reality, because objects or entities do not appear and disappear as we imagine from relying on the math! These sculptures represent a Fourier / Laplace transform method with the transform and its inverse. The form is aome kind of vorticular spiral the inverse just demonstrates that the Fourier/ Laplace consists of small products that do not exceed the bailout condition, doing the inverse recovers the original form.
I have to say i had no idea what the original form was, as i simply made up a geometric series of Fourier coefficients!
f=1*exp(z-c)+0.5*exp(2*z-c)+0.25*exp(3*z-c)
z=f*exp(-1*z)+c
October 26, 2011, 05:04:54 AM
Dilber MC Theme by HarzeM