polygons wavelengths frequencies

[FrankM]

It seems reasonable that wavelengths or frequencies can be applied as the dimensions of the lines that form a polygon, but not mixed. One would also expect that a mutually equivalent polygon would be created when the inverse of the dimensions expressed as a wavelength (or frequency) is calculated.

 

Has the mathematics of polygon wavelength frequency relationships been developed? What area of mathematics would cover this type of relationship? What search terms would cover the subject?

[swansont]

I have no idea what you're talking about here. Can you rephrase it?

 

Wavelengths and frequencies are relates to oscillatory phenomena, and that doesn't tend to cause polygons to leap to my mind.

[FrankM]

The oscillatory phenomena is an inherent aspect of our physical existence. One needs to think in terms of the mathematical and dimensional definitions that have been applied to describe waves and their associated frequency and the common factor that results in their inverse proportionality. When applied to geometric forms, the dimensional elements that are common to their inverse proportionality are mathematically embedded in the geometric relationships.

 

By applying wavelengths and frequencies as the numeric dimensions of polygons, one can mathematically and visually observe the effects of the inverse proportional relationship (IPR). For permitted polygon forms, the resulting wavelength frequency geometric pairs will have two basic results, one a parallel pair and the other an inverted pair. Two basic polygon forms demonstrate the two types of pairs.

 

When you define the lines of an equilateral triangle using a numeric value that represent a wavelength, one can mathematically create another triangle using the IPR that will have numeric dimensions that represent a frequency. Nothing surprising, it is a paired geometric relationship, one equilateral triangle dimensioned as wavelengths and its pair dimensioned as frequencies.

 

When you define the lines of a right triangle (RT) at 45 degrees using a numeric values that represent wavelengths, one can mathematically create another RT using the IPR that will have numeric dimensions that represent a frequency, however, the legs and hypotenuse values have to be "inverted". (swapped?)

 

In the real world there are two types of waves, electromagnetic (EM) and non-electromagnetic. Additionally, the velocity of both types of waves are effected by the permittivity of the material in which they are allowed to propagate. We do not concern ourselves with the permittivity factor until it is necessary to apply the mathematicial concepts to real world problems.

 

I have been examining the geometric relationship process as it relates to EM waves, as we usually apply an accepted permittivity value as a reference to establish "velocity". What is of interest in the 45 degree form is that "time", its duration, becomes a function of the angle.

[swansont]

Please move this thread to speculations.

[FrankM]

The process can be approached from pure mathematics or applied mathematics. There is nothing speculative about it.

[uncool]

I am basically a grad student in math, and what you have written makes no sense. How do you get frequencies from lines?

And P.S. Swansont is a mathematician...

=Uncool-

[FrankM]

Its a manner of abstract representation.

 

We know that a true physical wavelength will not look like a straight line, but we give it a dimension that can be represented as a straight line. If a 21.106 cm line had been inscribed on the Pioneer 10 plaque, most intelligent species would realize that it is a representation of the wavelength of a particular electromagnetic emission.

 

I see no difference in using a straight line to represent a numeric value for frequency. If you wanted to represent two frequencies as lines with different lengths, one could readily deduce that the harmonics would be the result of adding the two lines together or the difference between the two lines. It would be a graphical method of representing relationships between two frequencies.

 

If I want to use lines to represent the numeric values of wavelengths, I see no reason why I cannot use lines to represent the numeric value of the inverse relationship. The relative "length" of the lines are representative of numeric values, and I see no reason why these "lines" cannot be applied to geometric forms.

[FrankM]

The URL provides a pdf page which illustrates 3 triangle pairs. Most will be familiar with the numeric values in the first pair. The second pair gives the values when rotated to 45 degrees. The third pair illustrates the fundamental basis of the triangle pairs.

 

The vertical leg was chosen to be the fixed value. When the cross product, equation (1), is zero a triangle pair is related.

 

http://vip.ocsnet.net/~ancient/Relationships.pdf

 

How would one best present the process mathematically?