Geometry and trigonometry are two mathematical processes that can be used to define the characteristics of a right triangle. Trigonometry uses actual lengths and angle values, where the lengths are linear values. There is another related mathematical process where a right triangle is defined in transverse waveform notation and angle values. Transverse waveforms can be described in both linear (wavelength) and angular form (radians). The angular form of a wavelength, multiples or fractions, can be defined in radians, a full wavelength being 2π radians. When a wavelength is defined in terms of its period in time this notation can be referred to as radians/sec or frequency.

The key to this "new" process is a universal constant, although not defined as such within SI units, the wavelength associated with the precession emission of neutral hydrogen (commonly referred to as the 21cm value). Within the formulation in the attachment, the symbol for this value will be denoted as λ_H.

The small pdf attachment describes just the basic features of the geometric-mathematical process.

What I was hoping for when posting the "Mathematically Perfect Triangle" was to elicit aid in defining the underlying mathematical relationships of the geometric structure that produced the symmetry. The process to produce the symmetry requires one element of the right triangle be "constrained" and I choose the vertical leg for the constrained element in my example, and it will always be equal to one.

The use of the linear and angular notation associated with transverse waveforms is "unconventional" but it results in geometric/mathematical relationships that have unusual scope. You can translate (rotate) the angle to 45 degrees and still get symmetry, but you have recognize what is and is not a variable.

I suggest that the process that produced the symmetry is another variant of the geometric/trigonometric forms, which are plane, spherical and hyperbolic geometry, and their associated plane, spherical and hyperbolic trigonometry.

2 originally post by currere
How can a triangle be more perfect than a equilateral triangle?

An equilateral triangle is geometrically symmetrical. Pure geometric symmetry is not what is being presented in Mathematically Perfect Triangle.

There is a Yahoo! Group for amateur and professional mathematicians who are specifically interested in triangles. If I recall correctly, there was quite a protocol for people posting to the message board -- a message was composed of an abstract, with lemmas, conjectures, and proofs. And there were many, many posts, on a daily basis!!! Astonishing really. You should go to Yahoo! Groups and look for it.

currere:

What use is your mathematically perfect triangle?

In other words: does it tell us anything we didn't know before? Can we use it to get any new results? What is it good for?

I have not catalogued nor do I expect to identify all of the useful characteristics that can be deduced from the "constrained" geometric/mathematical relationships presented in the concept .

I do have a slightly longer paper, Universal.pdf (17k), attached, which I have distributed to a number of individuals in the scientific/academic community, which explains some of the features of the relationships in more detail.

I have identified a few of the mathematical terms and processes that apply to the concept presented in the two pdf articles. The equation set that is presented represents a 4 x 4 matrix, specifically a "symmetric matrix" (Hermitian). The solution used to solve the matrix is "iterative algebra". The wavelength equation is a "reversible transformation".

There is nothing simplistic about the terms I have quoted, as there are expanded definitions associated with each one. Using the terms in my favorite search engine I received the fewest hits for "iterative algebra", but more hits when I used "iterative linear algebra".

The actual numeric calculations are relatively simple, explaining them in "mathematical terms" creates a degree of complexity.

I have added a figure to my original attachment that depicts a method of representing time using a right triangle. In MathPerfect04.pdf, the radius of the outer circle represents the time vector that is equivalent to the SI second. The inner radius represents a unit of time that is defined by the geometric-mathematical relationships. The article is available at the following URL also,
http://www.vip.ocsnet.net/~ancient/MathPerfect04.pdf