1. The dimensions of a right triangle can be described using symbols to represent the three legs, such as x, y and z. We do know that the symbols for a given triangle must represent the same dimension type, and should have the same scaling factor when equated to a numeric value. Disregarding whether there is a practical use, it really shouldn't matter what dimension type is used to describe the legs of a right triangle just as long as they represent the same "unit of measure". A "unit of measure" can be a single dimension such as temperature, length, quantity, or a compound dimension, such as gallons per hour, kilometers per second or moderators per forum. Individual triangles structured from single or compound dimensions present no unusual mathematical problems. When triangle pairs are used to express relationships between "units of measure" the rules change depending upon the mathematical characteristics of the relationships.

Considering just a pair of right triangles, I would like to present four different sets of conditions.

(Case 1) Create two different triangles that use different "units of measure" that are proportional by definition or physical law. By linking a change in the elements of one triangle to the same elements in the second triangle there will be a proportional change. If you didn't know the value that defines the proportionality between the two triangles it can be extracted by division of like elements of the triangles.

(Case 2) Create two triangles that use "units of measure" that are inversely proportional. Examples would be the relationships between voltage, current and resistance, or time, distance and speed. In this case the value that creates the proportionality between the other two values can be obtained by either division or multiplication of like elements of the triangles, depending upon which "units of measure" were used to describe the triangle legs. Although there is an inverse relationship, no single parameter of the inverse relationship can be considered a "constant of proportionality" unless such is done by definition, which would change the rules, which comes to Case 3.

(Case 3) Create two triangles that use "units of measure" that are inversely proportional where the value that defines the proportionality is an invariant "constant of proportionality", which means that value cannot be used as a dimension that defines the legs of a triangle. Although this case is somewhat analogous to Case 1, the value that defines proportionality cannot be extracted by dividing or multiplying like elements of the triangles, but can be extracted by multiplying unlike elements of the triangles. This creates a condition where two triangles can be shown to be related when the cross products of the individual unlike elements of the two triangles give the same result.

(Case 4) Create two triangles that use "units of measure" that are inversely proportional where the value that defines the proportionality is an invariant "constant of proportionality", and one of the like horizontal or vertical leg elements of both triangles will be held as a constant. This creates somewhat of a dichotomy as it would seem to violate the validity of the value that is the "constant of proportionality", but in actuality it makes the relationships between the other elements that define the dimensions of the triangle pair a function of the angle.

Are there any references to mathematical work in this area?

I did pursue what I thought was an interesting set of dimensions that involved Case 4 relationships, but I think the concept of triangle pairs needs to be examined first.  2.

3. Huh   4. When wavelengths and frequencies are used to define the elements of a right triangle, over 2,000 years of Western mathematics and science are merged into a pair of geometric relationships that permit the basic units that describe the velocity of electromagnetic waves to be defined mathematically.  5. A right triangle pair, one dimensioned using wavelengths and the other dimensioned using frequency creates the condition wherein the constant of proportionality between an electro-magnetic wavelength and frequency is the product of the leg of one triangle and the hypotenuse of the other. When the two products are equal, each triangle is the inverse of the other.

However, the legs of the right triangles must be defined in a "unit wavelength" and a "unit frequency", but a unit frequency will not have a value of one.

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

By the 1870s the scientific establishment had the mathematical and technical knowledge to discern what is contained in the above article, but they wouldn't have known the source of the practical "unit wavelength", that becoming known in 1951.  Bookmarks
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