Can Area be Expressed in Terms Other Than Square Units? | Steve Wait – February 11, 2021

Preface

Having written prior about much of the following, further contemplation on this specific topic remains unavoidably necessary.

It is the exponent of 2 that clearly defines the square unit, that magic of 90° which affords the Pythagorean Theorem its achievement and just preeminence. The area of the parallelogram, brought to orthogonal compliance by Cavalieri, seems an excellent example of our insistence for conformity with the square. But this construct of square units, ingrained at a young age, makes difficult the notion of alternatives.

The idea of area absent a unit “squared” seems quite alien. So much so, I have found this concept requisite the constant suppression of what is so intuitive. Often, resistance seems futile, as many attempts are abandoned upon recognition of my subconscious efforts to reconcile in square terms. After great effort to relieve myself of this quadratic constraint, a question formulates; could exponents other than 2 offer valid context for area? This idea seems to have similarities with that of fractals.

Area basics are taught as the product of length times width, thus a square unit by exponent multiplication. However, exponent multiplication is unconstrained by the square as for any common base, only summation of powers is required. Values raised to the reciprocal power of 2 extract the square root. And the reciprocal of ℝ, the ℝ^th root of a value, does that provide a non-perpendicular linear quantity in the framework of area? This challenges the fundamental of multiplication being of an orthogonal nature.

This delivers me to the Pythagorean Theorem, the epitome of the square unit example. Like myself, most view the Pythagorean Theorem to be about area and not merely the sum of squares in the abstract. This, in my opinion, remains unchallenged. In fact, only reinforced by what follows. Using the Pythagorean Theorem as basis, well know its validity only for the right triangle, could not meager exponent manipulation offer solutions for triangles of lesser stature? And, if so, does raising the length of a triangle’s sides to a power other than 2 constitute consonant area? What might happen in the case of a^ℝ+b^ℝ= c^ℝ?

To attempt resolution, one must first arrive at a value for ℝ. My course of action in this regard employs (a^ℝ+b^ℝ)^(1/ℝ)= √a²+b²-(2ab x cos C). Having yet to put in terms solely of ℝ, if attainable, iterative solutions do affirm with the Law of Cosines and as such, ℝ is achieved for any given a : b ratio and angle C where C=cos^-1((-(((a^ℝ+b^ℝ)^(1/ℝ))²)+a²+b²)/2ab).

I should note that in circumspect I have used this approach prior. While not in the context of area, solving ℝ for measured data of a parabola has allowed me a means to assess eccentricity conformance with y = ax². The particular expression ℝ = (ln (y/a)) / ln x being used to that end.

This is applicable for the obtuse scalene, the right and some acute scalene triangles prior to the isosceles inversion. Below this point, for the remainder of the acute scalene, b^ℝ-a^ℝ = c^ℝ is required.

Beyond this two-dimensional confirmation of a^ℝ+b^ℝ= c^ℝ, in which the unit circle is also preserved, I put forward my idea of the Harmonious Pythagorean Tetrahedra as a means of further evaluation. This conjecture proposed as follows:

Given a base right triangle ABC with proportions a : b, a point D₁ exists above the plane where for any ratio between the limits of ¼ : 1 and 4 : 1, the surface area of ▽BCD₁= a², ▽ACD₁ = b² and ▽ABD₁ = c². Thus, will occur a four-sided polyhedron, being an oblique triangular pyramid, with right triangle base and corresponding lateral faces consonant with the Pythagorean Theorem.

As this offers point D₁ to be a unique solution (Note that the development, as described in my prior work, also provides a point D₂ below the plane of ◿ABC creating a bi-pyramid, symmetrical about the medial axis.) for the evolved Pythagorean compliant pyramid, will the same methodology hold for non-right triangle bases? Yes, in fact, unique three-dimensional solutions are realized for each, with their lateral faces of area derived from the ℝ exponent. Similarly, these too are constrained within sets of a : b limits. Thus, these tests offer optimism for the affirmative regarding the question posed in the title.

So, if not a square for an area with an exponent other than 2, representative of an angle other than 90°, then what? As diagramed below, consider this simple example of two isosceles triangles. Both triangles possess side lengths of 4 in. However, their areas differ. The challenge is to determine ∠ C for both. A single solution for each is possible. One can be quickly given away to be 90° as the math is fairly easy, while the other is requisite more rigor to conclude. Use was made of C = cos^-1(-(c²/2ab)+(a/2b)+(b/2a)) and the equations listed prior. For the obtuse triangle, the result is ∠C = 150° and a unit area represented as a rhombus, in this case, eight of them.

In this manner, it would seem that area can be expressed by exponents other than 2, for units not of the square. There may exist limitations, the most obvious being a value raised to the power of 1, which for the example given would yield a degenerate triangle (line) of ∠C = 180°. As stated at the onset, more evaluation is compulsory.

I intend to explore and hopefully better visualize real number exponents in a non-integer generalization framework of the Pythagorean Theorem. Perhaps this can be achieved in the complex plane.

Additionally, I will need to look at the fractal dimension connection with this topic.