Sequencing and Mapping Pythagorean Triples | Steve Wait – October 8, 2021

Of late, regarding the Harmonious Pythagorean Tetrahedra, I thought it prudent to consider both apex singularity solutions inclusively. The resulting object is a concave hexahedron where the base right triangle exists in the medial plane.

By manipulating the associated geometric construction, the requisite Pythagorean areas are distributed among the lateral faces of the upper and lower “tetrahedra”. In other words, I divided the area of each lateral face in the single-solution case (tetrahedron) and applied half to each in the two-solution case such that their sum achieves Pythagorean compliance for the developed hexahedra.

This exploration led to a fun tangent of interest as detailed below. Note that simply changing the common divisor of the equation below will produce different integer sequences and thus a different mapping of Pythagorean Triples. In this particular example I have chosen a common divisor of 2.

Established is a right triangle where a is the short cathetus, b the long and c the hypotenuse.

As an extension of 0 = c² – b² – a² and when applied to a Pythagorean Triple, the square root of c squared divided by two, less the same for b and again for a, yields the negative square root of an even integer.

√(c²/2) – √(b²/2) – √(a²/2) = -√+ℤ

For example, where: a = 3 , b = 4 , c = 5 

-√+ℤ  = √(5²/2) – √(4²/2) – √(3²/2) 

-√+ℤ  = √(25/2) – √(16/2) – √(9/2) 

-√+ℤ  = √12.5 – √8 – √4.5 

-√+ℤ  = 3.535533905932738 – 2.82842712474619 – 2.121320343559643 

-√+ℤ  = -1.414213562373095 

ℤ  = 2 

Solved for both primitives and multiples, these integers then sorted in ascending order produce the following sequence that when mapped provide an alternative “tree” of Pythagorean Triples.

2,8,8,18,18,18,32,32,32,50,50,50,72,72,72,72,72,72,98,98,98,128,128,128,128,162,162,162,162,162,200,200,200,200,200,200,242,242,242,288,

288,288,288,288,288,288,288,288,338,338,338,392,392,392,392,392,392,450,450,450,450,450,450,450,450,450,512,512,512,512,512,578,578,

578,648,648,648,648,648,648,648,648,648,648,722,722,722,800,800,800,800,800,800,800,800,800,882,882,882,882,882,882,882,882,882,968,

968,968,968,968,1058,1058, …

Mapping result:

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In the graph below, an incomplete population is depicted. Each integer occurrence is represented by the a/b ratio of a Pythagorean Triple. Curves indicate the ascending sequential paths of integer single occurrence with lower integer values corresponding to larger a/b ratios.

Tags: geometry