Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles

Authors

  • Chetansing Rajput M.B.B.S. Nair Hospital (Mumbai University) India, Asst. Commissioner (Govt. of Maharashtra)

DOI:

https://doi.org/10.24297/jam.v20i.9088

Keywords:

Metallic Ratio, Pythagorean Triangles, Pascal’s Triangle, Golden Ratio, Pythagorean Triples, Metallic Ratio Triads, Right Triangle, Pythagoras Theorem, Metallic Mean

Abstract

The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more “Metallic” than the Pentagons, Octagons or any other (n2+4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means.

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References

Rajput, Chetansing (2021). "A Right Angled Triangle for each Metallic Mean". Journal of Advances in Mathematics. 20: 32–33. https://en.wikipedia.org/wiki/Metallic_mean#cite_note-15

Rajput, Chetansing (2021). Golden Ratio. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 19–42. https://doi.org/10.24297/jam.v20i.8945

Rajput, Chetansing (2021). Metallic Ratios : Beyond the Golden Ratio; The Mathematical Relationships between different Metallic Means. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 158-166. https://doi.org/10.24297/jam.v20i.9023

Rajput, Chetansing (2021). “Metallic Means : Beyond the Golden Ratio, New Mathematics and Geometry of all Metallic Ratios based upon Right Triangles, The Formation of the Triples of Metallic Means, And their Classical Correspondence with Pythagorean Triples and p≡1(mod 4) Primes, Also the Correlation between Metallic Numbers and the Digits 3 6 9”, JOURNAL OF ADVANCES IN MATHEMATICS, 20, 250–266. https://doi.org/10.24297/jam.v20i.9056

Rajput, Chetansing (2021). Metallic Means and Right Triangles: The Geometric Substantiation of all Metallic Ratios JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. https://doi.org/10.24297/jam.v20i.9029

Rajput, Chetansing (2021). Golden Ratio and other Metallic Means: The Geometric Substantiation of all Metallic Ratios with Right Triangles. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 174-187. https://doi.org/10.24297/jam.v20i.9034

Rajput, Chetansing (2021). Metallic Ratio Triads, The Mathematical and Geometric Relations between different Metallic Means, Metallic Numbers and the Right Angled Triangles. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. ttps://doi.org/10.24297/jam.v20i.9077

Rajput, Chetansing (2021). Metallic Ratios and Pascal’s Triangle : Triads of Metallic Means in the Pascal’s Triangle. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. https://doi.org/10.24297/jam.v20i.9078

Rajput, Chetansing (2021). Metallic Ratios and the Digits 3 6 9, Mathematical Relations between different Metallic Means, And the Special Significance of Digits 3, 6, 9 in the realm of Metallic Numbers. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. https://doi.org/10.24297/jam.v20i.9076

Rajput, Chetansing (2021). Metallic Ratios, Pythagorean Triples & p≡1(mod 4) Primes : Metallic Means, Right Triangles and the Pythagoras Theorem. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. https://doi.org/10.24297/jam.v20i.9075

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Published

2021-07-15

How to Cite

Rajput, C. (2021). Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 312–344. https://doi.org/10.24297/jam.v20i.9088

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