1,720,964 research outputs found
Golden Ratio, Silver Ratio and other Metallic Means ; Geometric Substantiation of all Metallic Ratios
No full textThis paper introduces the concept of special right angled triangles those epitomize the different Metallic Ratios. These right triangles not only have the precise Metallic Means embedded in all their geometric features, but they also provide the most accurate geometric substantiation of all Metallic Means. These special right triangles manifest the corresponding Metallic Ratios more holistically than the regular pentagon, octagon or tridecagon, et
Metallic Ratios and Pascal’s Triangle : Triads of Metallic Means in the Pascal’s Triangle
Full text linkThis paper introduces the close correspondence between Pascal’s Triangle and the recently published mathematical formulae those provide the precise relations between different Metallic Ratios. The precise correlations between various Metallic Means can be substantiated with Pascal’s Triangle, as described herein
Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles
Full text linkThe 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
Metallic Ratios, Pythagorean Triples & p≡1(mod 4) Primes : Metallic Means, Right Triangles and the Pythagoras Theorem
Full text linkThis paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper illustrates the concept of the “Triads of Metallic Means”, and aslo the close correspondence between Metallic Ratios and the Pythagorean Triples as well as Pythagorean Primes
Metallic Ratios : The Mathematical Relationships between different Metallic Means
No full textThis paper introduces the precise mathematical relationships between different Metallic Ratios. Certain distinctive mathematical correlations are found to exist between various Metallic Means. This work presents the empirical formulae those provide the exact mathematical relationships between different Metallic Ratios.  
Metallic Means; The Geometric Substantiation of all Metallic Ratios
Full text linkThis paper introduces certain new geometric aspects of the Metallic Ratios. Each Metallic Ratio is observed to be closely associated with a special right triangle, which provides the precise fractional expression of that Ratio. This work explicates the geometric substantiation of Metallic Ratio, on basis of the right angled triangle which is the quintessential form of that particular ratio
Euler’s Number in Right Triangles: Euler’s Number in Right Triangles
No full textThis paper reveals an explicit and intriguing formula that yields the precise value of Euler’s number (e) from the side lengths of a right angled triangle
Metallic Ratio Triads, The Mathematical Relations between different Metallic Means, And Geometric Substantiation of Metallic Numbers with the Right Angled Triangles
Full text linkThis paper synergize the newly discovered generalised geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also illustrates the concept of the “TRIADS of Metallic Means”. The Metallic Means and their TRIADS can be geometrically substantiated, in an intriguing manner, as described in this paper
Metallic Means in Primitive Pythagorean Triples : Metallic Ratios substantiated in Pythagorean Triangles and other Right Angled Triangles
Full text linkThis paper elaborates the precise correlation between metallic ratios and the different families of primitive Pythagorean triples. This paper also discusses the explicit formulae those provide the mathematical relationships between different metallic numbers. Main purpose of this paper is to highlight the geometric substantiation of metallic means in Pythagorean triples and other right triangles, and the trigonometric expression of the metallic ratios
Metallic Ratios and 3, 6, 9, The Special Significance of Numbers 3, 6, 9 in the Realm of Metallic Meanseans
Full text linkThis work illustrates the intriguing relation between Metallic Means and the Numbers 3, 6 and 9. These numbers occupy special positions in the realm of Metallic Ratios, as elaborated herein
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