Abstract
This 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, etc
Highlights
An intriguing geometric relationship exists between the Fibonacci-like Integer Sequence and the corresponding Lucas Sequence, associated with each Metallic Ratio
Just like the Golden Ratio, such classical geometric synergism can be generalised for all other Metallic Ratios
Consider the right triangle having its catheti in proportion 1:δn
Summary
An intriguing geometric relationship exists between the Fibonacci-like Integer Sequence and the corresponding Lucas Sequence, associated with each Metallic Ratio. An outstanding synergy between these two right triangles provides for the phenomenal and precise expression of Golden Ratio.[1]. The prime objective of this work is to introduce and elaborate the concept of such special right angled triangles those provide the geometric substantiation all Metallic Means. This paper introduces and generalises the classical geometric relationships between the Integer Sequence and the corresponding Lucas Sequence, associated with any Metallic Ratio. The 1:2:√5 triangle and 3-4-5 triple are invariably formed together, but they exhibit an outstanding synergy between all their corresponding sides and angles, that provides for the phenomenal expression of Golden Ratio.[1]. All corresponding angles and catheti of these two right triangles synergize with each other to reflect the precise Golden Ratio, as follows.
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