Metallic Ratios and 3, 6, 9, The Special Significance of Numbers 3, 6, 9 in the Realm of Metallic Meanseans

This 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.


Introduction
The proponents of Vortex Based Mathematics will continue to make irrational claims, and their opponents will continue to debunk them on grounds of the Base-10 Number System. Let the both camps do their jobs with missionary zeal. Author's objective is just to appreciate the beauty of numbers and the special attributes of the numbers 3, 6 and 9, especially their unique patterns in the realm of Metallic Means.
Such intriguing pattern of the integers 3, 6 and 9 was introduced in the work mentioned in Reference [1].
The prime objective of this paper is to supplement that work and further illustrate the unique pattern of numbers 3, 6 and 9 in the domain of Metallic Numbers.
As a brief introduction, each Metallic Mean n is the root of the simple Quadratic Equation X 2 -nX -1 = 0, where n is any positive natural number.
If n is multiple of 3, like Bronze Ratio 3 or the Aluminium Ratio 6 or the Ninth Mean 9 and so on : here, the digital roots of Alternate terms of associated Integer Sequences and corresponding Lucas Sequences are 3, 6, or 9.

Mathematical Relations between different Metallic Means: The TRIADS of Metallic Ratios, and the Numbers 3, 6 , 9
If K, m and n are three positive integers such that n is the smallest of the three integers and Noticeably, n=6 forms such multiple triads: : Shaded Triads have been exemplified above.
And, just like n=6 exemplified above, every integer forms such multiple triads. Noticeably, every n th Metallic Mean can give precise values of various Metallic Means by the formula: Metallic Mean: mmax = kmax = (n 2 + n + 4)

Special Pattern regarding Numbers 3, 6 and 9 :
Consider the TRIADS of Metallic Ratios with various integer values of n, as shown below in Table 1.
Noticeably, in the following Table 1 : If n is NOT multiple of 3, the alternate values of m and k have their digital roots 3, 6, or 9.
And, if n is multiple of 3 : None of the associated ms and ks have their digital roots 3, 6, or 9.  : Alternate ms and ks have their digital roots 3, 6, or 9.
For n= 3, 6, 9 : None of the associated ms and ks have their digital roots 3, 6, or 9. : None of the associated ms and ks have their digital roots 3, 6, or 9. More remarkably, the number of Triads formed ( or the numbers of ms and ks ) increase noticeably for n = 6 and 9 For Even ns : the number of Triads exhibit noticeable rise at n = 6, 16, 26…….and so on.
For Odd ns : the number of Triads exhibit noticeable rise at n = 9, 19, 29…….and so on.
Moreover, it can be noticed from above Table : if n is multiple of 3, the Digital Root of | − | is 3, 6 or 9.
And, if n is NOT multiple of 3, the Digital Root of NONE of the |k − m| value is 3, 6 or 9. For Odd ns : the pattern based upon product of the Prime Factors of ( n 2 + 4 ) is observed.
In either case, the integers 3, 6 and 9 are conspicuous by their absence from these (m-n) or (k-n) values.
But, what's about integer 7 ? Consider another example with n = 34, as shown below.
Further, the Triads of Metallic Ratios with n = 9 and the multiples of nine 9 exhibit their own characteristic patterns.
The point is that the Numbers 3, 6 and 9 exhibit their very peculiar and distinctive attributes, in the dominion of Metallic Numbers.
Moreover, the idiosyncracy of 3, 6 and 9 is exhibited in several other such patterns of Metallic Means and their Triads.
And, if n is NOT multiple of 3 : the digital roots of [(m-n) + (k-n)] are invariably 3, 6 , or 9 , as shown below. Digital Root of (m-n) + (k-n) 9 3 6 6 3 9 On the last note, it is worth mentioning here that several other intriguing properties of Metallic Ratios and these TRIADS of Metallic  [12]. Further, all imperical formulae those provide the precise relations between different Metallic Means are described in the work mentioned in Reference [2].

Conclusion:
This paper illustrated certain intriguing patterns in the realm of Metallic Means, and the special attributes of Numbers 3, 6 and 9 therein. These integers 3, 6 and 9 are conspicuous by their peculiar numerical properties, particularly exhibited in the domain of Metallic Ratios.