And where pray tell have I ever said that "Sacred Geometry" or "Vibrational Resonances" ARE a mainstream topic in modern Mathematics? Red herring. Never said you did.
Why should I prove something that I've never claimed?
Because proving that "Sacred Geometry" with its "vibrational resonances" is foundational to modern advanced Mathematics MIGHT just provide a rational basis on which to assert it would be a likely topic for Space Aliens to want to communicate, which is what this whole thread was about, right? And the surest way to prove to EVERYBODY here that "Sacred Geometry" with its "vibrational resonances" is foundational to modern advanced Mathematics would be to show that Math Departments in the top-50 Universities are TEACHING courses in it! Conversely, absence of such courses would strongly suggest that "Sacred Geometry" with its "vibrational resonances" is NOT foundational to modern advanced Mathematics, which in turn would suggest that it would be silly for Space Aliens to travel across the galaxy just to communicate it to us.
I've only stated that MANY mainstream concepts ARE in fact derived from that which IS termed "Sacred Geometry". I've proved it with the Fibnocci numbers and related material, as the Golden Mean IS ONE item of so-called "Sacred Geometry" from the ancients.[emphasis added]
Related; yes (as I have repeatedly stipulated). Derived; no. As I explained to you previously, modern Geometry is based on (that means "derived from") Hilbert's axioms, not Euclid's, and not from ancient "Sacred Geometry" with its "vibrational resonances." Furthermore, during my entire undergrad career as a Mathematics major, I never saw a single thing "derived" FROM "phi" (the Golden section ratio). You either don't understand the meaning of the word "derived" as it is used in Mathematics, or you simply don't understand Mathematics. Fibonacci numbers are related to "phi", but they aren't "derived" from it. Thus, "phi" is not foundational (i.e., part of the underlying axiom system) for Fibonacci numbers, Mandlebrot sets, or anything else you like to link.
As I said earlier, you harp and screech about these so-called "vibrational resonances". I only provided a definition that I had found concerning "Sacred Geometry" and had no other comment concerning it. YOU are the one that is treating it as a major item here, not me....[emphasis added]
Yes, it's YOUR definition (YOU chose it; you're stuck with it). In Mathematics, we are required to use the ENTIRE definition of a thing, not just the portion that suits us. That's what sets Mathematics apart from less rational activities, like interpreting crop-circles, for example.
Fibonacci numbers are related to "phi", but they aren't "derived" from it. Thus, "phi" is not foundational (i.e., part of the underlying axiom system) for Fibonacci numbers Do you care to restate that Mr. Wizard?
From Phi and Mathematics
You can use phi to compute the nth number in the Fibonacci series (fn):
fn = Øn / 5½
This method actually provides an estimate which always rounds to the correct Fibonacci number.
You can compute any number of the Fibonacci series (fn) exactly with a little more work:
fn = [ Øn - (-Ø)-n ] / (2Ø-1)
Note: 2Ø-1 = 5½= The square root of 5
In fact, have you ever heard of Binet's Formula for the nth Fibonacci number ?
We have only defined the nth Fibonacci number in terms of the two before it: the n-th Fibonacci number is the sum of the (n-1)th and the (n-2)th. So to calculate the 100th Fibonacci number, for instance, we need to compute all the 99 values before it first - quite a task, even with a calculator!
A natural question to ask therefore is: Can we find a formula for F(n) which involves
only n and does not need any other (earlier) Fibonacci values? Yes! It involves our golden section number Phi and its reciprocal phi:
Here it is:
| Fib(n) = |
Phin – (–Phi)–n |
= |
Phin – (–phi)n |
|
 |
|
 5 |
5 |
where Phi = 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ... .
The next version uses just one of the golden section values: Phi, and all the powers are positive:
| Fib(n) = |
| Phin – |
(–1)n |
|
|
Phin |
|
|
|
5 |
Since phi is the name we use for 1/Phi on these pages, then we can remove the fraction in the numerator here and make it simpler, giving the second form of the formula at the start of this section.
| We can also write this in terms of 5 since Phi = |
1 + 5 |
and –phi = |
1 – 5 |
: |
|
|
2 |
2 |
If you prefer values in your formulae, then here is another form:-
| Fib(n) = |
1.6180339..n – (–0.6180339..)n |
|
|
2.236067977.. |
This is a surprising formula since it involves square roots and powers of Phi (an irrational number) but it always gives an integer for all (integer) values of n!
Here's how it works:
Let X= Phin =(1·618..)n
and Y=(-Phi)-n=(-1·618..)-n=(-0·618..)n then we have:
n: X=Phin : Y=(-Phi)-n: X-Y: (X-Y)/sqrt(5):
0 1 1 0 0
1 1·618033989 -0·61803399 2·23606798 1
2 2·618033989 0·38196601 2·23606798 1
3 4·236067977 -0·23606798 4·47213595 2
4 6·854101966 0·14589803 6·70820393 3
5 11·09016994 -0·09016994 11·18033989 5
6 17·94427191 0·05572809 17·88854382 8
7 29·03444185 -0·03444185 29·06888371 13
8 46·97871376 0·02128624 46·95742753 21
9 76·01315562 -0·01315562 76·02631123 34
10 122·9918694 0·00813062 122·9837388 55
.. .... .... ..
You might want to look at two ways to prove this formula: the first way is very simple and the second is more advanced and is for those who are already familiar with matrices.
Since phi is less than one in size, its powers decrease rapidly. We can use this to derive the following simpler formula for the n-th Fibonacci number F(n): F(n) = round( Phin / 5 ) where the round function gives the nearest integer to its argument.
n: Phin/sqrt(5) ..rounded
0 0·447213595 0
1 0·723606798 1
2 1·170820393 1
3 1·894427191 2
4 3·065247584 3
5 4·959674775 5
6 8·024922359 8
7 12·98459713 13
8 21·00951949 21
9 33·99411663 34
10 55·00363612 55
.. ... ..
Notice how, as n gets larger, the value of Phin/5 is almost an integer.
So longshadow, would you say that Binet derived the formula for the nth Fibonacci number from phi, or did he relate the formula TO phi? Perhaps you should study up a little on some of the links I've posted...
ie. Vibrational Resonances
Yes, it's YOUR definition
Why don't you get back to me when you find that definition, as you're really wearing out my patience on that...