Posted on 07/26/2002 11:24:55 AM PDT by Dengar01
Timing Suspicious On Mysterious Suburban Crop Circles
Could Eerie Mystery Have To Do With Movie Release?
Is it a case of mysterious crop circles -- or an elaborate movie hoax?
A soybean farmer in Naperville said the broken, concentric rings that appeared in a field off Diehl Road left him scratching his head.
"Have you ever heard of something so crazy?" Steve Berning said. "Unbelievable."
Berning said the circles appeared last weekend and damaged more than 10 percent of his 8-acre field.
The circles do resemble similar ones seen in England, but in this case, the timing of their appearance in the western suburb is a bit suspicious.
Two weeks from now, "Signs" hits the big screen. The movie starring Mel Gibson involves -- you guessed it -- mysterious crop circles.
William Leone, an investigator with the Mutual UFO Network, said soil analysis could determine whether the circles have human or extraterrestrial origins.
But Illinois Farm Bureau spokesman Dennis Vercler scoffed at that idea.
"Since I don't believe in UFOs -- at least not soybean-destroying UFOs -- I have to assume whoever did this did it intentionally as a malicious prank," Vercler said.
Meanwhile, Berning doesn't seem overly upset about the circles.
"There's some damage, which upsets me," Berning said. "But I'm more curious than anything. I"ll always be asking questions."
First, let me buy you a clue: real Mathematicians don't go around giving attribution every time they write down an equation somebody proved 200 years ago. Giving attribution the way you do is the signature characteristic of someone trying to impress people who don't know better.
Second, I have no interest in "bedazzling" you; in fact, I wasn't even thinking about you at all when I wrote it. I was thinking about the lurkers, and used it to illustrate my point, which was that e, i, pi, -1, and 0 are all important numbers in Mathematics, and are inter-related by the relation I posted. This is in contrast to your precious "phi" which notably is absent form Euler's relation.
Another interesting relationship that Euler discovered was Euler's phi function. [snip]
Another nice treatise on Euler's phi-function is provided at Euler phi-function[emphasis added]
FL, I hate to be the one to break the news to you: Euler's "phi function" ISN'T the same "phi" as the one in your precious "Sacred Geometry" with its "vibrational resonances." You do understand that Mathematicians use the same symbol to represent different things in different contexts, don't you?
No doubt you will protest that you never said it was.... which is true; but then the question becomes why would you post it in the first place unless you thought it was related to the issue at hand?
Euler's phi (or totient) function of a positive integer n is the number of integers in {1,2,3,...,n} which are relatively prime to n.You:
FL, I hate to be the one to break the news to you: Euler's "phi function" ISN'T the same "phi" as the one in your precious "Sacred Geometry" with its "vibrational resonances." You do understand that Mathematicians use the same symbol to represent different things in different contexts, don't you?He's dead, Jim!
Your extensive links on "phi" and it's relation to "Mandlebrot sets" and Fibonacci numbers, etc., merely reaffirms my previous objection: it, like all of "Sacred Geometry" is a Mathematical curiosity, a legacy of a mystical era in Mathematics, that today is nothing more than a hobby or curiousity for a few Mathematicians. [emphasis added]
You really aren't very good at reading comprehension are you?
I've put in the bold the two relevant words: "phi" and "it". As can be seen by anyone familiar with the workings of English grammar, the pronoun "it" refers back to the word "phi," NOT to Mandlebrot, or Finonacci, or anything else. Thus, my comment was about "phi" and not Fibonacci.
Please learn to read before misrepresenting what I wrote.
I mean, let's all go to see Signs and have a few beers (I mean a lot of beers) and then go to some farm and make a cool looking crop circles.
It sure would be a lot better than this pointless bickering that is going on.
I find the subject of crop circles to be very interesting.
It would be better to have a civilized and polite discussion of the topic rather than the current flame war that is going on.
Son, take it from me ... beer is good, but there are times when flame war is better.
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.
Well I do believe the Pythagorian Theorum still attributes to Pythagoras, and the Bernoulli Principle still attributes to Bernoulli. Fourier transforms still attribute to Fourier, and Maxwell's Equations still attribute to Maxwell. So don't try to be deceptive here, Dr. longshadow..
Giving attribution the way you do is the signature characteristic of someone trying to impress people who don't know better.
You must have forgotten about your own long list of attributations, such as in your comment, "and serious Mathematicians, like Peano, Cantor, Hilbert, Gödel, Whitehead, and Russell, didn't, and don't, spend their time working on it or worrying about what the value of "phi" is to a bazzillionth decimal place.
This is in contrast to your precious "phi" which notably is absent form(sic) Euler's relation.
Just because Euler didn't relate it in one of his famous formulas doesn't mean the number isn't important.
FL, I hate to be the one to break the news to you: Euler's "phi function" ISN'T the same "phi" as the one in your precious "Sacred Geometry" with its "vibrational resonances."
If you're trying to say that Euler's phi (or totient) function isn't the same as golden ratio phi, well no kidding Sherlock. PS: Will you explain to me why you keep trying to argue about "vibrational resonances"? Do you want to argue about oscillations at the atomic level, or resonance in general. I can't see what it has to do with anything I've said however...
You do understand that Mathematicians use the same symbol to represent different things in different contexts, don't you?
Duh, yep.
No doubt you will protest that you never said it was.... which is true; but then the question becomes why would you post it in the first place unless you thought it was related to the issue at hand?
Because there IS a relationship between the golden phi and the Euler phi function... :)
From On Approximate Harmonic Division of n by phi(n)
A Problem Proposal
Let phi(n) denote Euler's totient function phi(n), giving the number of natural numbers less than n and relatively prime to n. By number in this discussion, I mean natural number.
Consider the numbers n divided by phi(n) in approximately the golden ratio, i.e., numbers n minimizing |(k / EulerPhi(k)) - golden ratio phi|, where the expression ranges over all k with some fixed number of digits. In other words, for some r, n / EulerPhi(n) is a best approximation to phi, where n must be selected from the set of r-digit numbers. Equivalently, n is a r-digit minimizer of |(k / EulerPhi(k)) - phi|. Here, golden ratio phi = (1 + sqrt(5)) / 2. Written in increasing order, these numbers n, which I call the harmonious numbers, determine a sequence whose first few terms are
3, 9, 39, 117, 351, 507, 3417, 10251, 30753, 58089, 92259
(Note: These appear as sequence A065657 in the Online Encyclopedia of Integer Sequences by N. J. A. Sloane.)
For example, |(3 / EulerPhi(3)) - phi| = .118034 (approximately) is minimal for all one-digit numbers. |(117 / EulerPhi(117)) - phi| = .006966 (approximately) is minimal for all three-digit numbers.
Of course, n / phi(n) can only approximate the golden ratio, which is irrational.
Problems
- Can |(n / EulerPhi(n)) - golden ratio phi| be made arbitrarily close to 0? If yes, then find a function relating the accuracy epsilon to the number of digits of n.
- The listed terms have this property: for r = 1,...,5, all r-digit terms share the same set of prime factors. For example, all 4-digit terms have prime factors 3, 17, 67. Furthermore, all listed terms are multiples of 3. I conjecture that these hold in general.
Similar questions can be asked for the sequence of numbers n divided by n - phi(n) in approximately the golden ratio. (Consider instead the expression |(k / (k - EulerPhi(k))) - golden ratio phi|).
The first few terms of this sequence (the dual harmonious numbers) are
6, 10, 20, 40, 50, 80, 230, 460, 920, 5278, 10556, 21112, 36946, 42224, 68614, 73892, 84448
(Note: EIS Sequence A065758) Are all terms multiples of 2?
A Related Optimization
Consider the numbers n which, for some r, are r-digit maximizers of n/EulerPhi(n). The first few terms of this sequence are
6, 30, 60, 90, 210, 420, 630, 840, 2310, 4620, 6930, 9240, 25410, 50820, 76230.
(Note: EIS Sequence A065800) It is not hard to show that for r > 1, the first r-digit term of the sequence is the smallest r-digit primorial, if it exists. It remains to investigate the first terms when existence fails. It is also not hard to see that for r > 1, the r-digit terms are in arithmetic progression with common difference equal to the smallest r-digit term. For example, 210, 420, 630, 840 are in arithmetic progression with common difference 210. Obviously the r-digit minimizer of n/EulerPhi(n) is the largest prime of n digits.
I invite readers to communicate their solutions or comments by contacting me at the email address below. I will report any progress on these problems in this web page, and of course, acknowledge correct solutions.
Joseph L. Pe
iDEN System Engineering Tools and Statistics
Motorola Center
Schaumburg, IL
email:josephpe@excite.com
My list wasn't an "attribution." Do you actually not understand the difference between "attribution" and a list of names?
I've made no deception. My statement stands; the relation
doesn't customarily get attributed every time somebody writes it down, just as we don't customarliy give Pythagoras attribution every time we write down
It's not like they own the copyright to the equations.....
.... and the record will show I've made no claim regarding a doctorate.
Meaning what? They got paid?
BWAAAAAAAAAAAAAAAHAHAHAHA.
BWAAAAAAAAAAHAHAHA. You are truely laughable. They have a manual on how to create it in a few hours. Buy it and make one for yourself. Then spend another nickel and buy an improved personality.
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
| Fib(n) = | Phin – (–Phi)–n | = | Phin – (–phi)n | |
 |
|
|||
 5 |
5 |
| Fib(n) = |
|
||||||
|
|||||||
| 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): 5 )
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.
Perhaps you should study up a little on some of the links I've posted...
Meaning that anyone with half an ounce of common sense can see the irregularities and imperfections in the formation.
And for $19.99 I'll send you a blueprint for a UFO.
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