Originally posted by circlemaker
Dude... You can measure it with a stick and a string and it will still be 3.14159...
If the radius is 1, a half circle is pi. All numbers are relative to the distance between 0 and 1, including the length of a curve. It's pretty
simple. I've measured it myself.
You can't just throw fibonacci and phi at something and magically transform it.
The best connection I've found between phi with pi is this: 1/phi*2 = 0.309. When you're sine value is 0.309, the radian value is 0.314 (pi/10).
Yes, well - I think I have come to conclusion with this. As there is no straight line in nature, the lenght of the circles diameter - when
(maginarily) drawn will vary and thus approximate using the golden rule to the perfect straight line. When the (imaginary) line is measured by
following exactly the line, it is slightly longer than the exact approximation of pi value (3.141...). In a sense the value 3.144 does meet the
description of "circle's circumference to its diameter".
From The infernal prime spiral -page
At the beginning (Graph 1), we can only see 2 rotating arms in the spiral. But gradually (Graph 2 and 3), we can see 2 big arms in which we have
exactly 10 little arms.Why 10 ? I never used this number neither in the function nor in the program. Now, if we take x in degrees, we obtain a simple
Archimede spiral less and less dense (due to the prime numbers rarefaction). So, would it be an hidden property of pi ?
- Isaac Keslassy (11 Apr 1999), who has found that it wasn't a property of pi but a property of a range including pi (3.141 to 3.144) :
curious, no ?
[...]
- Erick Wong (03 Oct 1999), for his excellent explanations of all the phenomenas. Here are extracts from the mail:
The prime spiral on your web page is easily explained in terms of rational approximations to 2*Pi (this is why numbers close to Pi also work,
since they have the same approximants).
The best rational approximations for 2*Pi, given by the continued fraction 6+1/(3+1/(1+1/(1+1/(7+1/(2+...))))) are 6/1, 19/3, 25/4, 44/7, 333/53,
710/113, ...
Now, imagine plotting the numbers (6n+1) in radians. We get the spiral instead of a straight line because 6 is not exactly 2*Pi.
The spirals are counterclockwise because 6 < 2*Pi radians.
I think that 19/3 and 25/4 are not close enough to 2*Pi to be recognized, and 44/7 is much better. Well, Phi(44)=20 and that's why there are 20
arms (the two large "gaps" correspond to the spirals 44n+{10,11,12} and 44n+{32,33,34}, which are never prime. Now the spirals are clockwise because
44>2*Pi*7.
As we zoom out further, I expect to find spirals corresponding to the very good approximation 710/113. and ther should be exactly Phi(710)=280
arms.
So please forgive my clearly insane or ridiculous babblings.
A postulate arises: As phi is constantly found in the nature, it may be that the closest approximation of perfect circle the nature can produce could
be expressed using the formula 4/sqrt(phi).
edit on 12-12-2011 by JackTheTripper because: (no reason given)
) 
