golden mean

[AntonioLao]

Why the golden mean m is irrational could still remain a mathematical mystery. In certain respect it is one of the roots for the quadratic equation m+ m-1=0. This equation has two solutions: positive and negative but if the mean represents lengths and distances then only the positive solution is accepted as a physical quantity. Its approximate value is m≈0.618 033 989. Its reciprocal is 1.618033988. This clearly shows that the reciprocal of m is just m+1 and their product is unity: m(m+1)=1, expanding the parentheses by multiplying the factor m into each term inside give an expression equivalent to the quadratic equation given above.

[Profpat]

Not only is the math interesting but the Greek Parthenon and man himself follows the ratio of 1 to 1.6. A most beautiful ratio.

Best to you Antonio,

Pat

[mkirkpatrick]

As far as I am aware the pyramids were all built to this ratio.



regards michael.

[neutralino]

I'm not sure that it's a mathematical mystery: why shouldn't the golden ratio be irrational?

[AntonioLao]

why shouldn't the golden ratio be irrational

If I knew then it would not be personally mysterious. It is precisely the fact that it is irrational that it presents a mystery to me. I realized that the golden mean can be expressed as an irrational number greater than unity or one that is less than unity and yet their product is unity since they are the reciprocals of each other. furthermore, when expressed as continued fraction it tends to generate the Fibonacci sequence. This sequence appears in many natural growth processes. However, if structures conform to this golden mean, e.g. the Pyramid of Giza, etc. these seem to last forever while others would simply deteriorate quickly and be forgotten forevermore.

[Profpat]

Hi again Antonio;

I think that Pi and Phi are irrational numbers are a mystery in there own right. I agree with Pythagoras who, as I recall, was ticked off that the square root of 2 was an irrational number. I know I never liked the irrationality of irrational numbers when I was a kid taking arithmetic. And don't even bring up imaginary numbers.

Best to you sir,

Pat

[neutralino]

If I knew then it would not be personally mysterious. It is precisely the fact that it is irrational that it presents a mystery to me.

It appears my point didn't really come across properly. What I meant was that there are a whole lot more irrationals than there are rationals thus, to me, the fact that the golden ratio is irrational is no mystery. If it were a rational number, then it would be a little more mysterious, since rationals are more "special" since there are fewer of them.

[AntonioLao]

And don't even bring up imaginary numbers

What I dont understand is that why researchers keep using them? The latest are the phase factors of wave functions in Schroedinger's and Dirac equations of quantum field theories.

[AntonioLao]

then it would be a little more mysterious

It's relative to both of us since I'm standing at the side of rationality while you are standing at the side of irrationality. You understand them (irrationals) I dont while I understand them (rationals) you dont. The rationals are subfield of an abstract ring structure. They are commutative for both addition and multiplication but no operational inverses for multiplication.

[neutralino]

They are commutative for both addition and multiplication but no operational inverses for multiplication.

But the multiplicative inverse of a rational a/b is the rational b/a...?