Attempts to define Pi as a definite arithmetic progression?

[RSKueffner]

Howdy ho. No reason for a welcome around here, it's not about me it's about the Mathematical Anti-Telharsic Harfatum Septomin, eh!? (I hope at least one of you are familiar with that guy) Nonetheless, I've become obsessed with the transcendental property, and thusly therein my familiarization, I've come to ask this. Pi is transcendental, we got that, however, just being transcendental doesn't mean it is only defined by series or summations or integrals or anything above basic arithmetic. So, why hasn't there been more attempts to define Pi as a definite arithmetic progression? I've been having some fun exploring Pi in correlation to Phi and a truncated icosahedron, but haven't had much time as of late to finish it. Again, nonetheless, what are some of the more famous failures attempting to describe the exact nature of Pi other than a series, summation, integral or etc.?

 

[robert Ihnot]

Euclid of Alexandria (325 - 265 BC) is the one who proved that the ratio of C over d is always the same, regardless of the size of the circle Pi was defined by Euclid as the ratio of the diameter of a circle to its radius.

 

[mrandersdk]

of cause you mean the diameter to its circumference :-) or maybe the other way around circumference to its diameter can't remember in which way I should write it but i mean

circumference/diameter

 

[RSKueffner]

Of course, this much I understand. But those values are nonetheless irrational. Described by method of measurement. If we knew the exact value of the circumference of a euclidean circle, and described it as say 2, we wouldn't know the irrational value of 2pi. We wouldn't be able to describe it exactly without symbolic representation using pi, or one of my aforementioned methods for determining it's value. Does that make sense

 

[robert Ihnot]

Does that make sense?

Sure it does, but mathematics is able to handle this problem.

The value of pi is irrational, but that poses no great problem. Now with the Greeks they discovered that the square root of 2 is irrational. Since they were used to marking things off on a line, they had assumed that for a small enough unit any two lines could be expressed in the same basic unit.

HOWEVER, that was not true, some lines being incommenesurable with others. This was a revolutional discovery at the time. The narrow view of atomitism was that, "All is a plurity of 1."

I think your question about pi woud equally apply to the square root of 2 and of many, many other cases.

 

[RSKueffner]

Ah, and so you see, this much I understand as well. I'm not refuting that. For you see, you can square the Euclidean circle if you were to treat it in Hyperbolic space, or so I believe I've seen a proof on that aforementioned topic. Even so, the analogy doesn't work here. I'm not talking about a realistic, hands on problem. Such is the beauty of math. We may handle it with a thought experiment and test it with numbers. I don't need to draw a perfect Euclidean circle, I'm asking what some of the more famous attempts at describing the ratio known symbolically as Pi exactly. As an example. Take a geodesic grid. Generate a function for its volume with respect to F for the frequency. Take the limit as F approaches infinity and set it equal to 4πR²/3 (Sorry, not yet familiar with tex). You could then, hypothetically define Pi as this big transcendental nested radical, whatever it may be.

EDIT: My apologies, I have been occupied for a few days and incorrectly recollected that you made an analogy to squaring the circle. Same idea with the square root of two however.