New Math Formula

[Eugene]

The introduction of a new math formula is causing heated debate in science quarters the world over. Named "trianglature formula," it states simply: "sqrt (pi) times diameter of circle gives a triangle with precisely the same area as that of the given circle, where triangle base is circle diameter."

The formula by itself merely confirms the centuries old quadrature formula which "squares" the circle. Heat of the controversy is over the accompanying statement that either formula (squaring or triangulating) shows the ratio of pi to be arbitrary, which smacks in the face of the longheld academic assertion that the traditional pi is sacrosanct.

[Tormod]

Eugene, this is very interesting.

 

Can you provide us with some links?

 

Tormod

[Eugene]

The only link found on Google, Tormod. is your own "Hypography" (cutting edge is what I guess we like about it). But did confirm it for myself using the rational ratio 355/113 and traditional 3.1416.

 

What I see as far more interesting than the academic squabble is that the triangle given is the isosceles, thus a parallax with precise area as the circle it references. Project this meaning to our cosmological calculations of the celestial bowl and the implications would seem highly significant. In this regard there is recalled the words of the late Princeton astronomer Chas. A. Young: " It is not our powers of observation that are so much at fault as our mathematics."

 

Good to see we're continuing to emerge.

[Eugene]

***CORRECTION***CORRECTION***CORRECTION***

 

Sorry folks, I not only miscopied but confused the two formulas. Here is the way the trianglature formula should read:

 

Correctly stated the formula does not directly relate to area but rather gives an isosceles triangle with height of the radius and side which defines the quadrant by a true right angle. It is the quadrature formula which gives the area of a square with precisely the same as that of the given circle, i.e., sqrt (pi) times radius.

 

The trianglature formula is given by the equation: diameter/sqrt (2) = side of the isosceles as defining by its chord the quadrant right angle. The significance of this formula is that it defines a precise right angle, thus the vertex of the triangle giving an exacting parallax - relying of course upon the accuracy of pi.

 

Again, sorry for the confusion. This should set it straight.

[UncleAl]

1) The "squaring of the circle" refers to operations using the Greek formula of only straightedge and compass. The Greek formula is formally a second degree equation. It is rigorously proven that a circle *cannot* be squared using only first and second degree operations .

 

2) A circle *can* be exactly squared if we are allowed to use higher order curves - Hippia's quadratrix, Tschirnausen's quadratix, Ozam's quadratix, the cochleoid, the archimedian spiral... but that is cheating on the Greeks.

 

3) Pi is pi. As it unites algebra and analytic geometry through Euler's equation, e^[(i)(pi)]=-1, you can't go screwing around with it.

 

4) The ratio of a circle's circumference to it diameter is only pi in plane (curvaure = zero) geometry. In curved geometries (elliptic, hyperbolic) there are additional terms in the ratio. Pi remains pi. Exactly.

 

5) Since sqrt(2) is an irrational number... You've been had.

 

6) You can transform a circle into straignt linear motion witha mechanical linkage, nullright here, but that isn't squaring the circle by the book.

[Eugene]

Uncle Al, thanks for responding to this controversial subject, which the pi ratio certainly is. Hopefully I can contribute to it, but for the most part will be straddling the fence, not having the mathematics acumen of most.

 

1) I have to hold to the pi quadrature formula given above, by reason that it refers to correlation of the line and arc solely by ratio and not by the operations of calculus, which assign a specific length to the line. We cannot define the absolutely straight line, just as we cannot define the perfect circle. The best we can do is correlate the two by ratio to some common point of origin.

 

2) I don't know how to distinguish between the higher order and lower order curves, so clearly lost there. It does seem to me, however, that every curve, spiralled or plane, has some point of origin and that point mandating a lined radius.

 

3) Not meaning to act smart, but yes I can screw around with pi. We've been screwing around with it now for some 4 millennia and still don't have a handle on it. I do certainly reject out of hand such approximations as given by the Archimedes and other line-referenced methods.

 

4) No, sir, pi is not exact. Any pi ratio, rational or irrational, runs to infinity. While a pi ratio has been introduced that describes the finite condition, itself is not finite but runs to infinity like the others.

 

5) It is inconsequential that the sqrt (2) given by the trianglature formula is irrational - so is the sqrt (3) - yet both relate to and confirm the 2:3 ratio which is the closest possible relationship that can exist between one thing and another.

 

6) I beg to suggest that not all things are by the book.

[bbhattac]

Although, I find the so called "formula" is not a formaula at all since it lacks precision, I would like somebody who understands what the formula is, to verify one statement made in this posting. That is the value of pi is arbitrary according to this formula. Can one please verify that this formula holds when pi = 4 ? If it is arbitrary, then we are free to assign any value right ? At least that is the meaning of the english word "arbitrary".

 

thanks,

 

Bhaskar Bhattacharya

[Eugene]

Hi Bhaskar,

 

To which formula are you referring - or to both? The pi quadrature formula has been around a long time, but just recently shown to square the circle by any pi ratio (logically falling between 3 and 4). For example, let's take the irrational pi and a rational pi to show how the quadrature formula works (radius = 2).

 

sqrt (3.1416) * 2 = 3.5449077 side; side^2 = 12.5663706 area;

sqrt (355/113) * 2 = 3.5449078 side; side^2 = 12.5663716 area.

 

Clearly the pi ratio is not sacrosanct; indeed, the rational might even be given preference inasmuch as running to infinity in numerical sets, as opposed to the irrational for which no sets have yet been found.

 

Why do you say the formula is not a formula and lacks precision?

[Captain Obvious]

3.1416 is rational.

 

In fact, I dare you to write an irrational number without using a symbol. (Well, outside of the character set of 1-9 and maybe you can use a decimal point too).

 

From you descriptions of this equation it looks like you are being distracted by the tautology.

r^2 * PI = (r * sqrt(PI))^2

is valid for any precision of PI, not because PI is poorly defined, but rather its true for PI because it is true for all non negative numbers.

 

What you are forgetting is that the area of a circle is ONLY r^2 * PI when PI is its irrational value. We get 'close enough' by approximating PI.

[Mich]

I am so confused....

[Tormod]

In the meantime (for those of us who don't quite get the gist of this conversation), here is a couple of links:

 

A History of Pi

 

How to compute 1 billion digits of Pi

 

...and then we can probably fly through this one:

Pi Trivia Game

 

 

[Eugene]

And a great deal more, Tormod - going back some 4 millenia. Correlating the line and arc to a common point of origin is probably one of the most intriguing problems ever encountered by mathematics.

Thanks for the links....

[Eugene]

Capt. Obvious, I'd have to see the fraction 3.1416 derives from to show that it is a rational number.

 

If you are referring to the quadrature formula, sqrt pi (radius), it most certainly does give the side of a square with the same precise area as that of the given circle - and whether the pi ratio be rational or irrational.

 

Is this what you're referring to?

[Captain Obvious]

umm... sure... 3.1416 = 31416/10000

 

I suggest you try ridiculous numbers in your equation.

 

 

Pi r^2 = (sqrt(Pi) r)^2

 

Pi = 4, r = 1

4 * 1 = (2 * 1)^2 = 4

 

Pi = 25 r = 3

25 * 9 = (5 * 3) ^2 = 225

 

in fact for all positive values of x and all values of y

x * y^2 = (sqrt(x) * y)^2

 

And no, it does not work for arbitrary Pi, as your error is described by (x - Pi)/Pi where x is your arbitrary value of Pi.

 

There IS an exact value of Pi, we just don't use it, as we are willing to accept error. For instance 3.14 mostly correct, it only has error of around .05%.

 

Am I making any sense at all?

[Eugene]

Captain Obvious,

 

Yes, I think you make a great deal of sense, but for the most part confirming that pi is arbitrary, even by the use of ridiculous pi ratios. More rationally (if you'll excuse the expression) the ratio given above, 355/113, shows more succinctly that it is arbitrary by reason that we cannot discern empirically between it and the traditionally accepted irrational pi ratio.

 

The rub of the whole thing is IMHO that we have no documented proof for curvature of what we assume to be the straight line. And surely, just as there can be no such thing as the perfect circle, neither can there be the absolutely straight line. It's more than a philosophical question, for we do also have a finite pi ratio, 256/81, that derives from a finite mathematic progression, a discrete and irrefutable algorithm with finite upper and lower bounds. Though considerably wide of the irrational pi, it cannot be disregarded for so long as pi is shown to be arbitrary.

 

Thanks for showing the 3.1416 fraction. I find that to be extraordinarily interesting. Let's keep in touch - though I'm not always able to respond as readily as I may wish.

[Robust]

Eugene, t6he tringlature formula you give is actually a 2-part formulae:

 

r*sqrt2 *pi/4*sqrt2 is the complete formulae, where the 1st part gives chord distance to the quadrant. The 2nd part is a conversion factor giving distance on the arc. A most useful formulae I use extensively.

 

 

"All things number and harmony." - Pythagoras

[Robust]

Eugene, hello again! You mention that the trianglature formulae has created quite a stir in science quarters the world over. The only place I've heard it mentioned is on a somewhat remote science forum, and which is really more of a chat room than science. However, I too find the formulae most intriguing and applicable as well in that it correlates distance on the circle quadrant by both its arc and chord length - thus area of the closed continuum analyzed in such manner as to be defined by a whole number or ending decimal.

 

It's all a very intriguing prospect indeed and frought with much controversy, as one might expect. Any source info you or others might give on this subject would be greatly appreciated. I'll give a followup as I can.

 

 

"All things number and harmony." - Pythagoras