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This post was edited by Monkeboy on Sep 21 2009 03:33am

A New Approach to the Universal Constant "pi"

Attached you will find an original, I would like to believe, research study, concerning «A new approach to the area of a circle» and thus a new approach to the Universal Constant "pi".

The innovative aspect of the new approach is attributed to avoid the insurmountable

obstacles presented by the intervention of square roots and infinite series when we try to calculate the circumference as the

Upper Limit of the perimeter of a regular Polygon inscribed in the circle, according to the relevant "deceptive"

definition, i.e. sin15o = 1/4 {sqrt(6) - sqrt(2)}.

The New Theory considers the above way as an "impasse", which makes us to believe that the so derived number

pi' = 3.14159 2 6535.... is an irrational and trancendental number.

By the new way we approach the area of the circle as the "Lower Limit" of the area of the Superscribed Regular Polygon, coming from the square of side ao = 2R, and we use as main tool the "tangent", the values of which can be easily calculated, with the desirable accuracy, using as algorithm the formula

tan(2q) = 2tanq / {1-tanq} which defines the circle.

By this method we discover that it is impossible for the Lower Limit of the Superscribed Polygon, to be less than the

Limit 4 * 0.7854 = 3.1416.

More precisely the new method meets the Known number

3.14159 2 6535... and proves that in order to reach this number we have violated the sense of the circumference making the arbitrary assumption that: for very acute angles we can consider that

tanq = 1/2 tan(2q) instead of tanq < 1/2 tan(2q), according to the indisputable formula:

tan(2q) = 2tanq / {1-tanq} -> tanq < 1/2 tan(2q) always

and certainly 2sinq < 2arcq < 2tanq always.

However small the side-chord of the inscribed regular polygon, this chord is less than the respective arc and always there exists a minimum area between the chord and the arc, however great is the number N = 4 * 2n of the sides.

The above observation explains the irrational and trancendental character of the so defined circumference, and why "we can not see" the Upper Limit of an inscribed regular polygon.

On the contrary, through the New Theory, which dares the "transgretion", the lower Limit of the regular Superscribed polygon "is visible" and can not be other than 3.1416 R2 exactly.

Also, according to the New Theory, the area of the circle is the first real root of the algebraic equation:

X2 - 4X + 2.69674944 = 0, the second root of which is the 4k = m = 0.8584 R2 that is the area of the 4 equal curvilinear triangles, which lie between the circumference and the perimeter of the superscribed square of side ao = 2R, so that: ...

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This post was edited by Monkeboy on Sep 21 2009 03:34am

+50

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WOOT :smiley slap:

plus 1

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In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger one equals the ratio of the larger one to the smaller. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.[1] Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean.[2][3][4] Other terms encountered include extreme and mean ratio,[5] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[6] golden number, and mean of Phidias.[7][8][9] The golden ratio is often denoted by the Greek letter phi, usually lower case (φ).

The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:

\frac{a+b}{a} = \frac{a}{b} = \varphi\,.

This equation has as its unique positive solution the algebraic irrational number

\varphi = \frac{1+\sqrt{5}}{2}\approx 1.61803\,39887\ldots\, [1]

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

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