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DERIVING PI WITH OVERLAPPING SQUARES


Hold up any object in front of you. Fix your eye at a point on that object and rotate it in any direction. You will find that the point moves in a circle in 2 dimensional space and moves with the values of the sine function in 1 dimensional space. Calculating point position values around a circle is a key, fundamental math central to modern technology. This is why pi is so important.

Today this math is mostly used in processor chips destined for graphics cards. If you ever wondered how 3d games are displayed on a 2d screen, it is possible because a processor chip is calculating point positions making 2d polygons on a flat screen morph in such a way to appear 3d.

There are many ways to calculate pi. The equation below is one of the fastest ways I know to calculate pi to any finite digit. This equation can also be used to help calculate the sine and cosine functions.


Pi is the circumference of a circle with a diameter of 1 unit. Just as a square's perimeter can be found by multiplying 4 by its width, a circle's circumference can be found by multiplying pi by its diameter.



One can derive pi by using overlapping squares. If more and more squares geometrically overlap each other, then the perimeter of the totally overlapped area will equal ever closer to pi.

Below shows how to calculate pi step by step using this overlapping squares method. The equation on the right is a substitution version of the equation on the bottom left of each diagram. The operation being shown is in bold. X, Y and Z are the initial variables.


X = 1

Y = 1

Z = 4

(X) equals the area of the 1 unit square.

A = 2

(A) equals the area of the multiplied square.


X = 1

Y = 1

Z = 4

B = 1.414213562...

(B) equals the width of the square. The area of a square equals the square of its width.


X = 1

Y = 1

Z = 4

C = 0.414213562...

(C) equals the length of the ends of line (B).


X = 1

Y = 1

Z = 4

(Y) equals the length of the polygon's edge.

D = 1

(D) equals the proportion of the 1 width square over (Y). 1 is to (Y) as new (Y) is to (C).


X = 1

Y = 0.414213562...

Z = 4

(Y) equals the length of the polygon's edge.

Y = 0.414213562...

(Y) equals the length of (C) multiplied by the proportion of (D). In this case (C) is not multiplied at all.


X = 1

Y = 0.414213562...

Z = 8

(Z) equals the number of edges the new polygon has.

Z = 8

(Z) equals twice its original value.


X = 1

Y = 0.414213562...

Z = 8

Pi » 3.313708498...

(Pi) equals approximately the length of (Y) times (Z).


X = 1

Y = 0.414213562...

Z = 8

E = 0.585786437...

(E) equals the width of the 1 unit square minus the length of (Y).


X = 1

Y = 0.414213562...

Z = 8

F = 0.292893218...

(F) equals half the length of (E).


X = 1

Y = 0.414213562...

Z = 8

G = 0.707106781...

(G) equals the length of both (F) and (Y).


X = 1

Y = 0.414213562...

Z = 8

H = 0.207106781...

(H) equals the area of a rectangle with the length of (G) and the width of (F).


X = 1

Y = 0.414213562...

Z = 8

I = 0.414213562...

(I) equals the area of the 1 unit square minus the smaller tilted square.


X = 0.585786437...

Y = 0.414213562...

Z = 8

(X) equals the area of the smaller tilted square.

X = 0.585786437...

(X) equals the 1 unit square minus (I).


X = 0.585786437...

Y = 0.414213562...

Z = 8

A = 1.171572875...

(A) equals the area of the multiplied square.


X = 0.585786437...

Y = 0.414213562...

Z = 8

B = 1.082392200...

(B) equals the width of the square. The area of a square equals the square of its width.


X = 0.585786437...

Y = 0.414213562...

Z = 8

C = 0.082392200...

(C) equals the length of the ends of line (B).


X = 0.585786437...

Y = 0.414213562...

Z = 8

D = 2.414213562...

(D) equals the proportion of the 1 width square over (Y). (1) is to (Y) as new (Y) is to (C).


X = 0.585786437...

Y = 0.198912367...

Z = 8

(Y) equals the length of the polygon's edge.

Y = 0.198912367...

(Y) equals the length of (C) multiplied by the proportion of (D).


X = 0.585786437...

Y = 0.198912367...

Z = 16

(Z) equals the number of edges the new polygon has.

Z = 16

(Z) equals twice its original value.


X = 0.585786437...

Y = 0.198912367...

Z = 16

Pi » 3.182597878...

(Pi) equals approximately the length of (Y) times (Z).


X = 0.585786437...

Y = 0.198912367...

Z = 16

E = 0.801087632...

(E) equals the width of the 1 unit square minus the length of (Y).


X = 0.585786437...

Y = 0.198912367...

Z = 16

F = 0.400543816...

(F) equals half the length of (E).


X = 0.585786437...

Y = 0.198912367...

Z = 16

G = 0.599456183...

(G) equals the length of both (F) and (Y).


X = 0.585786437...

Y = 0.198912367...

Z = 16

H = 0.240108467...

(H) equals the area of a rectangle with the length of (G) and the width of (F).


X = 0.585786437...

Y = 0.198912367...

Z = 16

I = 0.480216935...

(I) equals the area of the 1 unit square minus the smaller tilted square.


X = 0.519783064...

Y = 0.198912367...

Z = 16

(X) equals the area of the smaller tilted square.

X = 0.519783064...

(X) equals the 1 unit square minus (I).


X = 0.519783064...

Y = 0.198912367...

Z = 16

A = 1.039566129...

(A) equals the area of the multiplied square.


X = 0.519783064...

Y = 0.198912367...

Z = 16

B = 1.019591158...

(B) equals the width of the square. The area of a square equals the square of its width.


X = 0.519783064...

Y = 0.198912367...

Z = 16

C = 0.019591158...

(C) equals the length of the ends of line (B).


X = 0.519783064...

Y = 0.198912367...

Z = 16

D = 5.027339492...

(D) equals the proportion of the 1 width square over (Y). (1) is to (Y) as new (Y) is to (C).


X = 0.519783064...

Y = 0.098491403...

Z = 16

(Y) equals the length of the polygon's edge.

Y = 0.098491403...

(Y) equals the length of (C) multiplied by the proportion of (D).


X = 0.519783064...

Y = 0.098491403...

Z = 32

(Z) equals the number of edges the new polygon has.

Z = 32

(Z) equals twice its original value.


X = 0.519783064...

Y = 0.098491403...

Z = 32

Pi » 3.151724907...

(Pi) equals approximately the length of (Y) times (Z).



Derived by: Crailin Mogged