International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
ISSN 2278-7763
- 77 -
Value of Pi - Were
Babylonians the most
accurate ?
International Journal of Advancements in Research &
Technology, Volume 2, Issue 5, May-2013 ?
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Mohankumar Shetty
23-Mar-13
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International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
ISSN 2278-7763
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Value of Pi – Babylonians were the most accurate
Babylonians had probably found the most accurate value of Pi as suggested by the value of diagonal m inus diam eter . The
following illustrations will substantiate the nearly 5000 year old value of Pi (3.125) to be the most accurate of all values used.
Copyright © 2013 SciResPub.value of Pi derived from the value of diagonal minus diameter
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International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
ISSN 2278-7763
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Figure:1 (a one unit square)
1.
How much perimeter or area of a circumscribed square around a circle is represented by diagonal minus diameter?
The area of the square is represented by the diagonals ac and bd together. Assuming that diagonal ac is reduced to
ao, the square becomes a triangle abd. Hence, while diagonal ac represents Whole Square abcd, diagonal bd
represents half of square abcd. Therefore, the perimeter and area represented by the diagonals outside the circle
plus the circumference and area of the circle should be equal to perimeter & area of the circumscribed square
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Figure:2 (a one unit square with one unit diameter circle inscribed)
2.
If the two diagonals represent full and half value of the square as explained above, the perimeter and the area
outside the inscribed circle should also be represented similarly by the lengths of diagonals outside the circle. The
length of the diagonal is (√2) 1.41421356237309 units. The diameter of the circle is one unit. Therefore, length of
diagonals outside the circle is 0.41421356237309 units. The calculations of perimeter and area of square outside the
circle with different combinations are as follows:
Description
Area
Perimeter
Brown Square
0.171573
0.585786
Blue square
0.042893
0.292893
Total Area/peri. outside the circle
0.214466
0.87868
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International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
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Square Area/Perimeter
1.000000
4.000000
Circle Area/Circumference
0.785534
3.121320
3.142136
3.121320
Pi
But, area * 4 <> circumference
(A) Hypotenuse/Diagonal = √ (1+1)
1.414214
( B ) Diameter =
2xr
1.000000
(C)
=
A-B
0.414214
(D)
=
C^2
0.171573
(E)
=
√ (D / 2)
0.292893
(F)
=
Ex2
0.585786
(G)
=
C/2
0.207107
(H)
=
G^2
0.042893
(I)
=
√ (H/2)
0.146447
(J)
=
Ix2
0.292893
(Square - Circle)
(Brown square Area Lost)
(Brown square Perimeter Lost)
(Blue Square Area Lost)
(Blue Square Perimeter lost)
Table – 1
What If ?
Erased from all 4 corners
Erased from one corner
c
d
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o
a
Figure - 3
Description
Brown Square 1
Brown Square 2
Figure - 4
Area Lost
0.042893
0.042893
Sides Lost
0.292893
0.292893
Blue Square 1
Blue Square 2
0.021447
0.021447
0.146447
0.146447
Total
0.128680
0.878680
1.000000
0.871320
3.485281
But, area * 4 <> sides
Table - 2
4.000000
3.121320
3.121320
Square
Circle
pi
3.
b
[ (1.4142135623730950488016887242097 - 1) * 1.5 ]
Description
Brown Square
Total
Square
Circle
Pi
Area lost
0.386039
sides lost
0.878680
0.386039
0.878680
1.000000
0.613961
2.455844
But, area * 4 <> sides
Table-3
4.000000
3.121320
3.121320
The values of Pi derived as per Table-1 (consisting of calculations of Figure-2) are pretty close to 22/7 and 25/8
respectively, both values adopted by ancient civilizations. It is also interesting to note that, in all the three
combinations, the perimeter of the square erased is identical. But area x 4 is not equal to circumference. Is this on
account of partial merger/rotational effect of the diagonals? What if both the diagonals are pulled out by half the
length to give effect of 90 degree rotation each to complete one cycle; or, whether the perimeter loses a bit of its
linear length on curving around the circle?
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International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
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4.
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From Table-4 (consisting of calculations of Figure-5), it is interesting to note that; when the area belonging to the
circle within the circumscribed square is bound, (by following the observations made in Para one) the value of Pi
comes to the same 3.121320; as was the case of perimeter of the circumscribed square. Both, the area and the
perimeter are producing identical answers independently, but not simultaneously. Yet, it is conclusive enough proof.
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Description
Area
Side
bda1 Triangle
0.353553
1.732050
bdc1 Triangle
0.426777
1.859330
Total
0.780330
3.591380
Pi
3.121320
3.591380
But, area * 4 <> sides
Description
Measure
bd
1.414213562
oa1
0.5
oc1
0.603554
ba1 & da1
0.866025
bc1 & dc1
0.929665
Table -4
5.
If both the diagonals ac and bd are pulled out from points a and b respectively (Figure 1) by half the length of the
diagonals to exhaust one full cycle, 75% of the area of the square is swiped out along with 75% (three sides ) of
the perimeter of the square. But, the area of the circle is 78% plus of the area of the circumscribed square. Where
from the extra area (0.03125%) comes?
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International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
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Figure – 6
6.
The extra area comes from the overlapping area of the square on the opposing diagonals (Figure-6) represented by
ocxd square. Hence, two diagonals ac and bd are equal to 1.414214 each and square root of ocxd square =
(1.414214 / 2). Sums of squares on these two diagonal = 4 square. Therefore, (1.414214 + 1.414214 + 0.707107)/
4 represents the square root of the area of a circle inscribed in the square and two
of the diagonals merging
with the third side of the square, represents square root of circumference of four circle as follows:
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(I) 3.535535 /4 = 0.88388375 & (0.88388375)2 = 0.78125
(II) (3.535535)2/4 = 3.125
& 0.78125 x 4
= 3.125
7. In the case of Archimedes method of measuring the perimeter of circumscribed/inscribed polygon, a
side of an inscribed polygon shall not have an apothem equal to the radius of the circle. If the apothem
of the inscribed polygon is equal to the radius of the circle, ends of the sides of the polygon should
obviously lie outside the circumference as both the polygons merge at this stage; thus resulting in value
of Pi being expressed a bit in excess of the actual value.
8. Therefore, as established by the calculations of value of diagonal minus diameter in terms of area and
perimeter (please refer to the annexures), the Babylonian value of Pi is the most accurate and it is a
rational value.
9. What was the necessity of people of Babylonian era or earlier to know the principles of the diagonal of
a square? It is obviously related to the construction of foundation of square shaped structures. Unless it
is ensured that the two opposite corners of the foundation had identical distance, the shape will not be
square. This might have further led to the understanding of the principles; which is now known as
Pythagorean Theorem. Babylonians had calculated precise value of square root of number 2 and hence
it is no wonder if they had found out the precise value of Pi
Based on the above observations, I had published a paper titled “Geometric Estimation of Value of Pi” through the Journal
IJOART Volume1 Issue2 July 2012. www.ijoart.org/docs/Geometric-Estimation-of-Value-of-Pi.pdf
Mohankumar Shetty
E-mail: [email protected]
Copyright © 2013 SciResPub.
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International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
ISSN 2278-7763
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Annexure-1
Annexure-2
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Annexure-3
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International Journal of Advancements in Research & Technology, Volume 2, Issue5, May-2013
ISSN 2278-7763
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