the 'pi' liberated
TRANSCRIPT
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Measuring the ImmeasurablePi - The cosmic constant3.148148148148148148148148148148148…
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Dedicated to the ONE
which has enriched me……
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“A mathematician has not thoroughlyunderstood his own work till he hasmade it so clear that he can go outand explain it effectively to thefirst man he meets in the street.”
Lagrange
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Abstract of the Concrete:3.148148148148148148148148148…
The world of mathematics has solemnly and sincerely
adopted the polygonal circumference as the highest
degree of circular approximation. A polygon with any
number of sides could never become a circle as the
characteristics of the polygon and the circle are
genetically distinct even at infinity. It is true
that pi of 3.141592653 is always averred by the
mathematicians as the ultimate expression of
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approximation and beyond that there could never be
any further progression towards absolute accuracy.
Therefore, this paper seeks to emancipate pi not only
from the apostles of approximation but also from the
shackles of irrationality and transcendence
attributed by them and consecrate pi within the
sanctum of circular singularity.
2. Apostles of approximation:3.148148148148148148148148148…
2.1. Euclid, (325 to 265 BC) the great geometric genius in his
“Elements” has laid down the axiom that the circumference of any circle
divided by its diameter is to yield the quantum of inasmuch as the ratio
of the diameter of a circle to its circumference remains always constant.
Archimedes, the great mathematician and scientist had proceeded on
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the premise that the circumference of the circle could only be
approximated to proximity of accuracy by optimizing the sides of a
polygon circumscribed and inscribed within circles.
2.2. Archimedes by increasing the number of polygons to 96 without the
advantages of decimal system and zero had secured with mere pincers
of fractions the highest approximation of between the fractions of 22/7
and 223/71 which are equivalent to 3.142857142 to 3.14084507 in
decimal system. At last after many more centuries, the deflecting needle
of pi has been settled at the present slot of 3.141592653….. As on date,
the flames of pi have consumed more than 5 trillion digits.
3.141592653 589793238 462643383 279502884 197169399 375105820 974944592 307816406 286208998 628034825342117067 982148086 …………….to 5 trillion….
Still there is smoke in the chimney of mathematics around this world as
much of the midnight oil is being burnt by the unblinking eyes petrified
in prayerful mood before the computing entities to know the last digit of
pi.
3. Science of Stonehenge:3.148148148148148148148148148…
3.1. The moment any point is subject to measurement on the plane of
space, the entire universe is instantly filled with rays the space of infinity
around the point of such measurement both exponentially and inversely
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with velocity of infinity. Therefore, whatever shape of finite is created on
the canvas of infinity, such a figure of the finite created on the plane of
human limitation unmistakably partakes the characteristics of the
infinite. Pi is born of this universal law of Pi where the relationship of the
relative with the absolute is inextricably linked up with each other in
more than a bond of placenta uniting the fetus with the mother.
4. Temple of Infinity:3.148148148148148148148148148…
4.1. No wonder pi becomes the very expression of the unity of the
finite with the infinite and infinite with the finite. Even after the advent
of the computer age, all the software developed and programmed close
on the scales of Archimedes, Ramanujan and other innovative methods
devised by the most brilliant mathematical minds to rationalize outside
the walls of transcendence, have failed in their mission because the
expedition to accuracy was altered by their search for the arc of
approximation.
4.2. It is a mathematical history that in Aug 2009,Japanese Supercomputer T2K Open Supercomputer
calculated to 2,576,980,377,524 digits in 73 hrs and
36 mts. In December 2009, it took a total of 131 days
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to compute 2,699,999,990,000 decimal digits for
calculation, conversion and verification. In Aug
2010, it took a total of the period from May 4 to
August 3 to compute 5,000,000,000,000 decimal digits
on a home computer designed for the purpose and
consuming 64 and 66 hrs for primary and secondary
verification. All these exercises have been rendered
null and void by this research which has determined
the orbit of accurately at 3.148148148148148148…. to
infinity. Now, it could be demonstrated that by a
mere touch of a scientific calculator, not only
embraces infinity but also returns to the singularity
instantly without any intervention of time and space.
4. Axioms of the Abacus:3.148148148148148148148148148…
4.1. If a circle is inscribed within a square
The side of the square and the dia
of the circle are one and the same.
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4.2. If a square is circumscribedby a circle
The hypotenuse of the square and
the dia of the circle are one and the same.
4.3. If a square is created out of a circle
The area of the circle and the area
of the square created out of the circle are
one and the same at different
diameter/side.
4.4. Within a circle itself
The area of the circle at “x”
diameter becomes the circumference of
the
circle at “y” diameter.
4.5. Within and without
The circumference of the circle with
its dia but without the quantum of pi
equals the circumference of the square.
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With pi with dia, the circumference
of the circle is more than the
circumference of the square.
5. Themes of Theorems:3.148148148148148148148148148…
5.1. If a circle is inscribed within a square, the
side of the square and the dia of the circle are one
and the same. When a circle circumscribes a square,
the hypotenuse of the square and the dia of the
circle are one and the same.
The seed of sprouts from the fraction of the ratios
of
(1) the area and circumference of the square,
divided by the area and circumference of the
circle,
(2) the area of the square divided by the area of
the circle, and
(3) the circumference of the square divided by
the circumference of the circle.
The schematic evolution of pi could be secured
through the path of trinity as laid out below;
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Table I
Side/
Dia
Path I Path
II
Path
III
Ratio
9 117
80.75
81
60.75
36 1.333333
333
8 96 72 64 48 32 24 1.333333
333
7 77
57.75
49
36.75
28
21
1.333333
333
6 60 45 36
27
24 18 1.333333
333
5 45
33.75
25
18.75
20 15 1.333333
333
4 32 24 16 12 16 12 1.333333
333
3 21
15.75
9
6.75
12 9 1.333333
333
2 12 9 4 3 8 6 1.333333
333
1 5
3.75
1
0.75
4 3 1.333333
333
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5.2. When the decimal of 1.333333333….. is divided by 9 which is the
finite quantum of the infinite, it results in the excess of 0.148148148…
which goes by the name pi if the circumference of the circle itself is 1 as
shown in the calculation below;
Therefore, = 1.3333333333……to the infinite 9 Excess = 0.148 148 148…… to the infinite If the above excess is added to the circumference of the circle 3 where
the diameter of the circle is 1, the quantum of works out to 3.148
148 148 …………………. repeating the strings of sequence148148148 148148148 148148148 148148148 148148148
148148148 148148148 148148148 148148148 148148148
148148148 148148148 148148148 148148148
148……….absolutely to infinity depending upon the circumference or the
diameter taken as the measure of reference as tabulated below;
Table II
Circumference Quantum of Pi Diameter of Circumference
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of
the circle (A)
the
Circle (B)
of
the circle
(A+B)
1 0.148148148 1 x 3 = 3 3.148148148
2 0.296296296 2 x 3 = 6 6.296296296
3 0.444444444 3 x 3 = 9 9.444444444
4 0.592592592 4 x 3 = 12 12.592592592
5 0.740740740 5 x 3 = 15 15.740740740
6 0.888888888 6 x 3 = 18 18.888888888
7 1.037037037 7 x 3 = 21 22.037037037
8 1.185185185 8 x 3 = 24 25.185185185
9 1.333333333 9 x 3 = 27 28.333333333
6. Locus of the Logic:3.148148148148148148148148148…
6.1. Mathematics is a science of spatial and numerical relationship.
There is an inextricable nexus between the square and the circle. If the
side of the square and the dia of the circle are one and the same, then,
either the ratio of the area of the square to its circumference or the ratio
of the area of the circle to its circumference (R1) multiplied either by the
ratio of the circumference of the square to its area or by the ratio of the
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circumference of the circle to its area (R2), always remains constant at 1
as it is the law of mathematical harmony manifest in the illustration
below;
Table III
d/s
Area/circSquare
Area/cirCircle
RatioR1
Cir/AreaSquare
Circ/areaCircle
RatioR2
ConstantR1 x R2
9 81/36
60.75/27
2.25
36/81 27/60.75
0.444444
1
8 64/32
48/24 2 32/64 24/48 0.5 1
7 49/28
36.75/21
1.75
28/49 21/36.75
0.571428
1
6 36/24
27/18 1.5 24/36 18/27 0.666666
1
5 25/20
18.75/15
1.25
20/25 15/18.75
0.8 1
4 16/16
12/12 1 16/16 12/12 1 1
3 9/12
6.75/9
0.75
12/9 9/6.75 1.333333
1
2 4/8
3/6 0.5 8/4 6/3 2 1
15
1 ¼
0.75/3
0.25
4/1 3/0.75 4 1
6.2. Therefore, against this background of inextricable and rather
inevitable relationship between the circle and the square, we can draw
an equation which would lay bare in all certainty. Not all the roads but
only three,
(1) one through the combined gateway of area and
circumference of both square and circle,
(2) another through the gateway of area of the square and
circle and
(3) yet another through the gateway of circumference of the
square and circle,
lead the seekers of Pi to measure the universal constant in all its
grandeur of accuracy. In the realm of mathematics in its entirety, it is
one of the simplest formulae where is related to a circle in its entirety as
such relativity is unalloyed by any correction unlike in the process of
approximating the area of a polygon to the quantum of pi by
Archimedes in the 3rd century BC or by Ramanujan in the 20th century.
(A) Through the combined gateway of area and circumference of square and circle:
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Formula I At side/dia 9
= 1.333333333
= 0.148148148….
(B) Through the gateway of area of square and circle: Formula II
=1.333333333 9= 0.148148148
9 =0.148148148
( C )Through the gateway of circumference of square and circle:
Formula III
9= 0.148148148
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9 =0.148148148
7. Postulates of Precision:3.148148148148148148148148148…
7.1. If a square is created out of a circle, the area of the circle and the
area of the square created out of the circle are one and the same at
different diameter/side. Therefore, the area of any square ought to be
considered as the area of any circle irrespective of the quantum of pi
used with difference only in the measurement of the side of the square
and diameter of the circle. Pi is not exactly the relationship between the
square and the circle but it is a relationship between the linear and the
spatial, space and time, the finite and the infinite. Within the circle itself,
pi is the relationship between the area of that circle apportioned into 3
parts at 120 and apportioned into 4 parts at 90 each. The
apportionment of the circle of 360 into 90 represents (4) the square
and the apportionment of the circle of 360 into 120 epresents (3) the
circle. This equation results in the fraction of 4/3 which is equivalent to
the decimal 1.333333333… constantly to infinity.
7.2. Even the circle whose area has been calculated by the present pi of
3.141592653 at dia 9 could be tested. Accordingly the area is 20.25 x
3.141592653 = 63.617255122. If this circle is apportioned into 120 and
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90 , the ratio results in 1.33333333 only. Even in the case of the
maximum of Archimedes, the area is 20.25 x 3.142857142 =
63.64285713 and if the area of this circle is apportioned into 120 and
90 as above the ratio results in 1.33333333 only making the fraction of
4/3 in the decimal 1.333333333… as always universal in its application
as displayed below;
Step I
Step II
= 1.333333333…
Step III
= 0.148148148….
8. Enchanting Exhaustion:3.148148148148148148148148148…
8.1. It could be stated with all certainty that of any circle if multiplied
by the pi-3.141592653 yields the area of the circle which actually is not
the entire area of the circle but the maximum area of that circle that
could be traversed in the method of exhaustion. Conversely, it could be
stated that the maximum area of any circle that could be traversed in
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the method of exhaustion if divided by of that circle, yields the quantum
of pi. Therefore, it is proved beyond any reasonable doubt that pi
reigning as on date is not at all related to the area of the complete circle
but is actually related to the complete area of a polygon.
8.2. In line with the proposition above, a circle of
any radius may be drawn. It may be divided into
120 each- into 3 equal areas. One part could be
subject to the method of exhaustion. Accordingly, if
the dia of the circle is 9, then r is 4.5. If its
area is worked out with the pi of 3.141592653, then
its area is 63.61725122 (20.25 x 3.141592653) which
if divided by 3 gives one of its segments as
21.20575041 corresponding to120 As the exhaustion
progresses, at the 18th step with 131,072 of
triangles, the radius of the circle used as the
height to work out the areas of triangles is reduced
to nullity as shown below in the table.
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Table IV
S
l
Base of
the
Triangle
2
Height of the
triangle
Area of
the
triangle
No
of
Total
Area
Progres
siv
total
1 3.897114
31
2.25 8.768507
21
2 17.537014
43
17.5370
14
2 2.25 0.60288568 1.356492
78
2 2.7129855
74
20.25
3 1.164685
70
0.15333378 0.178585
66
4 0.7143426
54
20.9643
42
4 0.587367
86
0.03849812 0.022612
56
8 0.1809004
87
21.1452
43
5 0.294314
08
0.00963485 0.002835
67
16 0.0453707
31
21.1906
13
6 0.147235
87
0.00240935 0.000354
74
32 0.0113517
96
21.2019
65
7 0.073627
79
0.00060237 0.000044
35
64 0.0028385
18
21.2048
04
8 0.036815
12
0.00015059 0.000005
54
128 0.0007096
63
21.2055
13
9
1
0
1
0.018407
71
0.009203
87
0.00003764
0.00000941
0.00000235
0.000000
69
0.000000
08
256
512
1024
0.0001774
16
0.0000443
54
21.2056
91
21.2057
35
21
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
0.004601
9
0.002300
9
0.001150
4
0.000575
2
0.000287
6
0.000143
8
0.000071
9
0.000035
9
0.00000058
0.00000014
0.00000003
0.00000000
0.00000000
0.00000000
20.25/4.5
0.000000
01
0.000000
00
0.000000
00
0.000000
00
0.000000
00
0.000000
00
0.000000
00
131,072
s
2048
4096
8192
1638
4
3276
8
6553
6
0.0000110
89
0.0000027
69
0.0000006
91
0.0000000
17
0.0000000
04
0.0000000
01
0.0000000
00
21.2057
46
21.2057
49
21.2057
50
21.2057
50
21.2057
50
21.2057
50
21.2057
50
8.3. Therefore if this area of 21.20575041 is multiplied by 3 it gives the
area of 63.61725122 for 360. And if this is divided by 20.25, it gives the
pi of 3.141592653. It is evident from the method above that only there
is no height but there remains a space as base which proves beyond
doubt that the area exhausted is not a circle but only a polygon.
8.4. It is evident from the above that pi as on date is related only to the
maximum area of the circle exhausted and not related to the entire area
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of the circle. It is the assertion of the mathematicians that pi could never
be accurate. It is their case that despite there is space in the base still pi
could not be stretched forward further as there is no radius available. It
is true that none could march forward beyond this point. But this
restriction is relevant only to this method and not to any other method
by which pi covering the circle in its entirety could be calculated.
Therefore, had other avenues where accuracy could be secured been
explored, then the value of pi would have been seen to increase till it
reaches to the higher degree of accuracy.
9. Symmetry of Trigonometry:3.148148148148148148148148148…
9.1. Trigonometry is built on the axiomatic principle that the length of
the arc is proportionate to the length of the radii while the vector
remains constant throughout the radii to infinity. If the radius of the
circle increases then correspondingly, the arc of the circumference also
increases.
9.2. There is perfect symmetry in the circle with all of its proportions and
therefore there ought to be symmetry between the area of the circle and
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circumference of the circle. Whatever may be quantum of pi, then the
dia declared in respect of the area and circumference of the circle ought
to be one and the same. In other words, the dia of the area cannot
afford to be in conflict with the dia of the circumference of the circle on
account of the quantum of the pi selected.
9.3. A circle with the radius of 4.5 is drawn and apportioned into 3 parts
of 120 each. If the unit of radius is 1, then one third of the
circumference is 2.098076211 as calculated below:
According to trigonometric functions;
Tan 120 = 1.732050808
Sin 120 = 0.866025403
Cos 120 = 0.5
(tan 120 + sin 120 ) cos 120
(1.732050808 + 0.866025403) 0.5 = 2.098076211
9.4. The arc of 120 with 4.5 as radius works out to
2.098076211 4.5 = 9.44134295 and for 360 , it is
28.32402885 (9.44134295 3). As per trigonometry
itself, the maximum of pi works out to 28.32402885 9
= 3.14711437. But unfortunately with the pi of
3.141592653, the circumference of the circle with the
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radius of 4.5 works out to 28.27433388 and the arc of
the circumference of the circle at 120 is
drastically reduced from 9.44134295 to 9.424777959.
9.5. Therefore, as per the axioms of trigonometry
itself the circumference of the circle at the arc of
120 at 4.5 radius works out to 9.44134295 and
accordingly the circumference of the circle at 360
is 28.32402885, it does not stand to reason as to how
the arc of the circumference of the same circle could
stand reduced to 9.424777959 from 9.44134295 at 120
and at 4.5 radius and accordingly the circumference
could also stand reduced to 28.27433388 from
28.32402885.
9.6. It is proved beyond doubt that the pi has been
restricted on the basis of the maximum area exhausted
in the method of exhaustion. Therefore, as pi has
been compelled to stand in relationship to a polygon,
such a pi is also bound to shrink the angle of 120
corresponding to the restriction inherent in its
polygonal relationship.
10. Pythagorean Perspiration:3.148148148148148148148148148…
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10.1. Pythagoras preached that everything in this
universe could be described as either whole numbers
or as ratios of the whole numbers. According to
Pythagoras the study of the nature and properties of
those numbers and fractions was to learn the true
essence of man, nature and the world. It was
contended by his disciples that his claim was wrong
as is not represented by any ratio of whole numbers.
Pythagoras was proved wrong and he also admitted the
same but he was right in respect of Pi which is a
universal constant explaining the essence of creation
and existence. It is demonstrated below that Pi is
the ratio of the fraction of whole numbers and it is
again divided by another whole number. The result is
repetition of sequence to infinity as mapped below;
Pi =
It does not stand to reason as to how such a natural sequence could be
considered as irrational and transcendent within the domain of reason.
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10.2. If the side of the equilateral right angle triangle is 1 which could be
equated to the dia of a circle, then pi of the equilateral right angle
triangle works out to
+ = Pi
+ = Pi
2 + 1.4142135562 = 3.14142135562
Therefore, as per the Pythagorean theorem, the pi of right angle triangle
works out to the pi of 3.14142135562 which lies in close proximity to
the present pi 3.141592653 proves its relationship only to the right
angle triangle and thereby to a polygon and not to the circle.
11. Signs of the Zodiac:3.148148148148148148148148148…
11.1. The ancients have divided the zodiac of 360 into 12 houses of 30
each. The entire zodiac was apportioned into 27 constellations for 9
planets including the nodes of moon. Each constellation spans a
distance of 13 .20. Every constellation has been divided into 4 quartets.
Accordingly each sign of 30 each houses 9 quartet of each
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constellation. In this schematic design, the nodes of moon have been
arranged in two triangles locked with one another as a head and tail.
They are not in a perfect position but slightly tilted within one another in
the following positions. South node is with 0 , 120 and 240 . North
node is with 66 .40, 186 .40 and 306 .40.
11.2. The angular distances are in the order of 0 -66 .40, 66 .40-120 ,
120 -186 .40, 186 .40-240 , 240 -306 .40 and 306 .40-360 .
Accordingly, there are 3 sides with 53 .20 and 3 sides with 66 .40
making up altogether into 360 as a whole. Actually the difference
between 66 .40 and 5 .20 is 13 .20 which accounts for each
constellation. In this schematic design therein lurks the element of pi in
all its simplicity as displayed below:
= 0.148148148…
12. Error Transparent:3.148148148148148148148148148…
12.1. Numerical methods involve finding approximate
solutions to mathematical problems. The axiomatic
principles of mathematics have held that errors of
approximation can result from two sources; error
inherent in the method, or formula used or local
28
truncation error and rounding off error. Inherent
error is the quantity of error which is present in
the statement of the problem itself, before finding
its solution. It arises due to simplified assumptions
made in mathematical modeling of a problem. It can
also arise when the data is obtained from certain
physical measurement of the parameters of the
problem. It is actually the error inherent in method
which has wreaked havoc in the calculation of
approximation as it is manifest from the steps of
calculation;
Maximum of Archimedes:
(i)Ascending
(ii)Descending
12.2. It is evident from the above that the amount of correction added is
0.111111111. But instead of adding the decimal of 0.142857142 to
3.111111111, it has been added to 3 to make it 3.142857142. What
remains to be added to in this case is not 3 but 3.111111111 which was
used as a denominator. If the quantum of correction is also counted for
addition to 3, then, it would result in 3 + 0.111111111 + 0.142857142 =
29
3.253968253. Therefore, the correction of 0.111111111 has been
delinked and only the quotient of 0.142857142 is added to 3 to make it
the maximum of pi – 3.142857142.
12.3. It is evident from the table below as to where the maximum of
Archimedes and the pi of the present exist in the chronology of the
side/dia. While that of the Archimedes lies at 28, the present pi rests at
28.2500532 and the pi proposed rests at 27 as side.
Table V
Side
Area/Circumf
Ratio Circumference/Area
Ratio
25
625/100 6.25 100/625 0.16
26
676/104 6.5 104/676 0.153846153
27
729/108 6.75 108/729 0.148148148
28
784/112 7 112/784 0.142857142
28.25
798.06/113.00
7.062475
113/798.0625 0.14159292
28.25005
798.06/113.00
7.0625125
113.0002/798.065325
0.141592669
28.250052
798.06/113.00
7.0625135
113.000208/798.065438
0.141592669
30
28.2500532
798.06/113.00
7.0625133
113.0002128/798.0655058
0.141592653
12.4. While what Archimedes had proposed is a descent
from (22/7) 3.142857142 to (223/71) 3.14084507, it
ought to have been an ascent from 3.142857142 to
3.160493827. Therefore, the value of Pi has got to
move up from 3.142857142 to 3.160493827 and not to go
down from 3.142857142 to 3.14084507 to find a slot
under 3.141592653. In other words, the value of ought
to have been fixed between 3.142857142 and
3.160493827. But in the case of the ongoing
3.141592653, it falls far short of the boundary.
Therefore, when it is not only logically
inappropriate but also mathematically inaccurate, is
it proper to measure the equatorial circumference of
our earth by the gauge of the on-going ?
13. Modulation of Ramanujan:3.148148148148148148148148148…
31
13.1. In the same way as discussed above, the formula
of Ramanujan also succumbs to the successive steps of
infirmity inherent in the method present in the
statement of the problem itself before finding a
solution. In the celebrated formula of Ramanujan to
approximate pi, it could be seen that the maximum of
Archimedes fraction 22/7 is inbuilt. The formula is
analogous in its methods to the following steps.
=
=
= 3.1415926525826461252…
13.2. The formula above corresponds to the same in the case of the
maximum of Archimedes as shown in the method below:
(i) Ascending
32
= 0.141592653
(ii) Descending
0.141592653 x 9 = x
Here the correction added is 0.138894816 but in the case of modified pi,
the correction is zero as shown below;
Ascending
= 0.148148148
Descending
0.148148148 x 9 = 1.333333333 x 3 = 4
The table below focuses light on the corrections inherent in the
approximation of pi in the case of Archimedes and Ramanujan and
makes it manifest that the pi proposed (0.148148148) is without any
correction in its method of perfection.
Table VI
Pi Ratio 1 Ratio 2 Numerator (R1)
Denominat (R2)
Correction
0.14285714
2
1.27x3.13=4 0.13889481
6
33
0.14159265
3
1.28x3.11=4 0.11111111
1
0.14814814
8
1.3333x3=4 zero
14. Hegemony of Testimony:3.148148148148148148148148148…
14.1. In order to appreciate the difference between the amount of
accuracy inherent in the re-determined and at present, a table is drawn.
It could be seen from the table that at the diameter of 9, the area of the
circle is 63.75 and at 18, the area is 255. Just as the span of 27 stars of
the zodiac fulfills the 360°, so also the wave of at every 27th sequence
repeats the quantum of circumference and area of the circle to full
number as it is evident from the table below; at the diameter of 27, the
circumference of the circle is 85 and the area is 573.75. At the diameter
of 54, the circumference is 170 and the area is 2295 and so on. It could
also be seen from the table below that while the value of 3.148148148
returns to full number in the case of both circumference and area, the
34
value of at present—3.141592653 is caught up in a net of fractal web
with an element of uncertainty as to what follows next.
Table VII
Dia/side Pi Modified
Circumferen
ce
3.148148148
Area
Pi Existing
Circumferen
ce
3.141592653
Area
1 3.148148148 0.787037037 3.141592653 0.785398163
2 6.296296296 3.148148148 6.28315306 3.141592653
3 9.444444444 7.083333333 9.424777959 7.068583469
4 12.59259259 12.59259259
5 15.74074074 19.67592593
6 18.88888888 28.33333333
7 22.03703704 38.5648148
8 25.18518518 50.37037037
9 28.33333333 63.75 28.27433388 63.61725122
12 37.77777777 113.3333333 37.69911184 113.0973355
18 56.66666666 255 56.54866775 254.4690049
24 75.55555555 453.3333333 75.39822367 452.389342
27 85 573.75 84.82300163 572.555
54 170 2295 169.6460033 2290.221
108 340 9180 339.2920065 9160.884
216 680 36720 678.584013 36643.5367
35
432 1360 146880 1357.168026 146574.146
864 2720 587520 2714.336052 586296.587
1728 5440 2350080 5428.672104 2345186.349
15. Oracle of the Octagon:3.148148148148148148148148148…
15.1. Every geometrical shape inscribed within the square has got a
definite relationship with the square. With the side of the square @ 9, a
table is drawn below. In the case of circle, both the figures of at present
and modified are shown. It is quite natural that as the figures have
been inscribed within the square, the dimensions of any other
geometrical figures ought to be lesser than that of the square.
Accordingly, in the case of regular octagon the area is 63 and the
circumference is 28. But in the case of the irregular octagon the area is
still 63 but the circumference is 28.9705. It is because in the regular
octagon, all the 8 sides are equal but in the case of an irregular
octagon, out of the 8 sides 4 are from the hypotenuse in line with
Pythagorean Theorem. However the sum total of area and
circumference of octagon whether regular or irregular ought to be less
than that of the circle as there is no geometrical shape in between a
square and a circle as a circle is inscribed within the square and that
square is circumscribed by the circle.
36
At 9 as dia/side;
Table VIII
Shape Area Circumference
Ratio Total
Square
81
36 2.25 117
Circle (-)Pi
60.75 27 2.25 87.75
Reg.Octagon
63 28 2.25 91
Irr.Octagon
63 28.9705627 2.17 91.9705627
Basket ofPi
Archimedes3.142857142
63.64285713 28.28571428 2.25 91.92857141
Reigning Pi3.141592653
63.61725122 28.27433388 2.25 91.8915851
Pi modified3.148148148
63.75 28.33333333 2.25 92.08333333
37
15.2. It is impermissible within the logical dictum of mathematics that
the sum total of the area and the circumference of the circle could never
be lower than that of the irregular octagon. It is evident from the above
that though the sum total of circle (91.89) with the ongoing
(3.141592653) is more than that of the regular octagon (91), it is less
than the sum total of the irregular octagon (91.97). Even the maximum
of Archimedes which works out to a total of 91.92857141 falls short of
the sum total (91.97) of the irregular octagon. But the sum total of the
circle (92.08) with the modified (3.148148148) is more than the sum total
of the irregular octagon (91.97) also. It is not only an unassailable
yardstick to uphold accuracy of the modified 3.148148148 but also
exposes the incurable infirmity inherent in the ongoing 3.141592653.
16. Vacuum in Velocity:3.148148148148148148148148148…
16.1. The person on earth in his inertial state is motionless. But earth
carries him around the sun at 1 lakh km/ph. Even though the person is
motionless with reference to himself, he is full of motion with reference
to the earth at 1 lakh km/ph around the sun. That means his supposed
38
inertia on the crust of earth is subsumed into the velocity of earth with
reference to the sun. In the same way, sun with all its planets with their
respective satellites goes in orbit around the galaxy at 8 lakh km/ph.
Now the velocity of the earth around the sun at 1 lakh km/ph is
subsumed within the velocity of sun at 8 lakh km/ph around the galaxy.
The person motionless on earth is actually whirling in the universe at 8
lakh km/ph. Therefore, as velocity increases with reference to ascending
symmetries the previous decreased velocity with reference to any
descending symmetry is subsumed into the next increased velocity with
reference to the ascending symmetry.
16.2. Even though the person on earth appears to be motionless thereby
in a state of inertia, considering the different velocities with reference to
different symmetries propagating into infinity, it is evident that person is
actually in motion thereby in alignment with the ultimate velocity which
is infinity governing and operating this universe. Pi ought to be
compatible with infinity. Then only Pi could operate the infinite cosmos
as a constant universal always. If Pi is finite, then the result would be
chaos instead of harmony. If the mass is whirling at the velocity of
infinity, it does not stand to reason as to how the energy content of such
mass alone could be restricted to the velocity of light. And in that sense
how Albert Einstein’s E = could withstand the stress of infinity will remain
a question for ever.
39
16.3. Just as there are different velocities subsumed within the ultimate
infinity so also there are different pis in relation to their geometrical
boundaries within the space of circle given below as illustrative which
are subsumed within the pi of the circle in alignment with infinity;
Pi of the right angle triangle = 3.14142135562
Pi of the polygon = 3.141592653
Pi of the octagon = 3.142857142
Pi of the trigonometry = 3.14711437
This is illustrative and not exhaustive.
They all operate within the Pi of the circle = 3.148148148 to infinity
17. Hinges of the Horizon:3.148148148148148148148148148…
17.1. There arises an irrevocable question in the realm of mathematics
as to whether Archimedes himself was displaced from the tub of pi. It
might be the first blunder of Archimedes analogous to the one
committed by Albert Einstein in the case of cosmological constant in the
year 1917. But fortunately, Einstein was alive to admit it, correct it and
acknowledge it as “the greatest blunder of my life”. But in the case of
40
Archimedes he was not aware of it as long as he was alive. But
unfortunately there were also none of the Friedmanns or the Edwin
Hubbles in the history of mathematics to contradict the line of thought
cherished by Archimedes. Instead when everyone in the queue fell in line
with Archimedes even though it is an error transparent on the face of
mathematics, still it has gained ground fostered by the majority who
justified the dictum to err on the right side of mathematics is also
human.
17.2. Perhaps had Archimedes not met with his tragic end in the hands
of invading enemies, in all probability sensing the incurable infirmity in
his conclusion Archimedes would have recalled his incorrect conclusion
and rectified the error apparent on the face of mathematics. But on the
contrary it is quite surprising to note for centuries subsequent to
Archimedes not only the masters of mathematics but also their
machines of impeccable precision have chosen only to enshrine the
incorrect conclusion of Archimedes by an inappropriate method of
approximation. This is equally applicable to Ramanujan also. Had not
the genius been pitted against the evils of illness, he could have smelt
the infirmity in the constitution of pi.
18. Sanctum of Singularity:3.148148148148148148148148148…
41
18.1. Space only has become time and circle has only become the
square. Therefore, within time we experience space. It is only the space
time unity and not that space and time are separate. is neither a
mathematical number nor a scientific number. All these years, it is we
who have invested the element of irrationality and transcendence into
the pristine purity of which spreads like the tress of a peacock across
the vault of space in its numerical excellence of 3.148148148
148148148 148148148 148148148 148148148 148148148
148148148 148148148 148148148 148148148 148148148
148148148 148148148 148148148 148148148 148148148
148148148 148148148 148148148…. and so on to infinity.
18.2. It is solemnly and sincerely averred that none of the contents made
in this research paper were borrowed from any of the resources both
ancient and modern, written or spoken but forged laboriously on the
anvil original in the laboratory of unprecedented thought as ordained
by the grace of my Maker. Everyone is at liberty to sow the seeds of pi
harvested in this land of enquiry across the latitude and longitude of this
planet without any restriction so that it may sprout again and again as
unflinching saplings to bear fruits of truth absolute.
18.3. The day when the whole world congregating in a conscientious
congress would rise in unison to accept the exact quantum of unalloyed
by any approximation, is not far behind but is round the corner. But how
42
many more centuries are to go before we become alive to the wakeful
accuracy from the sleep of approximation?
Written on behalf of the seeker of truth in everyone,
With all the regards,
Sivajayakumar.
E mail: [email protected]
Coimbatore, India.