ORIGINS OF PI & ITS EXACT VALUE
ORIGINS OF PI
According to Encyclopedia
Britannica (2024), the value of pi to 39 decimal places is 3.141592653589793238462643383279502884197.
GUSUMS however argues that this value is wrong and the exact value to 39
decimal places is exactly 3.142857142857142857142857142857142857142. So who is
right?
According to the same Encyclopedia
Britannica (2024), the current value of as we use it was based on the discovery
or the work by Archimedes. As of May 5/17/2024 at 7:32 AM, the value of pi that
is currently in use was 2273 years, 4 months, and 16 days, or 830,334 days old.
GUSUMS argues that Archimedes was actually right and that the value of pi is
exactly 22/7. However, GUSUMS disagrees that pi can be any other unit like the
current value or that falls 223/71
< π < 22/7 or with an average ratio of 3.1418. GUSUMS offers the following mathematical
proofs.
PI IS A CONSTANT
The First mathematical proof that GUSUMS provide to is that PI is a constant. This is an easy proof based on the GUSUM definition of pi. The GUSUMS defines pi as the circumference of a circle with a diameter of one unit regardless of the unit used. Thus, if you take a circle with a diameter of 1miles the circumference will be pi miles. If the diameter is 1 inch then the circumference, will be pi inches. The same applies to any other units. This is why GUSUMS states that PI*d is not the formula for calculating the circumference of a circle and we only get the circumference out of sheer lack is dividing or multiplying a unit by one gives the same answer. The following image shows this concept
The above circle is measured in
units called sticks, yet the circumference still remains pi. Thus, pi is just
the circumference of a circle with a diameter of 1. I have explained this more
clearly in the link on the origins of metrology. Thus, pi using an average value,
we are already wrong. If this is true, it will mean our values for most of our
planetary objects are wrong as Odhiambo (2024) demonstrated in showing the
approximate diameter of the sun, and the perihelion of other planetary objects
such as Venus
PI
To show that pi does not vary, I
will incorporate the concept of linear and circular circumferences by
inscribing and circumscribing hexagons in and outside the circle to identify their
relationships starting with a diameter of 1. Apparently, this is the same way
that Archimendes and the Babylonians used to identify pi. The linear circumference refers to the
perimeter of a hexagon inscribed in a circle. The circular circumference on the
other hand refers to the circumference of the circle in which the hexagon
is inscribed.
Circle 1
Circle 2
If we double the diameter, the linear circumference becomes 6 and the circular circumference also doubles to 2 pi.
Circle 3
If we triple the radius, the
linear circumference also triples to 9 while the circular circumference also triples
to 3 pis.
Circle 4
If I were to quadruple the
patterns would remain the same. However. Watch the decimal places as they mean
something.
7 IS THE DETERMINANT OF CIRCULAR
VALUES
The following relationship can be established between the linear and circular circumference
|
DIAMETER |
LINEAR CIRCUM(LC) |
CIRCULAR CIRCUM (CC |
LC:CC |
CC:LC |
|
1 |
3 |
3.142857143 |
0.954545455 |
1.047619048 |
|
2 |
6 |
6.285714286 |
0.954545455 |
1.047619048 |
|
3 |
9 |
9.428571429 |
0.954545455 |
1.047619048 |
|
4 |
12 |
12.57142857 |
0.954545455 |
1.047619048 |
|
5 |
15 |
15.71428571 |
0.954545455 |
1.047619048 |
|
6 |
18 |
18.85714286 |
0.954545455 |
1.047619048 |
|
7 |
21 |
22 |
0.954545455 |
1.047619048 |
|
8 |
24 |
25.14285714 |
0.954545455 |
1.047619048 |
|
9 |
27 |
28.28571429 |
0.954545455 |
1.047619048 |
|
10 |
30 |
31.42857143 |
0.954545455 |
1.047619048 |
|
11 |
33 |
34.57142857 |
0.954545455 |
1.047619048 |
|
12 |
36 |
37.71428571 |
0.954545455 |
1.047619048 |
|
13 |
39 |
40.85714286 |
0.954545455 |
1.047619048 |
|
14 |
42 |
44 |
0.954545455 |
1.047619048 |
|
15 |
45 |
47.14285714 |
0.954545455 |
1.047619048 |
|
16 |
48 |
50.28571429 |
0.954545455 |
1.047619048 |
|
17 |
51 |
53.42857143 |
0.954545455 |
1.047619048 |
|
18 |
54 |
56.57142857 |
0.954545455 |
1.047619048 |
|
19 |
57 |
59.71428571 |
0.954545455 |
1.047619048 |
|
20 |
60 |
62.85714286 |
0.954545455 |
1.047619048 |
|
21 |
63 |
66 |
0.954545455 |
1.047619048 |
|
22 |
66 |
69.14285714 |
0.954545455 |
1.047619048 |
|
23 |
69 |
72.28571429 |
0.954545455 |
1.047619048 |
|
24 |
72 |
75.42857143 |
0.954545455 |
1.047619048 |
|
25 |
75 |
78.57142857 |
0.954545455 |
1.047619048 |
|
26 |
78 |
81.71428571 |
0.954545455 |
1.047619048 |
|
27 |
81 |
84.85714286 |
0.954545455 |
1.047619048 |
|
28 |
84 |
88 |
0.954545455 |
1.047619048 |
|
29 |
87 |
91.14285714 |
0.954545455 |
1.047619048 |
|
30 |
90 |
94.28571429 |
0.954545455 |
1.047619048 |
Note that the ratio of linear and
circular circumference are constant and do not vary. This implies that pi is
constant and does not change. Now, if you look at the decimal places you will
notice that they appear as follows;
|
DIAMETER |
D/7 |
|
1 |
0.142857143 |
|
2 |
0.285714286 |
|
3 |
0.428571429 |
|
4 |
0.571428571 |
|
5 |
0.714285714 |
|
6 |
0.857142857 |
|
7 |
1 |
|
8 |
1.142857143 |
|
9 |
1.285714286 |
|
10 |
1.428571429 |
|
11 |
1.571428571 |
|
12 |
1.714285714 |
|
13 |
1.857142857 |
|
14 |
2 |
|
15 |
2.142857143 |
|
16 |
2.285714286 |
|
17 |
2.428571429 |
|
18 |
2.571428571 |
|
19 |
2.714285714 |
|
20 |
2.857142857 |
|
21 |
3 |
|
22 |
3.142857143 |
Based on the above we can note that
pie can be based on numerous factors.
For example, 22/7 is equal to pie, 44/14 is equal to pi, 66/21 is equal to pi,
88/28 is equal to pi, and so on and so forth. Thus, what differentiates
circular from linear values is that circular values are dependent on the
presence of 7. In addition, just like 66 or 88, 22/7 is a by-product as it is
the 22nd circular unit and not the first. Thus, the first circular
unit is 1/7 as a fraction and 7 as a whole number. This shows that 7 is the
base number of circular values. Since pi is not the first circular value, then
there ought to be another means of measuring the circumference of the circle. After
careful analysis, Odhiambo (2024) identified the following formula.
Circumference of a
circle = Diameter/base number of circular units + d*the base number of linear
values
Base number of
circular units =7
the base number of
linear values =3
Circumference of a
circle = Diameter/7 + d*3
For example; the
circumference of a circle with a diameter of 1 is equal to;
Circumference of a
circle when D is 1 = 1/7 + d*3
=3
The following
table confirms that this is true
|
D |
D/7 |
D*3 |
D/7+D*3 |
|
1 |
0.142857143 |
3 |
3.142857143 |
|
2 |
0.285714286 |
6 |
6.285714286 |
|
3 |
0.428571429 |
9 |
9.428571429 |
|
4 |
0.571428571 |
12 |
12.57142857 |
|
5 |
0.714285714 |
15 |
15.71428571 |
|
6 |
0.857142857 |
18 |
18.85714286 |
|
7 |
1 |
21 |
22 |
|
8 |
1.142857143 |
24 |
25.14285714 |
|
9 |
1.285714286 |
27 |
28.28571429 |
|
10 |
1.428571429 |
30 |
31.42857143 |
|
11 |
1.571428571 |
33 |
34.57142857 |
|
12 |
1.714285714 |
36 |
37.71428571 |
|
13 |
1.857142857 |
39 |
40.85714286 |
|
14 |
2 |
42 |
44 |
|
15 |
2.142857143 |
45 |
47.14285714 |
|
16 |
2.285714286 |
48 |
50.28571429 |
|
17 |
2.428571429 |
51 |
53.42857143 |
|
18 |
2.571428571 |
54 |
56.57142857 |
|
19 |
2.714285714 |
57 |
59.71428571 |
|
20 |
2.857142857 |
60 |
62.85714286 |
|
21 |
3 |
63 |
66 |
|
22 |
3.142857143 |
66 |
69.14285714 |
|
23 |
3.285714286 |
69 |
72.28571429 |
Please note that as insightful as
this is, it is not a new discovery. What GUSUMS has only managed to achieve is to utilize the existing technologies to compensate for errors made by ancient
mathematicians and civilizations such as
the Egyptians by utilizing the current technology to enhance the accuracy of the
values. Looking at the above values, we can note a pattern in circular
circumference as they are either whole numbers or contain decimals places containing; 0.142857142, 0.285714285, 0.428571428, 0.571428571,
0.714285714, & 0.857142857. It is with this that the GUSUMS pi rules were
created.
THE GUSUMS PI RULES
From the above demonstrations, we can note the following GUSUM
SYSTEM Pi Rule
1.
The value of pi is exactly 22/7 or 3.142857142857142857142…….
2.
For all circles that have diameters that are natural numbers and
multiples of 7, their circumferences will always be whole numbers.
3.
For all other circles with diameters that are natural numbers and they
are not divisible by 7, then the decimal places can only have the following
values and in the exact order: 0.142857142,
0.285714285, 0.428571428, 0.571428571, 0.714285714, & 0.857142857. This
means that;
§ The first decimal place can only start with
1,2,4,5,7 or 8.
§ 1 must always be followed by 4
§ 4 must always be followed by 2
§ 2 must always be followed by 8
§ 8 must always be followed by 5
§ 5 must always be followed by 7
§
7 must always be
followed by 1 and the pattern keeps repeating itself.
The importance of this concept is
that it can enhance the accuracy of measurements to a match greater accuracy.
For example, if we have a rough estimate of the diameter of a planet and the
diameter is a natural number, we can use its circumference to limit its
circumference to just 7 values with a margin of error of between 0.142857142
and 0.857142857. I demonstrated this in time measurements. Note that due to
conversation to smaller units, these values often appear further down the line
when smaller units are involved e.g. when there are seconds, milliseconds,
centi-seconds, etc. involved.
4.
If we can
divide a circle into 6 equal points and inscribe a hexagon inside it, then the
circumference will always be equal to the perimeter of the hexagon times 1.047619048.
Further Explanation: From the book
The general agreement is that the exact value
of pi has something to do with 7. This means that all circular measures are in
one way or another influenced by the circle and 7. In addition, another area in
which we have consensus is that units of time are based on multiples of 6 up to
a day. Thus a second is 1, 6*10 seconds is one minute, 6*10 minutes is one hour,
60 * 60 seconds is also an hour, and 4*6 hours is equal to a day. However, when
we get to a week, we introduce 7 as a base value since 7 days is equal to a
week.
Therefore, based on logical reasoning, we can only pair 6 and
7 where the two meet or can be perfectly converted to a perfect natural number.
This implies the connection of 6 and 7 occurs at the least common multiple of 7
and 6 which is 42. If this is correct then we should be able to identify both
the first linear and circular unit of time in using 42 or the multipliers that
connect them. The table below shows what happens when you divide examples of
the linear and circular multiples of time.
Results
|
TIME MULTIPLES |
LCM |
TIME MULTIPLES/LCM |
|
LINEAR
MULTIPLES OF TIME |
LCM |
RESULTANT
MULTIPLIERS |
|
1 |
42 |
0.0238095238095238 |
|
6 |
42 |
0.1428571428571430 |
|
12 |
42 |
0.2857142857142860 |
|
18 |
42 |
0.4285714285714290 |
|
24 |
42 |
0.5714285714285710 |
|
30 |
42 |
0.7142857142857140 |
|
36 |
42 |
0.8571428571428570 |
|
42 |
42 |
1.0000000000000000 |
|
48 |
42 |
1.1428571428571400 |
|
54 |
42 |
1.2857142857142900 |
|
60 |
42 |
1.4285714285714300 |
|
3600 |
4200 |
0.8571428571428570 |
|
CIRCULAR
MULTIPLES OF TIME |
LCM |
RESULTANT
MULTIPLIERS |
|
1 |
42 |
0.0238095238095238 |
|
7 |
42 |
0.1666666666666670 |
|
14 |
42 |
0.3333333333333330 |
|
21 |
42 |
0.5000000000000000 |
|
28 |
42 |
0.6666666666666670 |
|
35 |
42 |
0.8333333333333330 |
|
42 |
42 |
1.0000000000000000 |
|
49 |
42 |
1.1666666666666700 |
|
56 |
42 |
1.3333333333333300 |
|
63 |
42 |
1.5000000000000000 |
|
70 |
42 |
1.6666666666666700 |
Interpretation
The above results show that it is indeed
correct that it is the LCM of 6 and 7 which is 42 that unifies both the linear
and circular units of measurements. This is because when you divide the linear
multiples of time 42 you get sub-multiples of 7 and when you divide multiples
of 7 by 42 you get sub-multiples of 6.
The above illustration can be confirmed by the table below.
|
NUMBERS (N) |
LINEAR MULTIPLIERS (N/6) |
CIRCULAR MULTIPLIERS (N/7) |
|
1 |
0.166666667 |
0.142857143 |
|
2 |
0.333333333 |
0.285714286 |
|
3 |
0.5 |
0.428571429 |
|
4 |
0.666666667 |
0.571428571 |
|
5 |
0.833333333 |
0.714285714 |
|
6 |
1 |
0.857142857 |
|
7 |
1.166666667 |
1 |
|
8 |
1.333333333 |
1.142857143 |
|
9 |
1.5 |
1.285714286 |
|
10 |
1.666666667 |
1.428571429 |
|
11 |
1.833333333 |
1.571428571 |
|
12 |
2 |
1.714285714 |
|
13 |
2.166666667 |
1.857142857 |
|
14 |
2.333333333 |
2 |
|
15 |
2.5 |
2.142857143 |
|
16 |
2.666666667 |
2.285714286 |
|
17 |
2.833333333 |
2.428571429 |
|
18 |
3 |
2.571428571 |
|
19 |
3.166666667 |
2.714285714 |
|
20 |
3.333333333 |
2.857142857 |
|
21 |
3.5 |
3 |
|
22 |
3.666666667 |
3.142857143 |
|
23 |
3.833333333 |
3.285714286 |
From the above table, we can confirm that the
previous table was converting linear values to circular values and vice versa.
Another important thing to note is that pi forms when N is equal to 22. This
implies that Pi is not the factor but a byproduct as it only forms after 22
units which implies the Circular Multiplier that the area or the circumference
should be based on is what I refer to as the GUSUMS circular multiplier which
is 1/7 or 0.142857142857142857142857………
Another means of ascertaining this using the
circle. Based on the illustration of pi, we did not that a circle of diameter
has a linear circumference of 3 and a circular diameter of pi. Since 3 forms at
21 then the linear or circular multiplier of a circle should be 3/21 or 3.5/21.
Alternatively, since pi appears at 22 the linear or circular multiplier should
appear when 3.666666667 is divided by 22 or when pi is divided by 22. If they
appear, then the relationship of the linear and circular should be based on a
ratio of 21:22 or 22:21 and vice versa. Note that throughout all calculations
the value of pi has remained constant and that there are no competing values.
The following table shows the result of doing exactly that.
Results
|
LINEAR/CIRCULAR VALUE |
DIVIDER |
RESULTS |
|
3 |
21 |
0.142857143 |
|
3.5 |
21 |
0.166666667 |
|
3.666666667 |
22 |
0.166666667 |
|
3.142857143 |
22 |
0.142857143 |
|
21 |
22 |
0.954545455 |
|
22 |
21 |
1.047619048 |
|
1 |
7 |
0.142857143 |
|
1 |
6 |
0.166666667 |
The above results show what we expected. The GUSUMS
linear multiplier is indeed 0.166666666 and the GUSUMS circular multiplier is
0.142857142858….. In addition, the above values suggest that the linear and
circular values of a circle relate in the ratio of 21:22. Thus, to convert from linear to circular value
we multiplier the linear circumference by the GUSUM SYSTEM circular multiple of
1.047619048 and to change the circular value to a linear value we multiplier
the circumference of the circle by the GUSUM SYSTEM linear multiple of
0.954545454545454….
Using
Other Base Numbers and their Linear Multipliers
If the above is correct, then using other
linear base numbers should give the same results. For, now I will limit the
values to 10 to 12 digits. The results are shown below.
|
BASE NUMBERS-BN |
7 AS BASIS |
LCM |
BN/LCM |
7/LCM |
BN/7 |
7/BN |
|
1 |
7 |
7 |
0.142857143 |
1 |
0.142857 |
7 |
|
2 |
7 |
14 |
0.142857143 |
0.5 |
0.285714 |
3.5 |
|
3 |
7 |
21 |
0.142857143 |
0.333333333 |
0.428571 |
2.333333 |
|
4 |
7 |
28 |
0.142857143 |
0.25 |
0.571429 |
1.75 |
|
5 |
7 |
35 |
0.142857143 |
0.2 |
0.714286 |
1.4 |
|
6 |
7 |
42 |
0.142857143 |
0.166666667 |
0.857143 |
1.166667 |
|
7 |
7 |
7 |
1 |
1 |
1 |
1 |
|
8 |
7 |
56 |
0.142857143 |
0.125 |
1.142857 |
0.875 |
|
9 |
7 |
63 |
0.142857143 |
0.111111111 |
1.285714 |
0.777778 |
|
10 |
7 |
70 |
0.142857143 |
0.1 |
1.428571 |
0.7 |
|
11 |
7 |
77 |
0.142857143 |
0.090909091 |
1.571429 |
0.636364 |
|
12 |
7 |
84 |
0.142857143 |
0.083333333 |
1.714286 |
0.583333 |
|
13 |
7 |
91 |
0.142857143 |
0.076923077 |
1.857143 |
0.538462 |
|
14 |
7 |
14 |
1 |
0.5 |
2 |
0.5 |
|
15 |
7 |
105 |
0.142857143 |
0.066666667 |
2.142857 |
0.466667 |
|
16 |
7 |
112 |
0.142857143 |
0.0625 |
2.285714 |
0.4375 |
|
17 |
7 |
119 |
0.142857143 |
0.058823529 |
2.428571 |
0.411765 |
|
18 |
7 |
126 |
0.142857143 |
0.055555556 |
2.571429 |
0.388889 |
|
19 |
7 |
133 |
0.142857143 |
0.052631579 |
2.714286 |
0.368421 |
|
20 |
7 |
140 |
0.142857143 |
0.05 |
2.857143 |
0.35 |
|
21 |
7 |
21 |
1 |
0.333333333 |
3 |
0.333333 |
|
22 |
7 |
154 |
0.142857143 |
0.045454545 |
3.142857 |
0.318182 |
|
23 |
7 |
161 |
0.142857143 |
0.043478261 |
3.285714 |
0.304348 |
|
24 |
7 |
168 |
0.142857143 |
0.041666667 |
3.428571 |
0.291667 |
|
25 |
7 |
175 |
0.142857143 |
0.04 |
3.571429 |
0.28 |
|
26 |
7 |
182 |
0.142857143 |
0.038461538 |
3.714286 |
0.269231 |
|
27 |
7 |
189 |
0.142857143 |
0.037037037 |
3.857143 |
0.259259 |
|
28 |
7 |
28 |
1 |
0.25 |
4 |
0.25 |
|
29 |
7 |
203 |
0.142857143 |
0.034482759 |
4.142857 |
0.241379 |
|
30 |
7 |
210 |
0.142857143 |
0.033333333 |
4.285714 |
0.233333 |
The results above confirm what was expected. Pi
is not the key factor, and the defining factor of circular units is the base
number of 7. Most importantly, you cannot pi at 22. This implies that the 22
which is the numerator in Pi is irrelevant and can be changed as it is not the
basis but a bi-product of unifying the linear and circular measures. Thus, from
1 to 30 30 other units can take up its value and we can still calculate the
circumference or diameter of the circle. The most important figure is the GUSUMS
base number which is 7 or the Gregorian circular multiplier which is
0.142857142858.
Note that the ability of other numbers not
identifying as the natural numbers that can divide 360 to provide the same
value does not contradict the statement that the linear values only draw their
basis from the natural numbers that can divide Pi. This is because I identified
one as one of the natural numbers and every number can be divided
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