ORIGINS OF PI & ITS EXACT VALUE
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
