ORIGINS OF PI

 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 22^nd 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