A DISCOVERY,
WHICH COULD NOT BE
A DISCOVERY
AFTER 2000 YEARS
OF SCIENCE,
BUT
MAYBE YET IS!
© September 2013 by Joost Gielen – no part of this publication may be reproduced, stored in a retrieval system or transmitted
in any form or by any means without the prior written permission of the author.
Preface:
Yes , Pi = 3.14159265 … and several billions of decimal places !
But…. if Archimedes had not been so quick and had made the same approach as de-scribed in this publication, he would have undoubtedly chosen the number 3.1416407864998 … !
This fact implies the overwhelming thought that over the past 2,000 years, including the development of, for instance, science, had been totally different! Pi might have perhaps never existed! And even 3.1416407… would not have been needed for the calculations to circle and sphere!
This is a simple story about the slightly different number of Pi.
The statement here is: you have a sphere with surface 6, a volume of about 1,382 and….. radius and pi are unknown.
Calculating Pi from the data of a sphere has led to remarkable results. The sequences and numbers that I have found, do appear almost nowhere. All search engines, like Google, give virtually no results. The site Symbolab (Symbolab.com) which searches scientific articles and texts for numbers and formulas, gives almost no result for the formulas and numbers that I’ve found.
The OEIS, (Online Encyclopedia of Integer Sequences), a database of sequences, founded 50 years ago, with far more than 200,000 sequences, did not contain 6 or 7 of my found sequences and accessory formulas!
Some formulas pop-up here and there, which have the different value of Pi, but only to mention that this is a good approximation of Pi.
If my derivation is mentioned somewhere in math-books, I do not know. I couldn’t found out. If so, then it has been snowed-under and didn’t appear searching the Internet.
In the first chapter I propose a simple story, displacing myself in the time of those days, showing how things might have happened.
Then I give an interpretation of formulas with the knowledge of these days.
Small reflections on the implications of this approach are at my account, but I invite eve-ryone to give his thoughts the free run on the subject.
Could it be possible that we’ve chosen the “wrong” path?
And the big question is: why does our Pi differ from nature, from life? Joost Gielen
16-09-2013
JuPee
About eighty miles from Alexandria in Egypt, in the town Dibi, lived a boy called JuPee. It was around the year 250 BC.
From his 10th it was his duty to herd the goats every day. To pass the time he was al-ways looking around for round pebbles along the bed of the river. That fascinated him: the perfection of round shapes. At home he had been given from his father, who was a carpenter, a beautiful, perfectly crafted wooden box in which he kept his treasures.
The intention was that he gradually should learn the trade of his father and became a car-penter too.
In the meantime he was engaged in making round balls partly made of twigs. Over the years, he got very skilled; the last ball that he had made was very strong, beautifully rounded and … you could play football with it. The twigs were stripped of its bark, and he had them polished with sand and stone so they were all equally thick, straight and super smooth.
From his father he had learned to count and the boy managed to make a ruler too. He carved it gently and rubbed it with ashes and soot so that the small lines became clearly visible. His last, almost perfectly shaped ball, did he keep in his treasure box; pebbles were moved to a pouch.
When he was 15 his father came back from Alexandria with a lathe. JuPee’s job was mainly to operate the rope of the lathe for his father. His father made rounded wood sometimes with thickened parts such as rings and balls.
In the years that followed he learned to handle the lathe and in his spare time he tried to make balls. After a few years he made one that was so perfectly round, that it could be polished. To his own surprise did the ball fit exactly in his wooden box!
He could not even place the thinnest twig between the wall and the ball. The lid of the box just hit the ball when closed.
Intrigued by this coincidence, he began to measure the box.
Somewhere around the age of 16 he was once outside the village and encountered a fo-reigner who had fallen of his camel. Moaning loud in pain because of his broken leg. With the help of his father he had transported the man to their home. With instructions of the man (who apparently had some medical knowledge) they had splinted his leg.
The man, named Conon and from Alexandria, stayed with them for months and he edu-cated JuPee in mathematics in the meantime. Conon thought he was an intelligent boy with a natural aptitude for numbers and math. When Conon could travel again and went back to Alexandria, he invited JuPee to come over some day.
With the knowledge he started to measure the box as precise as possible.
For that he took his best ruler and calculated edges, area and volume of the box. That turned out to be a perfect cube! Did his father made it that special or was it coincidence? Knowing his father, it had to be on purpose: he was a perfectionist with a great sense of proportion. He was the best carpenter around! His self-made furniture was wanted, even in Alexandria.
JuPee’s measurements produced the following:
The side of the cube was 1.38 twig or thereabout.
Thus, the surface of one side was about 1.9044.
So the entire surface of the coffin was 6 x 1.9044 = 11.4264
The volume of the box was, after some calculations, 2.628072.
Those numbers told him that much. He also realized that they were not very accurate. So he went frantically to deal with all the possible abnormalities.
1:39, 1:38, 1:37, 1,385, 1,389, 1,388 to 1,381, etc.
1.3811, 1.3812, 1.3813, 1.3814, 1.3815, 1.3816 to 1.3819 etc.
1.38191, 1.38192, 1.38193, 1.38194, 1.31959, 1.38196 and then ….
then he noticed something!
At 1.38196 the numbers began to speak to him!
He tried the following:
1.381966 x 1.381966 gave 1.909830025156
and
1.381966 / 2 gave 0.690983
They looked alike!
He also found
1.381966 x 1.381966 x 1.381966 = 2.6393201605447367…
2.6393201605447367 / 2 = 1.31966008027236835…
Curiously, the number 8 disappeared but the numbers seemed sequences of the same. Would all the digits also apply to the range of 1.381966?
So this would be 1.381966008027236835?
After many calculations, he came to the following:
1.381966011250 x 1.381966011250 = 1.909830056250
1.381966011250 / 2 = 0.6909830056250
1.381966011250 x 1.381966011250 x 1.381966011250 = 2.639320225002
2.639320225002 / 2 = 1.31966011250
1.909830056250 x 2 = 3.81966011250
1.909830056250 x 4 = 7.6393202250
He now had three sequences: 909830056250, 1966011250 and 6393202250.
For months he was amazed about these three sequences but could not go further with it.
He could determine the circumference of the sphere by placing a piece of string around it and measured it. The circumference was approximately 4.34 twig. How to find the vo-lume and the surface of the sphere he did not know.
To determine the surface he had an idea: he had seen his mother spinning yarn but did find varying degrees of thickness too great to use, though his mother was very good at spinning the threads.
For his idea he needed much thinner wires. The few horses that he had seen in his life, reminded him of the long tails. After a few months he had sought and founded sufficient horses to cut enough long hairs from their tales.
He surrounded his sphere in an ingenious way with hundreds of long hair. When he had finished he took it gently of and measured the surface they occupied! After several at-tempts, he came to the conclusion that it had to be about 6 twig!
That reminded him of a box of 1 x 1 x 1: who also had an area of 6.
However, he still didn’t know how to determine the volume with these data.
So far, he had the following numbers:
box:
side : 1.381966011250
volume : 2.639320225002
surface side : 1.909830056250
surface box : 11.45898033750
sphere:
diameter : 1.381966011250
diameter / 2 : 0.690983005625
Area : 6
circumference : 4.34
volume : ?
Furthermore, he had tried all possible combinations and that gave him a.o. the following numbers:
6 x 1.381966011250 = 8.2917960675 1.381966011250 / 6 = 0.2303276685
6 x 2.639320225002 = 15.8359213500 2.639320225002 / 6 = 0.4398867041
6 x 1.909830056250 = 11.4589803375 1.909830056250 / 6 = 0.3183050093
6 x 11.458980337500 = 68.7538820250 11.458980337500 / 6 = 1.9098300562
6 x 0.690983005625 = 4.1458980330 0.690983005625 / 6 = 0.1151638342
6 / 1.381966011250 = 4.3416407865
6 / 2.639320225002 = 2.2733126291
6 / 1.909830056250 = 3.1416407865
6 / 11.458980337500 = 0.5236067977
6 / 0.690983005625 = 8.6832815730
1 / 1.381966011250 = 0.7236067977 2 / 1.381966011250 = 1.4472135955
1 / 2.639320225002 = 0.3788854381 2 / 2.639320225002 = 0.7577708763
1 / 1.909830056250 = 0.5236067977 2 / 1.909830056250 = 1.0472135955
1 / 11.458980337500 = 0.0872677996 2 / 11.458980337500 = 0.1745355992
1 / 0.690983005625 = 1.4472135955 2 / 0.690983005625 = 2.8944271910
etc.etc.
The numbers told him that much.
The only thing clear was that he had found another sequence:
416407865
This sequence came from:
6 / 1.909830056250 = 3.1416407865
and
6 / 1.381966011250 = 4.3416407865
He could not easily find the links and pondered how he could find the volume of the ball.
One day he was in a fen next to the river to wash himself.
He saw the water on the side rise when he stepped into it.
That got him thinking: first he only went up to his knees in it and saw the water go up slightly. Then he sat down in the water and the water went up even more.
He thought for a moment and then slid gently into the water until he could see how the water slowly rose.
Suddenly he knew! He jumped out of the water and shouted:
“Halllelujah; bingo JuPee!”
He ran home and searched his sphere and box.
With an old rag he went to the river, poked small holes in the rag, so he had a sieve to collect the finest sand. After drying, he filled his box, with globe, with sand.
Very carefully he took the ball out. After some knocking and gently shaking the sand he had distributed it as evenly as possible over the box.
Curious about the outcome he measured the distance to the sand.
It was somewhere between 0.72 and 0.73 twig.
Half the height of the box was 0.690983005625 he knew of his tables.
So the volume of the sphere was slightly larger than half the box!
Frantically he began calculating.
But in his table he had noted: 1 / 1.381966011250 = 0.7236067977
That had to be it!
His calculations gave the following:
1.381966011250 x 0.7236067977 = 1
0.7236067977 x 2 = 1.4473213954
0.7236067977 x 0.690983005625 = 0.5
0.7236067977 – 0.690983005625 = 0.032623792075
0.7236067977 + 0.690983005625 = 1.414589803325
0.7236067977 x 0.7236067977 = 0.5236067977
Again a sequence! 236067977
The following calculation almost made him drop down:
1.381966011250 x 1.381966011250 x 0.7236067977 = 1.381966011250 !!
He could not believe it and barely understand. Again and again he made the calculation, but then had to conclude that
the volume of the sphere was
the same number as the diameter!
He became dizzy of all the numbers and the way they were similar.
He now knew the volume of the sphere. Half the volume of the box was:
1.381966011250 x 1.381966011250 x 0.6909830056 = 1.319660112453
It was slightly smaller than the volume of the sphere.
He kept struggling how to do next.
The number six ghosted in his head and at a given moment he divided the volume of the box by 6.
So:
2.639320225002 / 6 = 0.439886704151
With that number he divided the volume of the ball:
1.381966011250 / 0.439886704151 = 3.14164078628295
HE HAD SEEN THAT BEFORE !!
The number 0.690983005625, half the edge of the cube, became divided by 3.
It gave 0.230327668541.
He divided the distance to the sand in the box without ball by this number en got:
0.7236067977 / 0.230327668541 = 3.14164078628295
Slowly something began to emerge; he started to see the links!
This was an important number!
1/6 part of the box times 3.14164078628295 was equal to the volume of the ball with di-ameter 1.381966011250.
The surface area of the ball was 6.
Divided by 3.14164078628295 gave 1.90983005638239
and that was equal to
1.381966011250 x 1.381966011250 in other words: diameter x diameter.
The circumference of 4.3416407865 divided by 3.1416407865 gave 1.381966011250.
To be quite sure of his case, he started making a wheel that was much larger than his wooden ball. So he could make better measures.
After weeks of tinkering he had made a large circle of wood that was almost perfectly round.
Now he sought for a flat piece of land or a flat rock where he could run his wheel on, to measure the circumference. Near the river he found what he was looking for.
For days he turned his wheel and gently rotated it with small marks on it and on the soil. He could determine the circumference quite well. After some recalculations he came to the conclusion that the number indeed matched 3.14xxx !
IT HAD TO BE 3.1416407864998 !!
Meanwhile, the town Dibi had gradually grown.
The reason was that there had been grown a good passable spot in the course of the river. The trade route from Cairo to Alexandria now passed by Dibi. So JuPee had no com-plains about clientele for his carpentry.
When he was busy with his wheel close to the river, an older man watching his operations came to him after a while and asked:
“What are you doing all the time?”
“Well,” said JuPee, “I try the circumference of the wheel to be calculated. This stands in a certain relation to the diameter of the wheel I think”.
“And how did you get this knowledge?” The man asked. “How can you calculate that good?”
“It has being taught to me by Mr. Conon who stayed here for a while cause of a broken leg. A few years ago” JuPee said.
“Oh, all right” the man said. “Good work young man, keep it up, there you can learn a lot.”
“Oh, but I already did” JuPee said, “I have …. ”
But the man had already moved on.
JuPee watched him.
The man, named Archimedes, slowly disappeared out of sight.
And JuPee?
He continued his calculations.
But was also married and had a whole batch of children. As a result, he had to work hard to earn a living. His calculations went into the background and the box with ball came to lie in a dusty corner.
After ten years he had the chance to go to buy a new lathe in Alexandria.
During those days, he came with his lathe under his arm along a large building with a huge patio. It was crowded with young men.
Suddenly he saw some young men with a big wheel rolling it cautiously over the patio. He walked up to it and asked:
“What are you guys doing?”
“We are studying mathematics and we are now measuring the circumference of the wheel. We try to determine the exact proportion to the diameter.”
“And you already know how much it is?” JuPee asked.
“It should be somewhere between 223/71 and 22/7, which is roughly between 3.141 and 3.143”
“I know what it…”
“If you do not mind, we go on. We should be concentrating ourselves.”
“Yes, but …”
“Come on carpenter, leave us alone. Go to work too.”
JuPee shrugged and left.
After a fruitful life with many children and grandchildren the time of departure from this world came for JuPee.
On his deathbed, just before he died, JuPee muttered:
“But I’m sure it should be that value.”
His wife asked gently:
“What value, my dear husband?”
“3.1416407864998”
PI and Radius UNKNOWN
Ball surface area 6, volume about 1,382
How to determine the balance between PI and the Radius?
As stated above, Pi is unknown. I.e. the ratio of circumference to the radius of the circle is not calculated. Pi doesn’t exist as a manner of speaking.
How can we find the volume of the sphere and how determine Pi? Or the radius? Or …. is pi really necessary?
As the story is clear, the volume is not so difficult to determine.
What is important in this context, is the fact that the number for the volume of the sphere is equal to the number of the diameter!
We first calculate a while with Pi as an example:
So: volume bol = diameter
Or: with now a days Pi:
4/3*pi*r3 = 2*r
which leads to:
r = √1.5/pi
r = 0.690988298942670953…
and
d = 1.381976597885341906…
and
d2 = 1.909859317102743998…
The relations in a cube with edge 1 and
a total surface area of 6 are:
The Pi-numbers translated to a cube will give the following:
(diagonal cube = diameter ball)
all numbers at row for the ball:
As to be seen the sequences do not correspond well.
radius: 0.6—90988298942670953…
and
diameter2: 1.—90985931710274399…
or
pi: 3.1—4159265…
and
circumference: 4.3—4160752…
The surface area of a cube with edges 0.797884560802 is
0.7978845608022 x 6 = —3.8197186341991176…
compared with
1.—381976597885341906…
A cube with edges 1.90985931710274399 has a volume of:
(1.90985931710274399…)3 = 6.—96633143757108923459…
compare this with:
1.381—976597885341906
So much for some numbers which are related to Pi.
As seen in the story of JuPee, he came to the number 1.38196601125 by searching for a proper ratio of the numbers that he had found.
In addition, he met up with the sequences that occurred in different numbers listed.
What better can you wish for?
Again some of the numbers found by JuPee at a glance:
0.6—90983005625
1.—90983005625
3.1—416407864998
4.3—416407864998
(1.—90983005625)3 = 6.—966011250
1.381—966011250
The JuPee-numbers translated to a cube giving this:
and some JuPee-numbers at row for the ball:
In determining the diameter of the sphere you come, by trying, ever closer to the correct value of the diameter.
If it suddenly shows that, at a diameter of 1.381966001125 …, sequences start to appear in multiple calculations, it’s apparent (especially in those days) that one immediately would opted for this outcome! Since the numbers all fitted too well to do not.
In other words, one would have accepted the values found as being right!
They were not looking for Pi, but for a method to calculate surface, volume and circum-ference of a circle and a ball.
From there on, one would then have linked numbers in any way related to each other.
One of those numbers would have been 3.1416407864998 … but…it would not have been special !!!
There would have been lots of ratios to make:
Eg.
Bol:
ratio surface – diameter:
6 / 1.381966001125 = 4.3416408183093174
ratio diameter – surface:
1.381966001125 / 6 = 0.2303276668541667
ratio surface – volume:
6 / 1.381966001125 = 4.3416408183093174
ratio volume – surface:
1.381966001125 / 6 = 0.2303276668541667
ratio perimeter – volume:
4.3416407864998 / 1.381966001125 = 3.1416408428286796
ratio volume – perimeter:
1.381966001125 / 4.3416407864998 = 0.3183050036679613
ratio perimeter – surface:
4.3416407864998 / 6 = 0.7236067977499667
ratio surface – perimeter:
6 / 4.3416407864998 = 1.3819660112501286
ratio surface – radius:
6 / 0.6909830005625 = 8.6832816366186347
ratio radius – surface:
0.6909830005625 / 6 = 0.1151638334270833
ratio perimeter – radius:
4.3416407864998 / 0.6909830005625 = 6.2832816190347000
ratio radius – perimeter:
0.6909830005625 / 4.3416407864998 = 0.1591525035215006
ratio volume – radius:
1.381966001125 / 0.6909830005625 = 2
ratio radius – volume:
0.6909830005625 / 1.381966001125 = 0.5
PI NOT NEEDED
And if we rely on the found numbers, we will see that we wouldn’t have needed Pi at all to make calculations for circle and sphere!
For the record:
For the question “a sphere with surface 6 and volume about 1382: what is the radius and what is Pi” we have found the radius!
So if we take the diameter found, as a starting point, we come to the number 3.1416407864998 …
Adjusted in the formula r = √1.5/pi
gives
d = 1.381966011250
r = 0.690983005625
0.690983005625 = √1.5/pi
pi = 3.1416407864998…
BUT….
as said here-above: we wouldn’t have needed Pi!
Due to the internet we find the following formulas:
After some calculations we get the following formulas for circle and ball if we take a di-ameter of 1:
(for ease we call 3.1416407864998… = PIjg )
= 6 * (φ +1)/5
= 6 * φ2/5
= 3 (3 +√5 )/5
= 9/5 + √(9/5)
= φ2 * 1.2
= √1.8 + 1.8
etc. etc. etc.
WE WOULD NOT HAVE NEEDED PI!
What would have been the consequence?
What would we have experienced?
What would science have encountered?
What is the reason that this has not been found by Archimedes
What if …
PERIMETER FOREVER
More than 2000 years ago a man, called Archimedes, went to calculate the ratio between diameter and circumference of a circle.
Simply said, he divided a circle in 6 parts, then 12 etc. After about 96 parts he could say that the ratio is somewhere between 223/71 and 22/7.
He writes it down on paper and it is cataloged as a proposition of Archimedes.
Many hundreds of years nothing happened in this area anymore.
This 2D concept will only be picked up somewhere around 1500 AD.
Then there seems to start a race: who knows to calculate the correct value of Pi. Who knows to find a formula which gives all digits of Pi. Who knows to find the end of Pi.
You could say, that science has had been sucked into the “screw” of Archimedes for more then 2000 years.
Archimedes was right? Yes.
Has science and math been exact as possible during the time? Yes.
Could it have been developed in another way? Yes!
Consider for example the unit of measure or the unit of weight.
Until the 18th century, just about every city had its own unit of measure. This was not very conducive for trade, evidently.
So there was chosen a new unit, which was recorded and preserved in Paris: the meter.
The definition of a ‘meter’ is changed several times in the course of history, because of more accuracy through the development of science, in relation to the standard definition used until then. The meter had not been changed (a single exception not mentioned), but made more accurate. (see Wikipedia)
With the advent of computers, lasers, atomic clocks etc. many units could be redefined. In other words they were adjusted. Did this have any effect on ordinary life? Not at all.
As time progresses and technology will become more precise units will be re-defined. Soon, one can say the meter is a much lower part of the speed of light at a temperature of element x or y with a deviation of 1/10th of a quark …….eg.
Unfortunately, this process of time, knowledge and technology has not been improving the values to Pi. Pi was almost directly determined! For the common man 3.1416 would have been fine. The only “improvement” that took place was to determine even more useless, billions of digits. Pi was to be defined as the ratio of circumference and diameter and it has been fixed for centuries.
Science has been doomed and burdened with the “sacred number Pi”.
The “sacred number Pi” which, according to my arguments, had not been needed if only we would have accepted a very small deviation around the 7th or 8th digit!
Only far in the 19th century a more precise Pi would have played a role (possibly) in the field of atomic physics and quantum physics. Or should we have had entirely different numbers for our calculations and constants?
Would √5 have taken the task of pi?
Or Phi?
Phi …. the neglected one of mathematics, the beautiful ugly duck of science beside Pi … Well, it would have given it a leading star role!
Imagine how Fibonacci, after finding his fabulous numbers, could have integrated these into mathematics with ease and jubilance.
Imagine that we could trace back formulas, links and connections to some basic concepts like numbers as root 5.
Imagine that perhaps the computer-era had started already 100 years ago….
Imagine how your own struggles at high school, with circles, spheres, complex pi formu-las and difficult derivations, would have been more clearly, logically and easily to under-stand.
Imagine how nature could have been explained in a much easier way. How “naturally” mathematics would have been.
Imagine ….
It looks a bit like the question :
how would my life have been
if I was born a girl instead of a boy
Answer could be: it hasn’t been that way; you are born a boy, so that question is irrele-vant. You can’t tell, so it has no use to think of. In short, you are a boy, basta! Learn to live with it.
Well …
With this described “discovery” it’s perhaps different.
Who knows what results of chemical or physical processes will be if we fill in this value of “just not pi”? Outcomes of calculations might be better or fall into place? Could we be looking for differences? Why does differ pi from nature?
NATURALLY
Of course I can be wrong and/or have made the wrong assumptions. This mainly due to lack of knowledge and understanding.
That does not explain why I can’t find information on this given “volume = diameter”.
Looking for the number 0.690988 and 0.690983 and 06.90951 (earlier examinations of mine), I did found on the Internet, after an intensive search, almost nothing but an article by Jean-Claude Perez. An article describing the ratio between the letters TCAG of DNA. Perez has found that one ratio is 0.690983!
Either: (3 – phi) / 2 !
The last article of J-C. Perez is included here as an appendix.
(complicated for non- insiders , but nonetheless informative)
Of course everyone has heard of Fibonacci, phi, golden ratio etc. Perhaps many know that in nature leaves are attached to the stem in a specific range to catch the maximum sunlight. How the seeds of a sunflower are embedded like spirals. How the scales of a pine cone have a specific rank-settlement. Etc. etc.
Do these arrangements that exists in nature, are really here cause of Phi or are other rea-sons at stake?
What Perez has been discovering here, can not be pushed aside as “unnatural”.
If something is natural, it is called DNA!
When I found his article I emailed him in 2012 to tell him that I had found his article in after checking my calculations on the internet. I asked him if he could use it for further investigations.
Recently, he has published his latest findings (see Appendix). A mention in the article of my discoveries, which he found on my blog site, turned me back to my findings and made me do more studies: this little publication is the result!
Of course it is absurd to call 2000 years of science into question. But no matter how long I turn it around and around, I can’t escape the awesome thought that, when in earlier time there was chosen for the simple
volume = diameter = 1.38196601125010515 …
the world would have developed differently!
………………………………………………………………………………………………
………………………………………………………………………………………………
CONCLUSION
I’m not a mathematician, not a scientist nor a philosopher.
I think I’ve found something peculiar.
It’s quite arrogant to say that science has been narrow-minded for over 2000 years.
It can’t be possible!
I don’t mean to be arrogant, just humble by the idea that I could be right.
If I’m not right…well, I’ve tried!
Joost Gielen
Attin, France
September 2013
