Approximations of Pi

Pi is the ratio of the circumference of a circle to its diameter.

It is “irrational number” and its decimal expansion therefore does not terminate or repeat.

The first 10 decimal places are:

3.14159 26535 …

Fractional approximations to Pi

There are few fractional approximations to Pi (that give very close approximation of the “pi”).
In order of increasing accuracy:

22/7 = 3.142 857…
333/106 = 3.141 509…
355/113 = 3.141 592 92…
52163/16604 = 3.141 592 387…
103993/33102 = 3.141 592 653 011…

Expressions used to estimate Pi

The following expressions can be used to estimate pi:

accurate to three digits:

${\sqrt {2}}+{\sqrt {3}}=3.146^{+}$

${\sqrt {15}}-{\sqrt {3}}+1=3.140^{+}$

accurate to 4 and 5 digits

(3/5) * (3+sqrt5) = 3.14164….

(6/5) * Phi²= 3.14150…

Sexagesimal System of counting

Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.

60 is superior composite with 12 division factors: 1,2,3,4,5,6,10,12,15,20,30.

Very close approximation of Pi expressed as sum of fractions in sexagecimal system of counting

3+(8/60) + (29/60²) + (44/60³) = 3.1415926
(exact value up to the same decimal point: 3.1415926 )

Drawing Pi with Ruler and Compass

A line of very close approximation of the Pi lengths can be drawn with a compass and ruler (straight edge).

Here we present two methods of drawing good approximations of Pi.

This construction was given by the Polish Jesuit priest Kochansky (Steinhaus 1999).

This compass and ruler method is based on approximation of Pi as:

3.141533…

In the figure below, OA=OF=1 , and the circle centered at A of radius 1.
Mark point E so DE=3.
EF = 3.141533…

2. This Construction was given by Alex Sokolowski

This compass and ruler method is based on approximation of Pi as:

(3/5) * (3 + sqrt5) = 3.141640….

3. Another, much more accurate, compass and ruler construction of Pi

Faster version of the above method:

Proof (based on properties of similar triangles and Pythagorean theorem)
Note: 0.875 is exactly 7/8

OF² + FE²= 0.5²OF/FE = OP/PD = 1/0.875
GR/FP=RD/DP = (0.875-FE)/0.875
HE=EG=1-OF-GR
IH=OF-HE

Solving these equations gives us:

OF=4/sqrt(113) = 0.376288347…

GR=(1-OF)²= 0.389016226…
HE=EG=OF(1-OF) = 0.234695427…

IH=OF²= 16/113= 0.145929…

Squaring the Circle

Draw two squares with side = 1 unit.
Draw sqrt(2) as diagonal of 1 square and sqrt(5) as diagonal of 2 x 1 rectangle. Point H is located at 0.5 * Pi distance from the middle of the left square G.
Half of GH generates side of the square with perimeter equal circumference of the circle with diameter – 1 unit.

This simple geometric construction (compass and ruler) method provides squared circle with perimeter accuracy of 0.0002.
For example, for a circle with 1 m diameter, the square generated with this method would have perimeter longer only by 0.2 mm from the circumference of the circle.

PS Decoding the Pi

If there was ever a magical number, it is pi, the mathematical constant whose value is the ratio of the circumference of a circle to its diameter. Pi appears in countless formulae in mathematics and engineering and is roughly 3.14159 though because it is an irrational number (meaning it will continue infinitely without repeating), it is forever the subject of efforts to calculate it to the most number of decimal places. Since 1995, the record has been held by supercomputers, which have progressed the record to 2577 billion decimal places. Now the record has been broken again, but this time by a desktop PC with a humble Core i7 CPU running at 2.93 GHz, with 6 GB RAM and 7.5 TB of disk storage. Congratulations to Febrice Bellard, who set a new record of 2,699,999,990,000 decimal places in December 2009. Febrice’s web site has all the available information.

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The following segment is a comment about Pi sent by our visitor

Pi = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375…

I have always been interested in astronomy and cosmology from the view point of ancient cultures, but that aside, one has to look at their overall understanding of calendars, time and dates, buildings, sacred monuments, mathematics, geometry, their city centers, trade routes, technology, language of priests and peasants, the language of their Gods and the way they civilized the population, war and kingship, battles and war tools, what they taught, their laws, traditions, dialects oral or written, influences that affect stability, artifacts and materials used, whom they went to for guidance or intervention, medicine and how they cultivated or divided the land, their desire or aspirations, skills of carpenters, masons, miners, access to water and spices used, establishment of trade routes, names of famous writes or advisers and leaders that supported the structure and community and how they protected themselves from outsiders etc, etc.

There are so many theories being put forward by G Hancock, R Buval and others and i am in no doubt that we have to come to the “table” with an open mind. We know that humans have existed for 100 million years and can draw the broken tree of our ancestors. We need to explain how different parts of Earth developed as it is clear that whilst the middle-east was in the bronze age, other parts were still existing in the stone age (3rd world status), but they may have supplied materials for bronze implements, without having the technology to make such products. The ice age may not have covered Europe with a ice sheet 2km thick, maybe some pockets remained where people lived in isolation, we do not know all the facts. We cannot trust GOVTs or scientists to tell us everything – they are not idiots. Look at the mess we are in now and they will deny all knowledge that they had some notion of the suffering – we are groomed not to tell the truth, because if we get away with it, that is all to our advantage for survival.

I do believe that each of us has to look at all the aspects as far back as possible, and see how the facts need to be interpreted for all humanity. This is our ancient heritage, whether this relates to Pyramids or Stone Circles, any artifact has to be placed as to identify what was the thinking process and what was its purpose. If we are divided in opinion we will fail to get to the truth.

Give you an example: we know that even nomadic hunters had to follow the food trail and moved from place to place in order to survive. Once we started lived in a fixed point, we had to get the food, water, tools, knowledge etc to come to us. We had to therefore interact, so whom did we interact with and why? Being fixed meant suffering bad and good conditions, so how did we survive the difficult months, how did we adopt to changing climate and natural disasters, disease and in fighting.

When did we first seem to know about the mechanics of the Earth, by looking at the stars and building an idea of time, by looking at the day/night cycles, months and seasons, so we knew when to sew and harvest, when to have children to give the best chance of survival, how to explain these movements of stars to new generations and give them a better chance of survival. When did we look at each aspect of life as a cycle, like the moon and planets nearest to our own Earth. How did we encode this information of cycles of time and space into great Epic Sagas developed into cosmic cycles and myths and legends so each generation could remember their heritage and understand social responsibility.

We talk of Plato and Atlantis, King Solomon’s mines, the Temple of Solomon, The Books of the Jews and Israelites and how the cosmos is 13.6999xxxxx billion years old, but despite all this know how and intellectual depth we have more problems today then our ancestors faced, more financial tsunamis each day and maybe its just my opinion but we have not found ONE DAY on Earth we celebrate peace, it eludes us, its not in our grasp, but we are more interested in wealth and achieving immortality. If the fact that we could achieve IMMORTALITY just to experience being inside every atom at the same time, and this was FREE, would the GOVT tell you or the priests tell you? absolutely NOT, because they will not be able to have control over you, but it is a state of happiness you can get by searching in the right places and not just for yourself but for all humanity, so if you achieved it yourself, would you share it? absolutely NOT, because people will treat you, as if you are mad.

We know that life is hidden inside a millet seed, so IMMORTALITY is also hidden inside a millet seed, would be a good way of looking, would you agree?

In all of humanity I know only TWO people who have IMMORTALITY, one died and lived again and the other never had experience of death, he never died and yet in all our search we do not understand one and the other is never mentioned, in fact he is only mentioned once of in Genesis and so you have to ask why these two, what was special about these two and how do all the other facts try to help us to achieve the same degree of wealth and understanding they had to become like them.

Let me leave you with this fact for your path to understanding:

1 = God
0 = Contains God (nothingness still has the presence of God)
8 = infinity or you can call it 10 to the power of infinity, so it contains everything.

4 x 27 = 108
108/5 = 21.6

3,6,12,24,27,36,54,60,72,108,144,180,216,252,288,324,360,540,720, 1296, 2592,4320, 5184 etc are TIME SPACE constants and hidden inside in PI = 3.141592654

NOW LOOK:

Pi = 3.141 592 653 5…

3 x 1 = 3
3 x 1 x 4 = 12
3 x 1 x 4 x 1 x 5 = 60
3 x 1 x 4 x 1 x 5 x 9 = 540
3 x 1 x 4 x 1 x 5 x 9 x 2 = 1080
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 = 6480
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 = 32400
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 x 3= 97200
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 x 4 = 129,600 (if approximated to the 9^th decimal place)
Number of seconds in a circle of 360 deg = 360x60x60 = 1,296,000 which is
exactly 10x of the 129,600.
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 x 3 x 5 = 486 000

SEE I TOLD YOU – Its all in the heart of a millet seed.

Note:

3 x 1 = 3
3 x 1 x 4 = 12
3 x 1 x 4 x 1 x 5 = 60
3 x 1 x 4 x 1 x 5 x 9 = 540 ::: 540/360=1.5
3 x 1 x 4 x 1 x 5 x 9 x 2 = 1080 ::: 1080/360=3
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 = 6480 ::: 6480/360=18
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 = 32400 ::: 32400/360=90
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 x 3 = 97200 ::: 97200/360=270

The Sumerians, Akkadians, Babylonians, Egyptians, Greeks, Romans and later Muslim scholars and the Western World adopted all these value and universal constants and we are just not sure how to use these to understand Divine Light and therefore achieve our goal. It seems we still are looking for Aliens and Atlantis – digging here and there and yet it is right in front of us, if we could only see!

GOOD LUCK and hope that one day we all live in peace and unity, because someone is trying hard for us NOT to achieve this and has so far succeeded for 10500 years!

PS Drawing Pi, Phi and e

Drawing 2 squares with side = 1 unit, and a circles with diameter 1 and 2 units, we can easily find key construction points for drawing Phi, Pi and e.

Note: Points LM, marked on this drawing, are 0.5* 2.72 units apart – a very good approximation of the base of natural logarithms, aka Euler’s number, e= 2.71828 ).
E. Pegg Jr. found this e approximation good to 7 digits:

Phi – another simple construction method

Draw a regular triangle with side AB = R = 1 and extend each side to intersect with the circle with R= (2/3)sqrt(3), drawn from the center of the triangle with radius passing through intersection of 2 blue circles with R=1.

Phi can be expressed in sexagecimal system of counting to high degree of accuracy as:

1+ 37/60 +4/60² + 55/60³ + 20/60⁴ + 29/60⁵ = 1.618033987
( compare to exact value rounded to the same decimal point: 1.618033989 )
Note: 1+ 37/60 +4/60² + 55/60³ = 1.61803

Ruler and Compass – “all in one”

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