this thread is for Rpenner

q > 0 → sin(tan⁻¹(p/q)) = p/√(p² + q²), cos(tan⁻¹(p/q)) = q/√(p² + q²)

sin(tan⁻¹(1/239)) = √2/338, cos(tan⁻¹(1/239)) = 239√2/338
sin(tan⁻¹(1/5)) = √26/26, cos(tan⁻¹(1/5)) = 5√26/26
sin(tan⁻¹(1)) = cos(tan⁻¹(1)) = √2/2

sin(a + b) = sin(a) cos(b) + sin(b) cos(a), cos(a+b) = cos(a) cos(b) - sin(a) sin(b)

So
sin(2 tan⁻¹(1/5)) = 2 sin(tan⁻¹(1/5)) cos(tan⁻¹(1/5)) = 5/13, cos(2 tan⁻¹(1/5)) = cos²(tan⁻¹(1/5)) - sin²(tan⁻¹(1/5)) = 12/13
sin(4 tan⁻¹(1/5)) = 2 sin(2 tan⁻¹(1/5)) cos(2 tan⁻¹(1/5)) = 120/169, cos(4 tan⁻¹(1/5)) = cos²(2 tan⁻¹(1/5)) - sin²(2 tan⁻¹(1/5)) = 119/169
sin(tan⁻¹(1/239) + tan⁻¹(1)) = sin(tan⁻¹(1/239)) cos(tan⁻¹(1)) + sin(tan⁻¹(1)) cos(tan⁻¹(1/239)) = 120/169, cos(tan⁻¹(1/239) + tan⁻¹(1)) = cos(tan⁻¹(1/239)) cos(tan⁻¹(1)) - sin(tan⁻¹(1/239)) sin(tan⁻¹(1)) = 119/169

Thus tan⁻¹(1/239) + tan⁻¹(1) = 4 tan⁻¹(1/5) or tan⁻¹(1) = 4 tan⁻¹(1/5) - tan⁻¹(1/239) (This is Machin's Formula.)
or π = 16 tan⁻¹(1/5) - 4 tan⁻¹(1/239) which is of some practical importance (at least historically) because direct calculation of tan⁻¹(1) from the Taylor series is much slower than tan⁻¹(1/5) or tan⁻¹(1/239).

π = 732 tan⁻¹(1/239) + 128 tan⁻¹(1/1023) - 272 tan⁻¹(1/5832) + 48 tan⁻¹(1/110443) - 48 tan⁻¹(1/4841182) - 400 tan⁻¹(1/6826318) is by one measure the "best" Machin-like formula which can be demonstrated by the same reasoning as above.
 
R penner I just like to thank you again for the math I am not denying anything about the properties of a Euclidean circle the relationship is clear 8/2.22 =3.60360 therefore 1/2.22= .45 will confirms "1 " this proves pi is accurate in a square Euclidean universe and it is irrational what I am noting is there is also another relationship in a non Euclidean circle when defined in a Euclidean space will equal 7.96/2.21= 3.60181 then .995/2.21 =.45 this will confirm less than 1 ".995" for the latter measurement this non Euclidean circle is described as 360/115.2 this is all am saying.
 
R penner I just like to thank you again for the math I am not denying anything about the properties of a Euclidean circle the relationship is clear 8/2.22 =3.60360 therefore 1/2.22= .45 will confirms "1 " this proves pi is accurate in a square Euclidean universe and it is irrational what I am noting is there is also another relationship in a non Euclidean circle when defined in a Euclidean space will equal 7.96/2.21= 3.60181 then .995/2.21 =.45 this will confirm less than 1 ".995" for the latter measurement this non Euclidean circle is described as 360/115.2 this is all am saying.

and also how this circle was isolated by co factor 1.25 I noticed this when I measured the increase of 450/360 in a non Euclidean space then I used this co factor to measure a transformation of a natural curve back to a straight line then back to a radian length so I got this 90 degree / 1.25 =72 then 72/ 1.25 = 57.6 both 72 and 57.6 or straight lines and 90 represents a curve so in my opinion it appeared to be some sort of universal growth ratio. there is obviously a lot more to this here is the generalise explanation and again this was just a mathematical observation that I have not yet fully investigated but I believe based on my intuition that may turn out to be significant. and also 57.6 *2 = 115.2,

450/360 I isolated the ratio from a set of transformations in spherical geometry when they described the increase of a straight angle triangle into a curve, a triangle expanding on a curve surface added up to be more than 180 degrees I believe this is in the hyperbolic space. This door was left open because no one could prove Euclid's fifth postulate true.
 
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Archimedes asserted that a circle's area and perimeter are intermediate between those of a regular polygon inscribed in that circle and a regular polygon circumscribed about it. What, specifically, do you find fault with in that statement?
I don't see how this conversation will improve if you don't answer direct questions about what your viewpoint is.
R penner I just like to thank you again for the math
I am not denying anything about the properties of a Euclidean circle
the relationship is clear
8/2.22 =3.60360 therefore 1/2.22= .45 will confirms "1 "
this proves pi is accurate in a square Euclidean universe and it is irrational
what I am noting is there is also another relationship in a non Euclidean circle when defined in a Euclidean space will equal 7.96/2.21= 3.60181 then .995/2.21 =.45 this will confirm less than 1
".995" for the latter measurement
this non Euclidean circle is described as 360/115.2
this is all am saying.
Even broken up into possible sentences, your post conveys no coherent line of reasoning or conversation. I don't see how this conversation will improve if you don't put real effort into communicating.
 
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I don't see how this conversation will improve if you don't answer direct questions about what your viewpoint is.

its correct because 180/pi* 180= area of a circle thus follows that this is also true ...pi * 180/pi ^2 = area of a circle
but I was trying to say I would have to admit it is a circle and that shows all points or equidistance but there something I never thought about this gives me an idea for further investigation on this issue so for now I would have to say I have no issue, but this though is very very interesting to me now I have more things to investigate I think I have to figure out the relationship. One perfect circle exist in Euclidean space that has an irrational ratio "pi" when another perfect circle exist from the ratio of a straight angle to a curve in Hyperbolic space which give you a rational pi. So I must say the definition is correct but thanks, you just made me realise something very interesting!! This is telling me something very significant ,now I must deem this as "work in progress" this might help me on my paper on infinities. So thanks for spending the time this issue is now resolved in my head now I just have to figure out what is all means and why I am getting these calculations if I find anything new for sure I will let you know my friend bye for now.
 
I don't see how this conversation will improve if you don't answer direct questions about what your viewpoint is.
Even broken up into possible sentences, your post conveys no coherent line of reasoning or conversation. I don't see how this conversation will improve if you don't put real effort into communicating.

Yes I know my appologies but I know you`re smart so I figured you could fill in the blanks if I don't have to do work I don't do it its a bad habit I don't sharping the saw often am a bit lazy. "I like" to cut corners through the hypotenuse to get the point lol bad joke I know lol "short cuts"
 
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So please don't feel disrespected by that fact, Because I appreciated the work you put in to answer my questions thoroughly but am satisfied now so thanks again for putting in the genuine effort. All definitions explaining concepts are now clear to me.
 
lol.

Wow. You don't need to "appologize" for your affliction any more: This is America! Even in North Carolina.
jack.gif
 
conclusion...
lets start by thanking hyperbolic geometry, this was the comparasent needed to compare a flat space such as Euclidean tranforming into a curve space such as its hyperbolic doppleganger. I will use the term quantum entagled as a metaphor to explain the relationship between a circle of Euclid and a circle defined by the ratio "Hala-kin" 1/5, derived from the helek measurement of hebrew astronomy measurements The helek (Hebrew חלק, meaning "portion", plural halakim חלקים) is a unit of time used in the calculation of the Hebrew calendar. The hour is divided into 1080 halakim. A helek is 31/3 seconds or 1/18 minute. The helek derives from a small Babylonian time period called a she, meaning '"barleycorn", itself equal to 1/72 of a Babylonian time degree (1° of celestial rotation).
360 degrees x 72 shes per degree / 24 hours = 1080 shes per hour.
The Hebrew calendar defines its mean month to be exactly equal to 29 days 12 hours and 793 halakim, which is one helek more than 29 days 12 hours and 44 minutes. It defines its mean year as exactly 235/19 times this amount.
"source wiki"
This I theorised must have been some sort of universal growth ratio showing the rate of transformations from a 2 dimensional flat plane such as Euclidean tranforming into its 3 dimensional doppleganger on a curve hyperbolic plane. So I measured for the rate I plotted axioms as follows... a line= a triangle= 180 degrees= a square = 2 triangles = 360 degrees =a circle... there is more but these axioms is suficient for the task. so while studgying a set of tranformations in hyperbolic space I noticed the constant 1.25 "Hala-kin" this is what you do, you measure back the amount of degrees 450 would take to get to 360 degrees 1/5 was the rate so "all units are in degrees" 360/4 = 90 degrees /1.25= 72/1.25= 57.6 so this follows that 57.6 would represent half of the line then 1/5 of the circle and finally 1/4 of a circle morhing into a curve. I mentioned the term quantum entangled earlier because it was a good metaphor to use to explain a comparsent relationship needed to perform a task "Quadrature of a circle" in the hyperbolic universe of "3.125" or simply "3.13" derived from the "helek" explained above. "universe 3.13 the herperbolic plane" now In my observations I concluded that there still exist one perfect circle but they are quantum entagled doppelgangers of the 2nd and 3rd dimensions so anything that is done on one circle will also be done on the other circle. They have different radius the Euclidean circle has smaller radian and area compared to its doppelganger that has larger radian and area. So the truth is pi can be both rational and irrational depending on how many dimensions of space is considered in its calculation, but it will remain irrational as long as the circle c/d is being investigated from a homogeneous perspective such as Euclidean. That makes perfect sense since you need to reach an infinite amount of points until pi can be defined, I see it as reaching the borders of the second dimension mathematically. Now the quadrature of a circle gives a task to construct a square with the same area as its circle counterpart. Now to do this purely from a Euclidean space without use of the doppelganger I will say no comment. But if you use the rational ratio of pi derived from the Hala-kin ratio it comes with a set of rational properties as well this circle measurements allowed me to do the technically impossible of squaring the circle once I did this on the doppelgnager I followed my postulate metaphor that stated that " In my observations I concluded that there still exist one perfect circle but they are quantum entagled doppelgangers of the 2nd and 3rd dimensions so anything that is done on one circle will also be done on the other circle. " This allowed me to predict that I can locate equivalent points if I measured the rate of change of radius quantities between the doppelgangers. So I isolated all the nessasary points and created the equivalent path on the Euclidean counterpart. the measurements with a straight edge and compass become extremely complex but in theory should be able to be performed with use of the doppelganger, following the logic of the postulate. So far I have isolated the Euclidean equivalent of the Hala-kin ratio. This can be used to calculate area of a circle witch is basically squaring the cirle, when ever you calculate the area of a circle you are tranforming it into its square equivalent, these are the tranformations of the formula. so because this ratio was detected that is more evidence that these circles or doppelgangers."

The square root of (H*r)*2*2


This is the formula for the Quadrator of a cirle on a Euclidean circle just substitute r for radius H is the equavalent ratio of Hala-kin on a Euclidean circle here is the quantity 1.2533141373155002512078826424055, lol I think thats why no one bothers with these things I think this is just and approximate value to defined it means a lot of math its too long for me to put up now but try out my formula let me know if I made any mistakes thanks this is supposed to give you the square equivalent of a circle. I also found equivalent parts in this pseudo hyperbolic circle that I can measure the speed of light with...
 
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The hour is divided into 1080 halakim. A helek is 31/3 seconds or 1/18 minute.
If 1 hour = 1080 heleks, and 1 hour = 60 minutes = 3600 seconds, then 1 helek = 4/3 seconds = 3 ⅓ seconds = 1/18 minutes.

One tiny space was missing, but it perverts your communication into falsehood and stands as a monument to your lack of love in your heart for precision and your readers.

http://en.wikipedia.org/wiki/Helek

But for all the precision of Babylonian astronomy, the ratio of the Earth's day to the lunar period and the lunar period to the year are neither fixed nor expressibly exactly as ratios of small integers. Knowing the difference between exact numbers and measured estimates of physical parameters is essential to ones ability to confront the predictions of theory with reality. At the precision to which the ration of tropical year to synodic month is now known, the Hebrew calendar calculations are known to be based on a ratio 235/19 which is larger than current bounds on the quantities allow.

D = 24 hours = 86400 seconds
M = 29 days 12 hours and 793 halakim = 2,551,443 ⅓ seconds
S = Synodic month ≈ 2,551,442 8/9 seconds
Y = 235 M/19 = 31,557,325 25/57 seconds
T = Tropical Year ≈ 31,556,925 3/16 seconds

Y/M = 235/19 which is only an approximation, a bit on the high side
T/S ≈ 365.2421896698 days / 29.530589 days ≈ 12628/1021 ≈ 235/19 - 1/6463

http://en.wikipedia.org/wiki/Tropical_year
http://en.wikipedia.org/wiki/Moon
 
If 1 hour = 1080 heleks, and 1 hour = 60 minutes = 3600 seconds, then 1 helek = 4/3 seconds = 3 ⅓ seconds = 1/18 minutes.

One tiny space was missing, but it perverts your communication into falsehood and stands as a monument to your lack of love in your heart for precision and your readers.

http://en.wikipedia.org/wiki/Helek

But for all the precision of Babylonian astronomy, the ratio of the Earth's day to the lunar period and the lunar period to the year are neither fixed nor expressibly exactly as ratios of small integers. Knowing the difference between exact numbers and measured estimates of physical parameters is essential to ones ability to confront the predictions of theory with reality. At the precision to which the ration of tropical year to synodic month is now known, the Hebrew calendar calculations are known to be based on a ratio 235/19 which is larger than current bounds on the quantities allow.

D = 24 hours = 86400 seconds
M = 29 days 12 hours and 793 halakim = 2,551,443 ⅓ seconds
S = Synodic month ≈ 2,551,442 8/9 seconds
Y = 235 M/19 = 31,557,325 25/57 seconds
T = Tropical Year ≈ 31,556,925 3/16 seconds

Y/M = 235/19 which is only an approximation, a bit on the high side
T/S ≈ 365.2421896698 days / 29.530589 days ≈ 12628/1021 ≈ 235/19 - 1/6463

http://en.wikipedia.org/wiki/Tropical_year
http://en.wikipedia.org/wiki/Moon

Yes it is only an approximation but it was enough to land me in the ballpark then from that axiom I was able to calculate the exact quantity.
 
I'd learn what an axiom is before attempting the clever stuff.
don't quote me for accuracy of grammar or anything else for that matter am just sharing my work so anyone can point out mistakes then I can fix them if possible but yes I may not have used all terms or definitions in the correct context but I know what I mean. This step here of peer review is meant to address any inaccuracies in my findings that I may have overlooked.
 
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And for the record I know the meaning of axioms just may have used it in the wrong context but I know what I was trying to illustrate to the reader maybe not as clear as I intended I will attempt to correct any and all errors this is just a rough copy and an introduction into my theory. so feel free to dissect my work I will appreciate all genuine effort.
 
I'd learn what a "theory" is before proceeding too.

You are really wasting your time here. It makes me sad.
 
don't quote me for accuracy of grammar or anything else for that matter am just sharing my work so anyone can point out mistakes then I can fix them if possible but yes I may not have used all terms or definitions in the correct context but I know what I mean. This step here of peer review is meant to address any inaccuracies in my findings that I may have overlooked.
And for the record I know the meaning of axioms just may have used it in the wrong context but I know what I was trying to illustrate to the reader maybe not as clear as I intended I will attempt to correct any and all errors this is just a rough copy and an introduction into my theory. so feel free to dissect my work I will appreciate all genuine effort.

If you were presenting anything seriously approaching a functional theory, you would take the time to clean up some of your admitted short comings in presentation. That along with rpenner's several attempts to straighten out some misconceptions or to be very generous miscommunications, does suggest to me that, unless you are willing to put the effort into your presentation to make it generally intelligible, you should either take it to Alternative Theories or back to your own drawing board until.., it has matured sufficiently for presentation here in Physics & Math.
 
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