Proving the lune of hippocrates using a rational pi.

Discussion in 'Physics & Math' started by Jason.Marshall, Jan 28, 2015.

  1. Jason.Marshall Banned Banned

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  3. danshawen Valued Senior Member

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    Now I see. That IS clever, Jason!

    A quick look reveals, this is likely an original approach. Most impressed.

    For those who missed it, Jason has basically made pi rational simply by making r irrational. A unit circle, it is not, but a rational pi result, it definitely is.

    I couldn't follow the circle squared completely, but I follow this one easily. Thanks.
     
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  5. rpenner Fully Wired Valued Senior Member

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    That wouldn't make π rational. That would, at best, make the circumference rational. But what he is actual doing is called lying about math.

    Let a circle be drawn with center A and radius r = AB. Extend the segment BA in to form the diameter BC. Construct the perpendicular bisector of BC and pick one of its intersections with the circle to be point D. Then DA ⟂ AC and AD = AC = r so in A, C, D we have the three corners of a square. Construct the fourth one, E (say by erecting lines perpendicular to CA and DA at C and D, respectively.) Mark F as the intersection of AE and CD. Then from C to D draw the arc exterior to the first circle with center F and radius CF.

    Since CF is a half the diagonal of the square ACED, it is the ½ hypotenuse of the isosceles right triangle with legs of length r, so it's length is ½ √2 r = r/√2.

    It follows that the area of the large full circle BCD is π r².
    Likewise, that the area of one quarter of the large full circle ACD is ¼ π r².
    Likewise, that the area of the small full circle ACED is ½ π r².
    Likewise, that the area of the square inscribed in the small circle ACED is r².
    Likewise, that the area of the triangle ACD is ½ r².
    Likewise, that the difference between the square inscribed in the small circle and the quarter of the large full circle CED is (1 − ¼ π) r².
    Likewise, that the difference between the small full circle ACED and the inscribed square is (½ π − 1) r².
    Likewise, that the area of just one of the four pieces of the small circle outside the square, the area between the arc CE and the line CE, is [(½ π − 1)/4] r².
    Likewise, that the area of any two of these is (¼ π − ½) r².
    Likewise, that the area of the lune between the short quarter-arc CD and the longer half-circle CED is the same as the the difference between the square inscribed in the small circle and the quarter of the large full circle CED plus the area of two of the four pieces of the small circle outside the square and is therefore (1 − ¼ π) r² + (¼ π − ½) r² = ( 1 − ½) r² = ½ r².

    Shorter without any use of π: Call the shape between an arc of a circle and its chord a D-shape. CD = (√2) CE. CD is joined by a line and an arc which is ¼ of a full circle. CE is joined by a line and an arc which is ¼ of a full circle. Thus D-shape "CE" is congruent to D-shape "ED" similar to D-shape "CD" and , and so the area of D-shape "CD" is equal to (√2)² times the area of either D-shapes "CE" or "ED" . Therefore the area of the triangle CED plus area of D-shape "CE" plus area of D-shape "ED" minusarea of D-shape "CD" must equal the area of triangle CED and therefore the area of triangles ACD.

    This is not a demonstration of the rationality of π. This is not a demonstration that is peculiar to a specific value of r. This is a shell game that says nothing about the area of circles or the value of π.

    And if you want to do math: http://math.ucr.edu/~res/math153/s10/history02a.pdf

    Area (corresponds to CF = √2 and AC = 2):
    \(\int_{-\sqrt{2}}^{+\sqrt{2}} ( \sqrt{2 - x^2} + \sqrt{2} - \sqrt{4 - x^2} ) dx = 2 \int_{0}^{\sqrt{2}} ( \sqrt{2 - x^2} + \sqrt{2} - \sqrt{4 - x^2} ) dx = 2 \times \left( \int_{0}^{\sqrt{2}} \sqrt{2 - x^2} dx \; + \; \int_{0}^{\sqrt{2}} \sqrt{2 - x^2} dx \; - \; \int_{0}^{\sqrt{2}} \sqrt{4 - x^2} dx \right) \\ = 2 \times \left( \left. ( \frac{1}{2} \sqrt{2-x^2} x + \sin^{-1} \frac{x}{\sqrt{2}} + C_1 ) \right|_{0}^{\sqrt{2}} \; + \; \left. ( \sqrt{2} x + C_2 ) \right|_{0}^{\sqrt{2}} \; - \; \left. ( \frac{1}{2} \sqrt{4-x^2} x + 2 \sin^{-1} \frac{x}{2} + C_3 ) \right|_{0}^{\sqrt{2}} \right) \\ = 2 \times \left( \left( 0 + \sin^{-1} 1 - 0 - \sin^{-1} 0 \right) + \left( 2 - 0 \right) - \left( + 1 + 2 \sin^{-1} \frac{\sqrt{2}}{2} - 0 - 2 \sin^{-1} 0 \right) \right) \\ = 2 \times ( \pi + 2 - 1 - \pi) = 2\)
    (which is correct considering this example corresponds to CF = √2 and AC = 2).

    Discussion: http://math.ucr.edu/~res/math153/s10/history02.pdf
     
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  7. danshawen Valued Senior Member

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    OK, maybe the idea isn't really all that original, however,

    If circumference = 2 × π × radius (1-1)

    The product of a rational number and an irrational number is another irrational number, but the product of two irrational numbers, particularly ones like

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    is not always irrational (just square it and you get the rational number 2).

    So, in equation (1-1) above, if pi is irrational, and the other two factors are not, then, yes, the circumference must likewise be irrational.

    Oh; I see the problem now. Never mind. Jason; notice that if you change the radius to be irrational, all bets are off as to whether the circumference still is. Is that what you actually intended to show in these last two threads?

    Thanks for helping us work through this, rpenner. It had been a long time since I saw the lune of Hippocrates, but I did get tired of ancient Greek math and philosophy a very, very long time ago. The Greeks were great at math and particularly geometry. The Romans were lousy at math, but evidently made up for it in the form of good roads, bread, armies, and aqueducts. Q.E.D.

    The Romans also invented lead pipes for their plumbing, which eventually destroyed their entire society. Gibbon didn't even mention this as a cause for their downfall, but it's a fact. Now you know. Thank goodness for PVC.
     
    Last edited: Jan 29, 2015
  8. rpenner Fully Wired Valued Senior Member

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    The OP is a pseudo-mathematician -- an autodidact who has plunged into a course of error and seemingly takes no correction.
    He has claimed to be close to retirement, so I despair of trying to talk him off the ledge over the internet.

    Having obsessed over \(\frac{25}{8} = 3.125\) (which is not as good an approximation of π as 22/7 or 355/113) he has made the unsupportable allegation that replacing π with 3.125 is a mathematically viable operation and not a mere approximation. Here as we see in this particular lune, the area is in rational ratio to r², not because the idea of using a rational approximation of π has merit, but because all terms in rational ratio to π r² cancel via subtraction, leaving only ½ r² as the solution.

    Here are some other threads where this user advocated this particular misunderstanding of the nature of π, and was eventually warned for it:


    Other threads just indicate that he is not skilled in this particular field:
     
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  9. rpenner Fully Wired Valued Senior Member

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    How do we know 3, 25/8 , 22/7 and 355/113 are successively better approximations to pi? Because they are better approximations to the first positive root of sin x = 0.

    \(\sin{x} = \lim_{N\to\infty} \sum_{n=0}^{N} \frac{(-1)^n \, x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \frac{x^{11}}{11!} + \dots \\ \sin \pi = 0 \)

    Using exact math and rounding to 8 decimal places we have:

    \( \begin{array}{r|cccc|c} N & 3 & \frac{25}{8} & \frac{22}{7} & \frac{335}{113} & \pi \\ \hline \\ 0 & 3.00000000 & 3.12500000 & 3.14285714 & 3.14159292 & 3.14159265 \\ 1 & -1.50000000 & -1.96126302 & -2.03109815 & -2.02612118 & -2.02612013 \\ 2 & 0.52500000 & 0.52226384 & 0.52420222 & 0.52404395 & 0.52404391 \\ 3 & 0.09107142 & -0.05519311 & -0.07675278 & -0.07522094 & -0.07522062 \\ 4 & 0.14531250 & 0.02312950 & 0.00569116 & 0.00692501 & 0.00692527 \\ 5 & 0.14087459 & 0.01617615 & -0.00171197 & -0.00044543 & -0.00044516 \\ 6 & 0.14113063 & 0.01661143 & -0.00124322 & 0.00002088 & 0.00002114 \\ 7 & 0.14111965 & 0.01659119 & -0.00126527 & -0.00000104 & -0.00000077 \\ 8 & 0.14112002 & 0.01659191 & -0.00126447 & -0.00000024 & 0.00000002 \\ \dots \\ \to\infty & 0.14112001 & 0.01659189 & -0.00126449 & -0.00000027 & 0.00000000 \end{array}\)

    We would get much faster accurate numerical results by computing the left side of the identity: \(\sin (\pi - x) = \sin x\), but that presupposes we can calculate π-x to arbitrary accuracy.
     
    Last edited: Jan 29, 2015
  10. danshawen Valued Senior Member

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    Being an autodidact isn't a crime (and we have a few of them on here). William F. Buckley was an autodidact home schooled by his mother, but he was hard to beat in a political discussion. Too bad, I don't think many of his opponents ever pointed that out to the general audience. I'm certain Buckley would have made an example of them for doing so.

    Chris Langan is also one of those, and made a top spot for himself on the Encyclopedia of American Loons with CTMU (Cognitive Theoretic Model of the Universe), basically Chris's philosophy about philosophy. I actually found CTMU to be both amusing and entertaining. He uses all of the tools of philosophy against philosophy itself, tearing it to pieces from the roots upward.

    Anxiously awaiting Jason's response. Don't be shy.
     
    Last edited: Jan 29, 2015
  11. Jason.Marshall Banned Banned

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    Oh; I see the problem now. Never mind. Jason; notice that if you change the radius to be irrational, all bets are off as to whether the circumference still is. Is that what you actually intended to show in these last two threads?

    Yes this is correct the secondary radius in my proof was not rational and not my claim of any significance but it can be deduced with my rational form of pi as well as my using my rational radian quantity = 57.6 ...I will reply to Rpenner with the details.
     
  12. Jason.Marshall Banned Banned

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    Therefore the area of the triangle CED plus area of D-shape "CE" plus area of D-shape "ED" minusarea of D-shape "CD" must equal the area of triangle CED and therefore the area of triangles ACD.
    I just need clarifiction in what exactly you mean here, this is what it seems to be saying in my terms a:2 + a:2 + a:1 = a:1 ??
    But I assume this is what you meant I just need to clarify
    a:2=a:2....a:1=a:1 ??
    That wouldn't make π rational. That would, at best, make the circumference rational. But what he is actual doing is called lying about math.
    "Lying!!" am doing no such thing, arent you not the expert? I seek peer review isnt it not obvious? Thank you for the feedback. So then if you at least agree that this shows that circumference is rational, then I can continue to say that is will be equal to 360 degrees safely so. Now this is my argument.... 360/115.2= 3.125=rational pi...115.2= CB
    ...CA=1=57.6=CB/2=r=Primary circle radius
    57.6^2 *3.125 =10368
    [sqrt{10368}]/1.25*.5= CF
    sqrt{(CF*2)^2*2} = CB= 115.2
    so therefore 360/CB= rational pi = 3.125
     
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  13. danshawen Valued Senior Member

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    This isn't such a bad idea.

    When I was 15, and literally, no one in math class was looking, I derived a continuing fraction for pi by subdividing a quarter circle into an infinite number of polygon sides, which approached the value of a quarter of the circumference as the number of sides --> ∞.

    To plot sinusoidal waveforms, we routinely utilize a unit = pi. In some respects, units are arbitrary, and this convention is used in math as well as physics. That being the case, there is nothing wrong with a counting system that is ordered in integer multiples of pi, or any other irrational number.

    One of the most startling effect of judicious selection of units in physics derivations, IMHO, was the one by John D. Norton at the University of Pittsburg who set the time interval = 1 in a derivation of E = mc^2, instead of the more usual and conventional c = 1. E = mc^2 dropped out in less than 4 steps (1/2 a page), plus it demonstrated that the c^2 in the formula is a side effect to relating distance to momentum and momentum to time in the same expression. This is something no other derivation (including Einstein's original) I know of can do, and he did it all without breaking a single math or physics rule, as many internet derivations do. If you see one that sets the mass of a photon to m, that derivation is not only wrong, but superfluous because that is basically what E=mc^2 = hν means anyway.


    You should by all means resume using both the orientation and direction of such simplifications wherever possible, Jason. Geometry is the best instrument for learning such lessons.
     
  14. Jason.Marshall Banned Banned

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    Am not sure if Rpenner was paying attention I was not claiming that my rational version of pi =25/8 =357/114.24=3.125 ...but what I was claiming is based on the constant of circumference being 360/CB=360/115.2 in the example he used above,you can deduce the diameter by using a factor of "1.25" described in a non Euclidean universe but the technique can also be translated in exactness in the Euclidean universe hence the prediction and discover of "H". But like always Rpenner is an excellent mathematician and again I would like to thank him for putting my work in the correct form I was about to post the Euclidean version but he beat me to the punch seemingly doing an experts version of my work Rpenner is great guy I cant thank him enough for the professional regurgitation of my work I myself is very sloppy and do not have the communication skills that seems natural to Penner but I will solve that problem inside a years time when I actually get the chance to assemble my intellectual skills in a formal way right now I have more pressing issues I must currently address. And as for you Danshawen I know for a fact I have not scratch the surface of your common sense intellect many of the things you say to me have put me in a position of humility based on the profound message you display so effortless in your available lines of communication.
    here is link where you can read my article about the " Quantum Entangled Doppelgangers"
    http://quadrature322.blogspot.ca/
     
    Last edited: Jan 30, 2015
  15. rpenner Fully Wired Valued Senior Member

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    You seem to be insisting that numbers behave like you want them to, not how they actually are defined to behave. This is a failing which makes all your related posts pseudo-mathematics.

    There is no rational version of π. There is no algebraic version of π. π is not the root of any finite polynomial with rational coefficients. However 3.125 is both algebraic and rational. It is the solution to the polynomial equation \(8 x - 25 = 0\). π, on the other hand, is the smallest positive solution to the transcendental equations \(\sin x = 0\) or \(e^{i x} + 1 = 0\).

    The only constant of circumference at issue is π which is defined in Euclidean geometry as the ratio of the circumference to a diameter of any circle. This definition applies also in analysis. This definition is modified in non-Euclidean geometry because not all circles are similar in non-Euclidean geometry but the π is the limit of the ratio of circumference to diameter in the limit of small circles.

    And when you say 25/8 =357/114.24=3.125 is somehow meant to be distinct from 360/115.2 you make this same failing because 25/8 = 357/114.24 = 3.125 = 360/115.2 = 3125/1000 . That you call the latter a "constant of circumference" means you are deep in a well of your own miseducation and digging deeper rather than climbing out.
     
  16. Jason.Marshall Banned Banned

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    There is no rational version of π. There is no algebraic version of π. π is not the root of any finite polynomial with rational coefficients. However 3.125 is both algebraic and rational. It is the solution to the polynomial equation \(8 x - 25 = 0\). π, on the other hand, is the smallest positive solution to the transcendental equations \(\sin x = 0\) or \(e^{i x} + 1 = 0\).

    Correct this is all true only in a Euclidean universe of two dimensions, which is derived from Archimedes method of inscribed polygons, this means only two dimensions can be allowed in the measurement. This is not my method I was actually able to measure a natural curve since I have access to three dimensions I don't need an infinite amount of steps to measure this c/d.
     
  17. Jason.Marshall Banned Banned

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    The only constant of circumference at issue is π which is defined in Euclidean geometry as the ratio of the circumference to a diameter of any circle. This definition applies also in analysis. This definition is modified in non-Euclidean geometry because not all circles are similar in non-Euclidean geometry but the π is the limit of the ratio of circumference to diameter in the limit of small circles.

    "Incorrect" Euclid defined circumference as 360 degrees, a perfect circle can also be defined by the definition all points equidistance from the centroid of the circle. This perfect circle can easily be constructed with a protractor with no contradiction of the definition, as well 360 degrees is a constant that is directly related to 2 pi, this is why 2 radians of a perfect circle can be derived by 360/pi.
     
    Last edited: Jan 31, 2015
  18. Jason.Marshall Banned Banned

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    And when you say 25/8 =357/114.24=3.125 is somehow meant to be distinct from 360/115.2 you make this same failing because 25/8 = 357/114.24 = 3.125 = 360/115.2 = 3125/1000 . That you call the latter a "constant of circumference" means you are deep in a well of your own miseducation and digging deeper rather than climbing out.

    Althought 25/8 = 360/115.2 ....the "25/8" you presented was applied in a different format a "Euclidean" one and 25 was never meant to equal "360" which is a constant of every Euclidean circle and in many ways equivalent to 2 pi. only diameter was undefined I defined it using your own example of your irrational pi quantity FC =[sqrt{10368}]/1.25*.5 ..... then sqrt{(FC*2)^2*2}=115.2 = diameter ...in perfect relation to the constant 360/115.2 witch is rational result of pi but it is not pi it is a "Helek" are you denying that all Euclidean circles do not have a constant circumference of 360 degrees that is which Euclid defined as such and admittedly accepted by you in your earlier post when you claimed my proof of the lune proves circumference but guess what quantity I used for circumference "360"? So to conclude my statement I will ad the definition pi "is the constant ratio of a circles circumference divided by its diameter" so if the circumference =360 and the diameter = 115.2 and you divided them like so 360/115.2 does this not satisfy the definition circumference /diameter =pi??
     
    Last edited: Jan 31, 2015
  19. Kristoffer Giant Hyrax Valued Senior Member

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    Bravo, Jason. You said something true. "There is no rational pi".
     
  20. Jason.Marshall Banned Banned

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    Just to clarify a couple things in the idea/theory I spoke about the other day while I was rushing I am aware that photons don't interact and this does not change the fact that the speed of light is a constant. To explain better would be to say the difference between the frequency intensity of electromagnetic fields will cause the photon to merge on the higher frequency but this cannot be observed but this leads to the photons becoming permanently quantum entangled. So when I say "expand in the 4th dimension" I just mean the photons now become ambiguous to spacetime and merge with the fabric of spacetime itself causing them to transcend the speed of light with out any violations of the laws of physics.
     
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