A couple of weeks ago I posted pictures of a letter that I received from one R.S.J. Reddy claiming that the exact value of of $latex pi$ was $latex frac{14-sqrt{2}}{4}$ which equals 3.1464466… and I asked all of you wonderful readers, and listeners, to help me find out where he went wrong.  Well, you all came through and what follows is a a general consesus of where the work of R.S.J. Reddy left the rails.

The letter itself contained 3 methods that Reddy used to derive his new value of $latex pi$, the Siva Method, the jesus Method, and the Hippocrates method, but before we tackle how those methods incorrectly derived the value of $latex pi$ I am going to mention a couple of general problems with this value.

General Mistakes

The first general mistake comes from Colin W.(@ColinTheMathmo on twitter where he sent this to me) who mentioned that the new value of $latex pi$ just happens to violate the work of Archimedes who proved long ago that the circle constant has to occur between $latex 3 frac{10}{71}approx 3.14084dots$ and $latex 3 frac{1}{7}approx 3.14285dots$. If that is not enough both Stephen M. and Steve W.(@swildstorm) chipped in with the problem that $latex frac{14-sqrt{2}}{4}$ happens to be an algebraic, not transcendental, violating Lindemann’s proof that $latex pi$ is not a root of a non-constant polynomial equation with rational coefficients as $latex frac{14-sqrt{2}}{4}$ is a root of the polynomial $latex 2x^{2}-14x+frac{97}{4}=0$.

 

Siva Mistakes

The Siva method for calculating $latex pi$ is the first one given in Reddy’s letter so it makes sense to tackling it first, even if many of the people who commented on the previous post thought that it was probably derived from the value from the Jesus method.  The Siva method depends on splitting a circle into 32 different pieces of which 16 are of one type and the rest of another, finding the area of these pieces, and then summing them all back together. The problem with this method was unanimous amongst all who commented(Kate N., Matthew, Philip W., Daniel R., Stephen M., C. Sean B., Eddie, and Colin W.) the values derived for the $latex S_{1}$ and $latex S_2$ areas. While there was not agreement on what the areas should be everyone agrees that not explaining how they were derived is suspicious, and that the values found are clearly wrong. If you want proof of that, well Matthew was kind enough to explain:

The first computation contains “Area of the S1 segment” and “Area of the S2 segment,” with no explanation given for the formulae. The formulae are incorrect. (They’re algebraic multiples of d^2 and should be transcendental.)

Jesus Method

The Jesus method for calculating $latex pi$ relies on straightening out a circle, creating an equation by adding up the three diameters and setting it equal to the circumference, and then use that equation along with circumference$latex = pi*$diameter to derive $latex pi$. Here Mark D., Matthew, Stephen M., and C. Sean B. all chimed in and, once again, all agreed on the problem; this time it was the claim of integer multiple relationship between the excess from straighten a circle using diameter lengths(which he calls esc) and the excess from straightening a square using its diagonal length.(which he calls esp). In fact, he claims that $latex esc=frac{esp}{8}$ and then offers no explanation of his reasoning for this relationship. C. Sean B. pointed out that Reddy did not even get this relationship right when deriving it from his own claims.. He had $latex esp=4-2sqrt{2}a$ and $latex esc=(pi -3)a$, where a is the length of a side of the square, and for these values $latex frac{esp}{esc}ne 8$ instead $latex frac{esp}{esc}approx 8.2742$. Oops.

Hippocrates Method

This method is derived from Hippocrates ‘s work on squaring circles, of which he did succeed in squaring  a lune(a geometric figure bounded by the arcs of two circles which looks rather like the crescent of the moon from which it gets its name). This time Matthew, Stephen M., and C. Sean B. all worked on figuring out what went wrong and it seems that they did, Reddy based his work on the previous incorrect methods. In fact as Stephen M. notes:

The Hippocrates’ method is correct until he says ‘With the guidance of the formulae given by the author where a circle is inscribed with the square…’ So this proof is explicitly based on the Siva proof. The very next formula is incorrect.

The Response

It is important to mention that it appears that the author of this paper also commented on the previous post. Of course I haveno way to prove the identity of this commenter, but the name and writing style matches. I will post his comment here in its entirety:

sir, my approach to the problem basically was to associate 3.1464466..exactly with that of the lenghts of the line segments of siva method, jesus method etc. the derivation aspect of formule were immaterial to me due to my ignorance in mathematics( i am a student of zoology). dr. gerry leversha , editor of the “mathematical gazette” (uk, 2003, july, page 368), has said ” …a trailblazing contribution to the canon…” my work is in your hands and it can be disproved and NOT by comparing with the transcendental number 3.1415926..which is that of polygon. rsj reddy author

Thank You

I want to thank all of you that participated in this little project. I wanted to do this because it seemed like it would be fun, and it was, and because this author seems to like to send these letters around and a nice takedown of the work should appear somewhere online, and here seemed as good a place as any. I want to leave you with a picture, it is a photo of the cover of a 32 page booklet by the same R.S.J. Reddy that happened to arrive as I was typing up this post. The cover alone is a nice predictor the expected value of mathematics within.