An Approximation of Pi from Monster Group Symmetries

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An Approximation of Pi from
Monster Group Symmetries
Jim Cullen

An Approximation of π from Monster Group Symmetries

In the mathematics of group theory, the Monster Group is a simple group and the largest of the sporadic groups, having an order of 2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. Most other sporadic groups are found within the Monster as subquotients. You can think of the Monster as a collection of vertex or rotational symmetries within a space of 196,883 dimensions. Connections have been found, between the coefficients of the j-functions of certain elliptic curves, and the symmetries within the Monster Group, revealing some interesting relations between transcendental values such as e and π, and regular integers emerging from the latticed symmetries of the Monster. You can Google for more info on these groups, class invariants, groups orders, and more. Some I can explain, but much of it I can not grasp.

One of the intersting results from the study of the Monster is the amazing near-integer values when inspecting the behavior of some class polynomials. This page is actually a very narrow glimpse into the Monster Group but mention of it is a reminder of where these types of near-integer values originate. We'll look at one of them here, one of Hilbert's made famous by Ramanujan's π approximation which made use of the following relation:

e^π√163= 262537412640768743.9999999999992500726 ...

which is a surprising result since both e and π are transendental numbers, meaning their digits never repeat, show no patterns, and are for all practical purposes, random. This result, so close to an exact integer, is no accident. This is something that you can try for yourself on a calculator that can handle twenty or thirty digits or so. Windows calculator in scientific mode will work nicely. The near-integer result of e^π√163 is sometimes called Ramanujan's Constant ( a misnomer really though he did work on closely related values ), which we will here denote by R_c, as opposed to the estimate of the constant which we will denote as R_e. Note that Ramanujan's Constant R_c is almost exactly 262537412640768744, and I'll spoil the surprise by telling you that the number 744 will be important just about any time it appears. A little sleuthing will reveal that:

since R_c = e^π√163 ≈ R_e = 262537412640768744 = 262537412640768000 + 744

and 262537412640768000 + 744 = 640320³ + 744

then e^π√163 ≈ 640320³ + 744

and it's soon realized that one can construct an approximation to π by taking the natural logarithm of both sides of the resulting equation, and then dividing both sides by √163, giving what many will recognize as the formula 'taken from one of Ramanujan's notebooks':

π ≈

ln( 640320³ + 744 )
√163

The accuracy of the approximation is remarkable, 30 digits past the decimal point are accurately reproduced ( the final digits that are incorrect are printed in red ):

π ≈ 3.1415926535897932384626433832797266...

The surprises do not stop there. What if we could make a better estimate of Ramanujan's Constant? This would give us a better estimate of the value of π, and we've already improved this once by adding the 744 to 640320³. Without the 744, we would still have a decent estimate, accurate to 15 digits past the decimal point: 3.1415926535897930164..., where the incorrect digits are again shown in red print. Our estimate R_e is slightly larger than Ramanujan's Constant R_c by an amount:

R_e - R_c = 7.49927402801814311120646... × 10^-13

which looks suspicious already but, watch what happens when we multiply this slight difference by 640320³:

640320³ · ( R_e - R_c ) = 196883.99999999936018469021...

Remember that 196883, which is equal to 47 · 59 · 71, is the number of dimensions in the Monster Group, and 196883 is equal to the product of the three largest prime divisors of the order of the group. The above result is almost exactly this number plus one. Again, this is no accident. Taking this new revelation into account, we can now write our second estimate R_e2 of Ramanujan's constant R_c as:

R_e2 =

640320³ + 744 -

196884
640320³

and we construct a new π approximation that works out to be:

π ≈

ln( R_e2 )
√163

= 3.141592653589793238462643383279502884197169398648...

This time the approximation of π is accurate to 44 digits past the decimal point. The approximation is elegant also for the fact that only 26 digits go into the calculation in order to achieve 45 digits of accuracy in the result. Ramanujan's estimate was accurate to 31 decimal digits with only 13 digits going into the formula for the calculation. A sort of law of diminishing returns applies to these kinds of calculations so, at some point, further calculation will eventually lead to more work than it's really worth. As a reference, when constructing approximations of π in the area of 10 to 20 decimal digits, it's rare to see more than three or four digits accuracy above the number of digits in the calculation.

What if we continue this kind of analysis one step further? In the last step, we corrected R_e by subtracting the amount 196884/640320³ and called this improved estimate R_e2. We can assume that we are still in error by some small amount and we will find that this time our estimate is just a pinch too low:

R_e2 - R_c = -2.43704431819258908663555197... × 10^-27

There are a couple ways to handle this but one way is to multiply the difference by 640320⁶ to obtain the following result:

640320⁶ ( R_e2 - R_c ) = -167975455.999999312125824988...

Since we have another near-integer, we can round this number off and use it to construct our third estimate R_e3 of Ramanujan's Constant R_c:

R_e3 =

640320³ + 744 -

196884
640320³

167975456
640320⁶

Again we construct a new π approximation formula that works out to be:

π ≈

ln( R_e3 )
√163

= 3.141592653589793238462643383279502884197169399375105820974947569...

and has an amazing accuracy of 59 digits past the decimal point; 60 digits of accuracy total, including the '3' up front, before the decimal point. In the approximation formula we made use of 42 digits so we have still come out way ahead though our lead is beginning to wane. I don't intend to continue on with the fourth step; 60 digits of accuracy is good enough for what I'm concerned with. However ( I guess it's fair to peek ) the next approximation would be R_e4 and would be equal to:

R_e4 =

640320³ + 744 -

196884
640320³

167975456
640320⁶

180592706130
640320⁹

which approximates R_c to within a margin of 4.5875×10^-56 and would result in an approximation of π that is accurate to 75 total decimal digits:

3.14159265358979323846264338327950288419716939937510582097494459230781640627252...

Those who are interested may want to continue with further corrections to see what the results may be. You will however note that the strings of '9's will get shorter but the integers larger. Things will begin to deteriorate rather quickly using the current method. The value of the coefficient for R_e5 is barely identifiable as the integer 217940004309743, though applying this fifth step improves the approximation of R_c to a margin of just 1.5648×10^-71, and improves the approximation of π to 90 decimal digits. This will make it more difficult to make elegant corrections; the expressions for these corrections will get larger also for the same increase in accuracy. At any rate, there are other interesting things to look at here where we have left off adding digits to our approximation of π.

Improvements to the π Approximation Formula

The first thing I need to mention is that I did not follow the literature precisely when it came to working out these approximations. In the above examples of the use of Hilbert class polynomials and j-invariants, the series expansion of the modular j-function is:

j(q) = 1/q + 744 + 196884q + 21493760q² + 864299970q³ + 20245856256q⁴ + 333202640600q⁵ + ...

with an inverse:

j^-1(q) = 1/q + 744·q^-2 + 750420·q^-3 + 872769632·q^-4 + 1102652742882·q^-5 + 1470561136292880·q^-6 + ...

and the closely related expansion:

q - 744 - 196884·q^-1 - 167975456·q^-2 - 180592706130·q^-3 - 217940004309744·q^-4 - ...

and so we expect to 'see' certain characteristic integers pop out in our successive approximations to π, and for the first two steps we did; we found the integer values 744 and 196884 in our π appproximation formula. I wholly expected to see the number 21493760 appear when calculating the value of R_e3. Instead, my approximation ended up making use of another integer value, 167975456. This value did not really pop up out of the blue since it only appeared because of the way I handled my approximation formula. The integers 167975456, 180592706130, and the others that follow them, can be easily shown to be related to the other values. For 167975456 we have:

167975456 = 744 · 196884 + 21493760

and so on. One thing to realize is that the expression for R_e3 is actually a ratio of integers in disguise. The expression:

R_e3 =

640320³ + 744 -

196884
640320³

167975456
640320⁶

is equivalent to a large integer divided by 640320⁶ as follows:

R_e3 =

18095625621654408239886517800903679512343056257063456
640320⁶

The problem then becomes one of expressing the value of this large integer in an efficient, compact way. A good start is to realize that this can be expressed in terms of 640320³ as in many other expressions on this page:

R_e3 =

( 640320³ + 248 )³ - 100130719031538199405536
640320⁶

The integer in the numerator can be further expressed in more compact form:

R_e3 =

( 640320³ + 248 )³ - 381396 · 640320³ + 152722464
640320⁶

resulting in an approximation of π, accurate to 59 decimal digits, that is given by the expression:

π ≈

ln[ ( 640320³ + 248 )³ - 381396 · 640320³ + 152722464 ] - 6 · ln[ 640320 ]
√163

This expression is still complex and requires many digits of input to calculate. There are other ways to process the expression but, it takes great effort to reduce the number of digits required to calculate π to this level of precision. Even the use of Ramanujan's Constant and other symmetries within the Monster group has its limits.

We'll stop here in order to take a look at some of the work others have done. I will give a few specific examples of other π approximation formulas that are related to the one I am working on, just to see how my results compare to theirs. You may also want to Google the subject since the links and information I'll give below is in no way comprehensive. Symmetries of The Monster Group and their applications is a very open topic that just about anyone can delve into and make new discoveries. Some of the math, the subject of group theory in particular, can be very difficult so a good mathematical background and a lot of study of previous work by others may be required.

Other Related π Approximations from Monster Symmetries

It's not too difficult to find other approximations to π using j invariants. Taking a look at the solutions to j[(1+i√n)/2], there are many others of interest at small values of n that give decent π approximations:

π ≈

ln[ 2¹⁸ · 15³ + 744 ]
√43

ln[ 960³ + 744 ]
√43

That gives π ≈ 3.1415926535898315... for 13 total digits of accuracy, and:

π ≈

ln[ 396⁴ - 104 ]
√58

That gives π ≈ 3.141592653589793239411... for 18 total digits of accuracy, and:

π ≈

ln[ 2¹⁵ · 165³ + 744 ]
√67

ln[ 5280³ + 744 ]
√67

That gives π ≈ 3.141592653589793239572... for 18 total digits of accuracy, and:

π ≈

ln[ ( 396⁴ - 104 )² - 8744 ]
√232

That gives π ≈ 3.141592653589793238462643384129... for 27 total digits of accuracy, and:

π ≈

ln[ ( 396⁴ - 104 )³ - 13116 · 396⁴ + 1075296 ]
√522

That gives π ≈ 3.1415926535897932384626433832795028846424... for 37 total digits of accuracy.

Taking a look at the work others have done, I found one example on Mathworld's Pi Approximations page. The same formula from Ramanujan that I have used was extended in a different way by Warda (2004) with the result:

π ≈

ln[ ( 640320³ + 744 )² - 2 · 196884 ]
2 · √163

giving the resulting approximation to π :

π ≈ 3.1415926535897932384626433832795028841971693992820...

that is accurate to 46 digits past the decimal point. Another possible extension I've worked out is this one ( and I have not found a suitable simplification of this yet ):

π ≈

ln[ (( 640320³ + 744 )² - 393768 )² + 85975040 · 640320³ - 17018389032 ]
4 · √163

which is equivalent to:

π ≈

ln[ 640320¹² + 2976 · 640320⁹ + 2533680 · 640320⁶ + 561444608 · 640320³ + 8507424792 ]
4 · √163

and results in an approximation of π that is accurate to 79 total digits:

π ≈ 3.141592653589793238462643383279502884197169399375105820974944592307816406286207727...

Another approximation, this one a Class 2, is:

π ≈

ln[ 5280³ ( 236674 + 30303 √61 )³ + 744 ]
2 · √427

giving the resulting approximation to π :

π ≈ 3.14159265358979323846264338327950288419716939937510586006...

that is accurate to 52 digits past the decimal point. A good set of pages to read more about these topics can be found at the website of Titus Piezas III, specifically Part 1 of the Index called Assorted Identities. Note that there are two pages; the second is linked to from the bottom of the first.

Other π Approximation Formulas from Various Sources

Here I will give some π approximations that I have stumbled across or have come up with on my own. The formula will be given along with the numerical results and an indicator of the total number of digits of accuracy. No incorrect digits in red print will be shown; only the correct digits are given. For example, if an approximation gives the correct digits 3.1415 for π, then I will give the formula along with the digits 3.1415 and the notation (5d) to indicate five correct digits; one digit before the decimal ( the '3.' ) and four digits past the decimal point ( '.1415' ). A note on radicals or 'square roots'. Square roots or n'th roots will always apply only to the number following the radical sign, unless a set of parenthesis indicates that the radical applies to a larger portion of the expression.

One very good one comes from Plouffe, Borwein, and Bailey (2003) which is a nice little formula to use in applications where high accuracy is not required and π is not available natively. This approximation is actually a rearrangement and a simplification of a π approximation from the Monster Group, ln( 5280³ + 744 ) / √67 but is included here as a compact and useful alternative:

π ≈ ln( 5280 ) · √( 9 / 67 ) = 3.14159265 (9d)

From the same source is this approximation which leaves out the 744 in the expression 640320³ + 744 in the Ramanujan approximation, and uses the equivalent 3 · ln( 640320 ) in place of ln( 640320³ ). I earlier mentioned this approximation:

π ≈ 3 · ln( 640320 ) / √163 = 3.141592653589793 (16d)

Another nice approximation, which I believe comes from a trigonometric manipulation, is the following ( source not certain since it's been around a long time ):

π ≈ ( 2143 / 22 )^1/4 = 3.14159265 (9d)

The following approximations are two of my own, submitted to the Pi Approximation Contest Center. Mine certainly won't win any contests but, they were good enough to be included on the page:

π ≈ ln( √820 - 544 / 99 ) = 3.1415926535 (11d)

π ≈ ( √580674 - 456 )^1/5 = 3.141592653589 (13d)

Finally, here are a few more of my better π approximations that were not submitted to the Pi Approximation Contest Center, since they just weren't elegant enough or accurate enough to bother making them public:

π ≈ ln( 1643 / 60 - √18 ) = 3.14159265 (9d)

π ≈ [ ln( 1865 ) + 9481 ]^1/8 = 3.141592653 (10d)

π ≈ ln( 10691 / 462 ) = 3.141592653 (10d) ( previously found by de Laurent de Jerphanion )

π ≈ [ 66 + √2972 - √8012 ]^1/3 = 3.1415926535 (11d)

π ≈ [ 3019 + ln( 933 ) - 8·ln( 2 ) ]^1/7 = 3.14159265358 (12d)

π ≈ [ 9472 + √9742 - √6752 ]^1/8 = 3.141592653589 (13d)

π ≈ ( 248999 / 37 )^1/7 - 8 / 21 = 3.141592653589 (13d)

π ≈ [ ( 924197 )^1/6 + 8 / 62273 ]^1/2 = 3.14159265358979 (15d)

π ≈ [ ( 924209 )^1/4 + 13 / 25756 ]^1/3 = 3.14159265358979 (15d)

You can find more π approximation formulas from: Mathworld's Pi Approximations page, Simon Plouffe's A few approximations of Pi, the Pi Approximation Contest Center, Prime Puzzles Best Approximations to Pi with Primes page, and John Heidemann's Rational Approximations to Pi page. You may also want to take a quick look at Wikipedia's Numerical Approximations of π. From these links you should be able to find further links to more information.

Reference Sequence: The First One Thousand Decimal Digits of π

π ≈ 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9 4 0 8 1 2 8 4 8 1 1 1 7 4 5 0 2 8 4 1 0 2 7 0 1 9 3 8 5 2 1 1 0 5 5 5 9 6 4 4 6 2 2 9 4 8 9 5 4 9 3 0 3 8 1 9 6 4 4 2 8 8 1 0 9 7 5 6 6 5 9 3 3 4 4 6 1 2 8 4 7 5 6 4 8 2 3 3 7 8 6 7 8 3 1 6 5 2 7 1 2 0 1 9 0 9 1 4 5 6 4 8 5 6 6 9 2 3 4 6 0 3 4 8 6 1 0 4 5 4 3 2 6 6 4 8 2 1 3 3 9 3 6 0 7 2 6 0 2 4 9 1 4 1 2 7 3 7 2 4 5 8 7 0 0 6 6 0 6 3 1 5 5 8 8 1 7 4 8 8 1 5 2 0 9 2 0 9 6 2 8 2 9 2 5 4 0 9 1 7 1 5 3 6 4 3 6 7 8 9 2 5 9 0 3 6 0 0 1 1 3 3 0 5 3 0 5 4 8 8 2 0 4 6 6 5 2 1 3 8 4 1 4 6 9 5 1 9 4 1 5 1 1 6 0 9 4 3 3 0 5 7 2 7 0 3 6 5 7 5 9 5 9 1 9 5 3 0 9 2 1 8 6 1 1 7 3 8 1 9 3 2 6 1 1 7 9 3 1 0 5 1 1 8 5 4 8 0 7 4 4 6 2 3 7 9 9 6 2 7 4 9 5 6 7 3 5 1 8 8 5 7 5 2 7 2 4 8 9 1 2 2 7 9 3 8 1 8 3 0 1 1 9 4 9 1 2 9 8 3 3 6 7 3 3 6 2 4 4 0 6 5 6 6 4 3 0 8 6 0 2 1 3 9 4 9 4 6 3 9 5 2 2 4 7 3 7 1 9 0 7 0 2 1 7 9 8 6 0 9 4 3 7 0 2 7 7 0 5 3 9 2 1 7 1 7 6 2 9 3 1 7 6 7 5 2 3 8 4 6 7 4 8 1 8 4 6 7 6 6 9 4 0 5 1 3 2 0 0 0 5 6 8 1 2 7 1 4 5 2 6 3 5 6 0 8 2 7 7 8 5 7 7 1 3 4 2 7 5 7 7 8 9 6 0 9 1 7 3 6 3 7 1 7 8 7 2 1 4 6 8 4 4 0 9 0 1 2 2 4 9 5 3 4 3 0 1 4 6 5 4 9 5 8 5 3 7 1 0 5 0 7 9 2 2 7 9 6 8 9 2 5 8 9 2 3 5 4 2 0 1 9 9 5 6 1 1 2 1 2 9 0 2 1 9 6 0 8 6 4 0 3 4 4 1 8 1 5 9 8 1 3 6 2 9 7 7 4 7 7 1 3 0 9 9 6 0 5 1 8 7 0 7 2 1 1 3 4 9 9 9 9 9 9 8 3 7 2 9 7 8 0 4 9 9 5 1 0 5 9 7 3 1 7 3 2 8 1 6 0 9 6 3 1 8 5 9 5 0 2 4 4 5 9 4 5 5 3 4 6 9 0 8 3 0 2 6 4 2 5 2 2 3 0 8 2 5 3 3 4 4 6 8 5 0 3 5 2 6 1 9 3 1 1 8 8 1 7 1 0 1 0 0 0 3 1 3 7 8 3 8 7 5 2 8 8 6 5 8 7 5 3 3 2 0 8 3 8 1 4 2 0 6 1 7 1 7 7 6 6 9 1 4 7 3 0 3 5 9 8 2 5 3 4 9 0 4 2 8 7 5 5 4 6 8 7 3 1 1 5 9 5 6 2 8 6 3 8 8 2 3 5 3 7 8 7 5 9 3 7 5 1 9 5 7 7 8 1 8 5 7 7 8 0 5 3 2 1 7 1 2 2 6 8 0 6 6 1 3 0 0 1 9 2 7 8 7 6 6 1 1 1 9 5 9 0 9 2 1 6 4 2 0 1 9 8 9 . . .

* The red sequence of six nines is a special point in the decimal representation of π called the Feynmann Point, after physicist Richard Feynmann. The sequence, beginning at the 762'nd digit and continuing to digit 767, is the first occurrence of four, five, and six consecutive identical digits within the digits of π. The next such sequence of six consecutive identical digits ( also a sequence of six nines ) begins at digit 193034 and continues to digit 193039. Further, the Feynmann Point is the first occurrence of a string of three, four, five, and six consecutive nines. The first 'double' nine is found at digits 44 and 45. You can search for other digit or character strings in π online here or here.

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