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An Approximation of Pi fromMonster 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

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

which is a surprising result since both

and 262537412640768000 + 744 = 640320

then e

and it's soon realized that one can construct an approximation to

π ≈ | ln( 640320^{3} + 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 ):

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

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

Remember that

R_{e2} = | 640320^{3} + 744 - | 196884640320 ^{3} |

and we construct a new

π ≈ |
ln( R_{e2} )√163 |
= 3.141592653589793238462643383279502884197169398648... |

This time the approximation of

What if we continue this kind of analysis one step further? In the last step, we corrected

There are a couple ways to handle this but one way is to multiply the difference by

Since we have another near-integer, we can round this number off and use it to construct our third estimate

R_{e3} = | 640320^{3} + 744 - | 196884640320 ^{3} | + | 167975456640320 ^{6} |

Again we construct a new

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} = |
640320^{3} + 744 - |
196884640320 ^{3} |
+ |
167975456640320 ^{6} |
- |
180592706130640320 ^{9} |

which approximates

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

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:

with an inverse:

and the closely related expansion:

and so we expect to 'see' certain characteristic integers pop out in our successive approximations to

and so on. One thing to realize is that the expression for

R_{e3} = | 640320^{3} + 744 - | 196884640320 ^{3} | + | 167975456640320 ^{6} |

is equivalent to a large integer divided by

R_{e3} = | 18095625621654408239886517800903679512343056257063456640320 ^{6} |

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

R_{e3} = | ( 640320^{3} + 248 )^{3} - 100130719031538199405536640320 ^{6} |

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

R_{e3} = | ( 640320^{3} + 248 )^{3} - 381396 · 640320^{3} + 152722464640320 ^{6} |

resulting in an approximation of

π ≈ | ln[ ( 640320^{3} + 248 )^{3} - 381396 · 640320^{3} + 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

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

Other Related π Approximations from Monster Symmetries |

It's not too difficult to find other approximations to

π ≈ | ln[ 2^{18} · 15^{3} + 744 ]√43 | = | ln[ 960^{3} + 744 ]√43 |

That gives

π ≈ | ln[ 396^{4} - 104 ]√58 |

That gives

π ≈ | ln[ 2^{15} · 165^{3} + 744 ]√67 | = | ln[ 5280^{3} + 744 ]√67 |

That gives

π ≈ | ln[ ( 396^{4} - 104 )^{2} - 8744 ]√232 |

That gives

π ≈ | ln[ ( 396^{4} - 104 )^{3} - 13116 · 396^{4} + 1075296 ]√522 |

That gives

Taking a look at the work others have done, I found one example on

π ≈ | ln[ ( 640320^{3} + 744 )^{2} - 2 · 196884 ]2 · √163 |

giving the resulting approximation to

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[ (( 640320^{3} + 744 )^{2} - 393768 )^{2} + 85975040 · 640320^{3} - 17018389032 ]4 · √163 |

which is equivalent to:

π ≈ | ln[ 640320^{12} + 2976 · 640320^{9} + 2533680 · 640320^{6} + 561444608 · 640320^{3} + 8507424792 ]4 · √163 |

and results in an approximation of

Another approximation, this one a Class 2, is:

π ≈ | ln[ 5280^{3} ( 236674 + 30303 √61 )^{3} + 744 ]2 · √427 |

giving the resulting approximation to

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

Other π Approximation Formulas from Various Sources |

Here I will give some

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

From the same source is this approximation which leaves out the

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 ):

The following approximations are two of my own, submitted to the Pi Approximation

Finally, here are a few more of my better

You can find more

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. |

© October 24, 2009 - Jim Cullen - all rights reserved.