Nested radical approximation of PI and PI day on the forum

[pier4r]

Reading the formula of this paper: https://web.archive.org/web/201107062156...ers/pi.pdf (A nested radical approximation for π , version of Jul 6 2011 in the web archive) I remembered that here plenty of such a formulas are shared every year (more or less).

Could it be that π ≈ sqrt(7 + sqrt(6 + sqrt(5))) was found by some users of the forum independently? If I am not wrong for PI day (and even when it is not PI day) a lot of those formulas are posted and computed.

All this was started by this tweet: https://twitter.com/abakcus/status/16864...78688?s=20

[EdS2]

Is this a happy coincidence or is there some mathematical or geometric way to help explain the close fit? It's adrift by about 13 parts in a million, using only 3 digits. Or, 3 digits and 8 operations, depending on how we look at it.

[EdS2]

Wolfram Alpha says
sqrt(7 + sqrt(6 + sqrt(5))) = 3.14163254...
or as a continued fraction
[3; 7, 16, 1, 1, 10, 1, ...]
whereas Pi is
[3; 7, 15, 1, 292, 1, ...]

When your only tool is a hammer, every screwdriver looks like a chisel, so:
sin sqrt(7 + sqrt(6 + sqrt(5))) = -0.00003989091...
or as a continued fraction
- [0; 25068, 2, 1, 2, 1, 4, 4, 1, 2, 8, 5, 2, 1, 4, 1, 1, 4, 1, 8, 22, 1, 1, 4, 17, 1, 3, 1, ...]

Or indeed, as if it made any difference:
tan sqrt(7 + sqrt(6 + sqrt(5))) = 0.00003989091...
[0; 25068, 2, 1, 2, 1, 4, 3, 1, 1, 1, 13, 2, 1, 6, 4, 5, 2, 2, 1, 4, 4, 1, 38, 1, 2, 1, 1, 2, ...]

[Thomas Klemm]

(08-04-2023 07:05 PM)EdS2 Wrote:  Is this a happy coincidence or is there some mathematical or geometric way to help explain the close fit?

It is listed among others in the The Contest Center's \(\pi\) Competition.

There are others like:

√√√√√√√√√√√√√√√√√√√√√√√√√√√√8 + √√√√√√√√√√√√14 + √√√√√68

It contains 5 digits, but matches pi to 9 decimal places, so it is considered an outstanding approximation.

Or then the root of:

\(
6x^6−4x^5+5x^4+2x^3−2x^2+3x−5083=0
\)

which has just ten digits of coefficients and leads to the fourteen-digit approximation:

3.1415926535898031685143792

I could be wrong, but this looks more like a selection that's mostly pleasing to our eyes.

I also like Ramanujan's

\(
\frac{9}{5}+\sqrt{\frac{9}{5}}=3.1416 \cdots
\)

It is easy to calculate on an HP calculator:

1.8
\(\sqrt{x}\)
LAST x
+

[Albert Chan]

Someone had summarized tricks from HP forum memebers, for approximations of pi, log(2) ...

(03-23-2020 04:34 PM)Albert Chan Wrote:  LN(2) = 2 * probability of integer part of RND/RND is odd

My ln(2) entry was in it. Based on above date, pdf was created on year 2020, or later.
Does anyone know the link to the pdf?

[pier4r]

(08-05-2023 11:05 AM)Thomas Klemm Wrote:  It is listed among others in the The Contest Center's \(\pi\) Competition.

Nice link

E: ehy! There is also Gerson listed there. I knew it that this community did found several original (? Or at least not previously recorded) approximations.