Nested radical approximation of PI and PI day on the forum
08-02-2023, 08:39 PM
Post: #1
 pier4r Senior Member Posts: 2,245 Joined: Nov 2014
Nested radical approximation of PI and PI day on the forum
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

Wikis are great, Contribute :)
08-04-2023, 07:05 PM
Post: #2
 EdS2 Senior Member Posts: 554 Joined: Apr 2014
RE: Nested radical approximation of PI and PI day on the forum
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.
08-05-2023, 10:08 AM
Post: #3
 EdS2 Senior Member Posts: 554 Joined: Apr 2014
RE: Nested radical approximation of PI and PI day on the forum
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, ...]
08-05-2023, 11:05 AM (This post was last modified: 08-05-2023 11:06 AM by Thomas Klemm.)
Post: #4
 Thomas Klemm Senior Member Posts: 1,887 Joined: Dec 2013
RE: Nested radical approximation of PI and PI day on the forum
(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
+
08-05-2023, 04:55 PM
Post: #5
 Albert Chan Senior Member Posts: 2,359 Joined: Jul 2018
RE: Nested radical approximation of PI and PI day on the forum
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?
08-06-2023, 07:28 PM (This post was last modified: 08-06-2023 07:29 PM by pier4r.)
Post: #6
 pier4r Senior Member Posts: 2,245 Joined: Nov 2014
RE: Nested radical approximation of PI and PI day on the forum
(08-05-2023 11:05 AM)Thomas Klemm Wrote:  It is listed among others in the The Contest Center's $$\pi$$ Competition.