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The Computation of Pi by Archimedes

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The Computation of Pi by Archimedes

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23 Nov 2010 (Updated )

Archimedes wrote 3 1/7 > pi > 3 10/71. This is how he did it.

Editor's Notes:

This file was selected as MATLAB Central Pick of the Week

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File Information
Description

Before there was floating point, or a way to write zero, or algebraic notation, Archimedes bounded the value of pi by estimating the perimeter of regular polygons inside and outside the circle. His computation is repeated and explained here using MATLAB.

Acknowledgements

Variable Precision Integer Arithmetic and Fractions Toolbox inspired this file.

MATLAB release MATLAB 9.0 (R2016a)
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Comments and Ratings (9)
15 Jan 2016 Bill McKeeman

Thank you danf. I think your change should be incorporated in the code. I can no longer do this, so MathWorks will have to intervene.

Comment only
15 Jan 2016 danf

danf (view profile)

Am I the only one? but this code is failing.
----------------------------------------
version= 2015B
error= Assignment between unlike types is not allowed.
Error in ComputationOfPiByArchimedes (line 222)
cf(end+1) = r;
I am running in "run and advance" mode.

The reason is that cf is vpi and r is a vpi (by propagation). But on the 15th or 16th iteration r is assigned 1 (r=1).
So I changed to r= vpi(1) and it corrected the problem.
Still I wonder. After all, this is a pick of the week, so it was working.
When did it break?

Comment only
02 Mar 2012 Zachary Danziger  
21 Feb 2012 Derek O'Connor

This is a lovely piece of work and is a credit to the author Bill McKeeman. This is a great example to use when explaining successive approximations and the difference between rational and floating point numbers.

With reference to Bill's and John D'Errico's comments I would like to make two points:

(1).

Unfortunately Mathworks's websites are so far behind the times that it makes publishing anything that includes mathematics very difficult, and anything such as Bill's work, nearly impossible, as he notes in his 20 Jun 2011 comment.

Mathworks should look at how Stack Exchange handles typesetting text, mathematics, code, and diagrams. Here are examples of my posts to Math Stack Exchange:

Square Root: http://math.stackexchange.com/questions/90435/

Sign of Perm: http://math.stackexchange.com/questions/65923/

Boolean Inverses: http://math.stackexchange.com/q/111357/14701

The last example has a link to Mathworks's Answers

http://www.mathworks.co.uk/matlabcentral/answers/16192-inversion-of-a-boolean-matrix

so you can can switch from Stack Exchange to Mathworks and back, to see the stark contrast.

(2).

I agree with John's two uses of FEX, but I would add a third: to see BAD code. As John knows well, such code can lead to very lively discussions indeed. These discussions can be very fruitful for all concerned, except perhaps the writer of the code.

It is for this third reason that I deplore Mathworks's policy of allowing code submissions and the discussion surrounding the submission, to be deleted by the author of the code. If authors knew that they could not delete their code, then they might be more careful about what they submit.

24 Jun 2011 Ben Petschel

Ben Petschel (view profile)

Very nice work.

23 Jun 2011 Bill McKeeman

MathWorks just fixed the bug that was messing up the text. So, I'll leave the entry where it is. Thanks Matthew.

Comment only
20 Jun 2011 Bill McKeeman

Unfortunately, MathWorks has made changes in the published form of this submission which make it harder to read. I plan to delete the entry in a few days.

Comment only
26 Nov 2010 Nicolas Cusseau  
24 Nov 2010 John D'Errico

John D'Errico (view profile)

There are two uses for the File Exchange that I love. One is to provide high quality code. The second is to teach others about some interesting part of mathematics and hopefully, write it using well written MATLAB code. This submission eminently qualifies. You can read it and learn something from what Bill has done.

Updates
24 Nov 2010 1.1

Typo in title page

26 Nov 2010 1.2

Rewrote some of the descriptive material.

27 Nov 2010 1.3

Improved wording.

01 Dec 2010 1.6

Minor fixes, sent wrong file last submission.

03 Dec 2010 1.7

Apologies for many updates. Wordsmithing again.

18 Feb 2012 1.8

Fixed type in Greek integer

26 Feb 2012 1.9

Still trying to change typo in Greek number

29 Feb 2012 1.12

Previous version did not have vpi or fr on the MATLAB path -- caused errors.

16 Mar 2012 1.13

The submitted zip file still did not have the right contents. Trying again.

23 Aug 2012 1.19

Added @vpi and @fr class directories

22 Jan 2016 1.20

Fixing bug noted by danf in comments.

03 Mar 2016 1.20

Added a live script

04 Mar 2016 1.20

Added a live script

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