Image:PiArchimedes.png

From Wikipedia, the free encyclopedia

Image
File history
File links

Size of this preview: 424 × 599 pixels
Full resolution‎ (595 × 841 pixels, file size: 7 KB, MIME type: image/png)

This is a file from the Wikimedia Commons. The description on its description page there is shown below. Commons is a freely licensed media file repository. You can help.

[edit] Summary

calculating pi like Archimedes with polygons

[edit] Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation license, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation license".

This file is licensed under the Creative Commons Attribution ShareAlike license versions 2.5, 2.0, and 1.0

You may select the license of your choice.

[edit] Explanation

Archimedes of Syrakus prooved the identity of the two numbers a and b, where a is the quotient of the perimeter and the diameter of a circle and b is the quotient of the area and the square of the radius of the same circle. He did not yet use the name pi for this number, but he described how this number could be calculated using inscribed and circumscribed polygons. This is presumably the oldest numerical approximation in mathematical history. Archimedes himself calculated pi using polygons with up to 96 vertices and got his famous result:

$3{,}1408450 \dots = 3 + \frac{10}{71} <\pi< 3 + \frac{10}{70} = 3{,}1428571 \dots$

In modern mathematical language the approximation uses an unit circle (r = 1) and starts with simple polygons with small vertice numbers like n = 2, 3, 4 or 6 and the respective edge lengths $s 2 = 2$ , $s_3 = \sqrt{3}$ , $s_4 = \sqrt{2}$ or $s 6 = 1$ .

Then the vertice number is doubled using the line $ρ n = M S$ and in two rectangled triangles the theorem of Pythagoras ( $AM^2 = MS^2 + AS^2, 1 = \rho_n^2 + s_n^2/4, \rho_n = \sqrt{1-s_n^2/4}$ and $AC^2 = AS^2 + SC^2, s_{2n}^2 = s_n^2/4 + (1-\rho_n)^2 = s_n^2/4 + 1 - 2 \rho_n + \rho_n^2$ ). The result is the edge length $s 2 n = A C$ :

$s_{2n}=\sqrt{2-2\sqrt{1-\frac{s_n^2}{4}}}$

With this formula an approximation of pi can be calculated using elementary arithmetic (addition, subtraction, multiplication, division and square root). The following table contains the begin of this approximation starting with a "two-angle". It gives the numbers n (vertice number), the distance $1 - ρ n$ between line mid S of AB and perimeter of the circle, the edge lengths $s n = A B$ and $S_n = A'B' = s_n \cdot 1 / \rho_n$ of the inscribed and circumscribed polygon and the respective areas $a n = n s n ρ n / 2$ and $A n = n S n / 2$ . These areas should enclose pi and they should converge to pi.

The calculation is numerically unstable because the subtraction produces a cancellation of digits and can be used to demonstrate cancellation.

 $n$               $1 - ρ n$          $s n$           $S n$              $a n$                   $A n$ 
2          1.000e+00  2.00e+00       Inf  0.00000000000000               Inf
4          2.929e-01  1.41e+00  2.00e+00  2.00000000000000  4.00000000000000
8          7.612e-02  7.65e-01  8.28e-01  2.82842712474619  3.31370849898476

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Dimensions	User	Comment
current	09:00, 22 November 2006	595×841 (7 KB)	Brf	({{Information \|Description=Graphic to explain the calculation of pi after Archimedes \|Source=own work \|Date=2006/11/22 \|Author=brf \|Permission=own work \|other_versions=non }})
	08:43, 22 November 2006	595×841 (7 KB)	Brf
	14:58, 21 November 2006	595×841 (6 KB)	Brf	(calculating pi like Archimedes with polygons)

File links

No pages on the English Wikipedia link to this file. (Pages on other projects are not counted.)

Image:PiArchimedes.png

From Wikipedia, the free encyclopedia

[edit] Summary

[edit] Licensing

[edit] Explanation

File history

File links

Views

Navigation

Interaction

Search