File:Brent Salamin pi calculation geometrical larger.png
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Summary
| Description | Higher-resolution version of an image illustrating the Brent-Salamin iterative method for calculating pi. The image makes extensive use of the "power of a point theorem" to multiply the lengths of two line segments. The radius of the rose-coloured circle is one, and centered at the origin where the two red axes cross. The quantities an and bn are laid out along the x-axis, both beginning at the origin. The next point an+1 is the center of the circle on the x-axis that includes an and bn, whereas bn+1 is the length of the tangent from the origin to that circle. The diameters of the ever-halving grey circles give 1/pn. t0 begins at 1/4 (the height of the small green circle near the origin) and this height is iteratively shaved down by the wide horizontal green bar, which represents the sum Σ Δtn; thus the final t is given by the vertical green bar at the right. The contributions Δtn are determined from the height above the x-axis of the cyan-green circles, which are tangent at their bases to the corresponding line from 1/pn and intersect the x-axis at an and bn. The large purple circle is drawn such that it is tangent to the line y=1/4, its height above the horizontal green bar is t, and it intersects the same bar at the origin and at x=2a. The depth below the horizontal green bar equals π, shown here in gold. Thus, a very exact value of π can be constructed using only a compass and straight-edge. This image is a PNG replacement for earlier JPEG version. SVG format will be possible once Blender or the Gimp export in that format; stay tuned! ;) |
| Date | |
| Source | Own work |
| Author | WillowW |
| Other versions | A quarter-size GIF version can be found at Image:Brent Salamin pi calculation geometrical smaller.gif. |
This image was created with Blender.
Licensing
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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. |
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 22:14, 28 January 2008 | | 1,600 × 1,200 (140 KB) | WillowW | {{Information |Description=PNG replacement for earlier JPEG version. SVG format will be possible once Blender or the Gimp export in that format; stay tuned! ;) |Source=self-made |Date=28 January 2008 |Author= WillowW |Permission= |other_ |
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