<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Overview</title> <link rel="stylesheet" href="../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.75.2"> <link rel="home" href="../../index.html" title="Boost.Octonions"> <link rel="up" href="../octonions.html" title="Octonions"> <link rel="prev" href="../octonions.html" title="Octonions"> <link rel="next" href="header_file.html" title="Header File"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="../octonions.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../octonions.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="header_file.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> <a name="boost_octonions.octonions.overview"></a><a class="link" href="overview.html" title="Overview">Overview</a> </h3></div></div></div> <p> Octonions, like <a href="../../../../quaternion/html/index.html" target="_top">quaternions</a>, are a relative of complex numbers. </p> <p> Octonions see some use in theoretical physics. </p> <p> In practical terms, an octonion is simply an octuple of real numbers (α,β,γ,δ,ε,ζ,η,θ), which we can write in the form <span class="emphasis"><em><code class="literal">o = α + βi + γj + δk + εe' + ζi' + ηj' + θk'</code></em></span>, where <span class="emphasis"><em><code class="literal">i</code></em></span>, <span class="emphasis"><em><code class="literal">j</code></em></span> and <span class="emphasis"><em><code class="literal">k</code></em></span> are the same objects as for quaternions, and <span class="emphasis"><em><code class="literal">e'</code></em></span>, <span class="emphasis"><em><code class="literal">i'</code></em></span>, <span class="emphasis"><em><code class="literal">j'</code></em></span> and <span class="emphasis"><em><code class="literal">k'</code></em></span> are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span> (or <span class="emphasis"><em><code class="literal">j</code></em></span> or <span class="emphasis"><em><code class="literal">k</code></em></span>). </p> <p> Addition and a multiplication is defined on the set of octonions, which generalize their quaternionic counterparts. The main novelty this time is that <span class="bold"><strong>the multiplication is not only not commutative, is now not even associative</strong></span> (i.e. there are octonions <span class="emphasis"><em><code class="literal">x</code></em></span>, <span class="emphasis"><em><code class="literal">y</code></em></span> and <span class="emphasis"><em><code class="literal">z</code></em></span> such that <span class="emphasis"><em><code class="literal">x(yz) ≠ (xy)z</code></em></span>). A way of remembering things is by using the following multiplication table: </p> <p> <span class="inlinemediaobject"><img src="../../../../../octonion/graphics/octonion_blurb17.jpeg" alt="octonion_blurb17"></span> </p> <p> Octonions (and their kin) are described in far more details in this other <a href="../../../../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../../../../quaternion/TQE_EA.pdf" target="_top">errata and addenda</a>). </p> <p> Some traditional constructs, such as the exponential, carry over without too much change into the realms of octonions, but other, such as taking a square root, do not (the fact that the exponential has a closed form is a result of the author, but the fact that the exponential exists at all for octonions is known since quite a long time ago). </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2001 -2003 Hubert Holin<p> Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="../octonions.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../octonions.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="header_file.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>