ceph/src/boost/libs/math/doc/html/math_toolkit/oct_overview.html

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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.oct_overview"></a><a class="link" href="oct_overview.html" title="Overview">Overview</a>
  28 </h2></div></div></div>
  29 <p>
  30       Octonions, like <a class="link" href="../quaternions.html" title="Chapter&#160;9.&#160;Quaternions">quaternions</a>, are a relative
  31       of complex numbers.
  32     </p>
  33 <p>
  34       Octonions see some use in theoretical physics.
  35     </p>
  36 <p>
  37       In practical terms, an octonion is simply an octuple of real numbers (&#945;,&#946;,&#947;,&#948;,&#949;,&#950;,&#951;,&#952;), which
  38       we can write in the form <span class="emphasis"><em><code class="literal">o = &#945; + &#946;i + &#947;j + &#948;k + &#949;e' + &#950;i' + &#951;j' + &#952;k'</code></em></span>, where
  39       <span class="emphasis"><em><code class="literal">i</code></em></span>, <span class="emphasis"><em><code class="literal">j</code></em></span>
  40       and <span class="emphasis"><em><code class="literal">k</code></em></span> are the same objects as for quaternions,
  41       and <span class="emphasis"><em><code class="literal">e'</code></em></span>, <span class="emphasis"><em><code class="literal">i'</code></em></span>,
  42       <span class="emphasis"><em><code class="literal">j'</code></em></span> and <span class="emphasis"><em><code class="literal">k'</code></em></span>
  43       are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>
  44       (or <span class="emphasis"><em><code class="literal">j</code></em></span> or <span class="emphasis"><em><code class="literal">k</code></em></span>).
  45     </p>
  46 <p>
  47       Addition and a multiplication is defined on the set of octonions, which generalize
  48       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
  49       associative</strong></span> (i.e. there are octonions <span class="emphasis"><em><code class="literal">x</code></em></span>,
  50       <span class="emphasis"><em><code class="literal">y</code></em></span> and <span class="emphasis"><em><code class="literal">z</code></em></span>
  51       such that <span class="emphasis"><em><code class="literal">x(yz) &#8800; (xy)z</code></em></span>). A way of remembering
  52       things is by using the following multiplication table:
  53     </p>
  54 <p>
  55       <span class="inlinemediaobject"><img src="../../octonion/graphics/octonion_blurb17.jpeg"></span>
  56     </p>
  57 <p>
  58       Octonions (and their kin) are described in far more details in this other
  59       <a href="../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../quaternion/TQE_EA.pdf" target="_top">errata
  60       and addenda</a>).
  61     </p>
  62 <p>
  63       Some traditional constructs, such as the exponential, carry over without too
  64       much change into the realms of octonions, but other, such as taking a square
  65       root, do not (the fact that the exponential has a closed form is a result of
  66       the author, but the fact that the exponential exists at all for octonions is
  67       known since quite a long time ago).
  68     </p>
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  72 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
  73       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
  74       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
  75       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
  76         Distributed under the Boost Software License, Version 1.0. (See accompanying
  77         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>)
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