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<link rel="up" href="../octonions.html" title="Chapter 17. Octonions">
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.oct_overview"></a><a class="link" href="oct_overview.html" title="Overview">Overview</a>
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</h2></div></div></div>
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<p>
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Octonions, like <a class="link" href="../quaternions.html" title="Chapter 16. Quaternions">quaternions</a>, are a relative
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of complex numbers.
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</p>
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<p>
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Octonions see some use in theoretical physics.
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</p>
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<p>
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In practical terms, an octonion is simply an octuple of real numbers (α,β,γ,δ,ε,ζ,η,θ), which
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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
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<span class="emphasis"><em><code class="literal">i</code></em></span>, <span class="emphasis"><em><code class="literal">j</code></em></span>
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and <span class="emphasis"><em><code class="literal">k</code></em></span> are the same objects as for quaternions,
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and <span class="emphasis"><em><code class="literal">e'</code></em></span>, <span class="emphasis"><em><code class="literal">i'</code></em></span>,
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<span class="emphasis"><em><code class="literal">j'</code></em></span> and <span class="emphasis"><em><code class="literal">k'</code></em></span>
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are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>
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(or <span class="emphasis"><em><code class="literal">j</code></em></span> or <span class="emphasis"><em><code class="literal">k</code></em></span>).
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</p>
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<p>
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Addition and a multiplication is defined on the set of octonions, which generalize
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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
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associative</strong></span> (i.e. there are octonions <span class="emphasis"><em><code class="literal">x</code></em></span>,
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<span class="emphasis"><em><code class="literal">y</code></em></span> and <span class="emphasis"><em><code class="literal">z</code></em></span>
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such that <span class="emphasis"><em><code class="literal">x(yz) ≠ (xy)z</code></em></span>). A way of remembering
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things is by using the following multiplication table:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../octonion/graphics/octonion_blurb17.jpeg"></span>
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</p>
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<p>
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Octonions (and their kin) are described in far more details in this other
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<a href="../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../quaternion/TQE_EA.pdf" target="_top">errata
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and addenda</a>).
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</p>
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<p>
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Some traditional constructs, such as the exponential, carry over without too
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much change into the realms of octonions, but other, such as taking a square
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root, do not (the fact that the exponential has a closed form is a result of
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the author, but the fact that the exponential exists at all for octonions is
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known since quite a long time ago).
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</p>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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