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| <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> |
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| <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>) |
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