/usr/share/doc/axiom-doc/hypertex/numoctonions.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Octonions</div>
  <hr/>
The Octonions, also called the Cayley-Dixon algebra, defined over a
commutative ring are an eight-dimensional non-associative algebra. Their
construction from quaternions is similar to the construction of quaternions
from complex numbers (see <a href="numquaternions.xhtml">Quaternion</a>).
As <a href="db.xhtml?Octonion">Octonion</a> creates an eight-dimensional
algebra, you have to give eight components to construct an octonion.
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" onclick="makeRequest('p1');"
    value="oci1:=octon(1,2,3,4,5,6,7,8)" />
  <div id="ansp1"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p2" class="subbut" onclick="makeRequest('p2');"
    value="oci2:=octon(7,2,3,-4,5,6,-7,0)" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
Or you can use two quaternions to create an octonion.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" onclick="makeRequest('p3');"
    value="oci3:=octon(quatern(-7,-12,3,-10),quatern(5,6,9,0))" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
You can easily demonstrate the non-associativity of multiplication.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="handleFree(['p1','p2','p3','p4']);"
    value="(oci1*oci2)*oci3-oci1*(oci2*oci3)" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
As with the quaternions, we have a real part, the imaginary parts i, j,
k, and four additional imaginary parts E, I, J, and K. These parts 
correspond to the canonical basis (1,i,j,k,E,I,J,K). For each basis
element there is a component operation to extract the coefficient of 
the basis element for a given octonion.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="handleFree(['p1','p5']);"
value="[real oci1, imagi oci1, imagj oci1, imagk oci1, 
imagE oci1, imagI oci1, imagJ oci1, imagK oci1]"/>
  <div id="ansp5"><div></div></div>
 </li>
</ul>
A basis with respect to the quaternions is given by (1,E). However, you 
might ask, what then are the commuting rules? To answer this, we create
some generic elements. We do this in Axim by simply changing the ground
ring from
<a href="db.xhtml?Integer">Integer</a> to
<a href="dbpolynomialinteger.xhtml">Polynomial Integer</a>.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" onclick="makeRequest('p6');"
    value="q:Quaternion Polynomial Integer:=quatern(q1,qi,qj,qk)" />
  <div id="ansp6"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p7" class="subbut" onclick="makeRequest('p7');"
    value="E:Octonion Polynomial Integer:=octon(0,0,0,0,1,0,0,0)" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
Note that quaternions are automatically converted to octonions in the
obvious way.
<ul>
 <li>
  <input type="submit" id="p8" class="subbut" 
    onclick="handleFree(['p6','p7','p8']);"
    value="q*E" />
  <div id="ansp8"><div></div></div>
 </li>
 <li>
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    onclick="handleFree(['p6','p7','p9']);"
    value="E*q" />
  <div id="ansp9"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p10" class="subbut" 
    onclick="handleFree(['p6','p10']);"
    value="q*1$(Octonion Polynomial Integer)" />
  <div id="ansp10"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p11" class="subbut" 
    onclick="handleFree(['p6','p11']);"
    value="1$(Octonion Polynomial Integer)*q" />
  <div id="ansp11"><div></div></div>
 </li>
</ul>
Finally, we check that the <a href="dbopnorm.xhtml">norm</a>, defined as 
the sum of the squares of the coefficients, is a multiplicative map.
<ul>
 <li>
  <input type="submit" id="p12" class="subbut" onclick="makeRequest('p12');"
    value="o:Octonion Polynomial Integer:=octon(o1,oi,oj,ok,oE,oI,oJ,oK)" />
  <div id="ansp12"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p13" class="subbut" 
    onclick="handleFree(['p12','p13']);"
    value="norm o" />
  <div id="ansp13"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p14" class="subbut" onclick="makeRequest('p14');"
    value="p:Octonion Polynomial Integer:=octon(p1,pi,pj,pk,pE,pI,pJ,pK)" />
  <div id="ansp14"><div></div></div>
 </li>
</ul>
Since the result is 0, the norm is multiplicative
<ul>
 <li>
  <input type="submit" id="p15" class="subbut" 
    onclick="handleFree(['p12','p14','p15']);"
    value="norm(o*p)-norm(o)*norm(p)" />
  <div id="ansp15"><div></div></div>
 </li>
</ul>
Issue the system command
<ul>
 <li>
  <input type="submit" id="p16" class="subbut" 
    onclick="showcall('p16');"
   value=")show Octonion"/>
  <div id="ansp16"><div></div></div>
 </li>
</ul>
to display the list of operations defined by 
<a href="db.xhtml?Octonion">Octonion</a>.
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axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / numoctonions.xhtml
