Initial population.

1 files changed, 86 insertions, 0 deletions

diff --git a/src/input/octonion.input.pamphlet b/src/input/octonion.input.pamphlet
new file mode 100644
index 00000000..4deb8e82
--- /dev/null
+++ b/src/input/octonion.input.pamphlet
@@ -0,0 +1,86 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input octonion.input}
+\author{The Axiom Team}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{License}
+<<license>>=
+--Copyright The Numerical Algorithms Group Limited 1991.
+@
+<<*>>=
+<<license>>
+
+)clear all
+
+-- the octonions build a non-associative algebra:
+
+oci1 := octon(1,2,3,4,5,6,7,8)
+oci2 := octon(7,2,3,-4,5,6,-7,0)
+oci3 := octon(-7,-12,3,-10,5,6,9,0)
+
+oci := oci1 * oci2 * oci3
+(oci1 * oci2) * oci3 - oci1 * (oci2 * oci3)
+
+-- the following elements, together with 1, build a basis over the ground ring
+
+octon(1,0,0,0,0,0,0,0)
+i := octon(0,1,0,0,0,0,0,0)
+j := octon(0,0,1,0,0,0,0,0)
+octon(0,0,0,1,0,0,0,0)
+octon(0,0,0,0,1,0,0,0)
+octon(0,0,0,0,0,1,0,0)
+J := octon(0,0,0,0,0,0,1,0)
+octon(0,0,0,0,0,0,0,1)
+
+i*(j*J)
+(i*j)*J
+
+-- we can extract the coefficient w.r.t. a basis element:
+imagi oci
+imagE oci
+
+-- 1 and E build a basis with respect to the quaternions:
+-- but what are the commuting rules?
+
+qs := Quaternion Polynomial Integer
+os := Octonion Polynomial Integer
+
+-- a general quaternion:
+
+q : qs := quatern(q1,qi,qj,qk)
+E := octon(0,0,0,0,1,0,0,0)$os
+
+q * E
+E * q
+q * 1$os
+1$os * q
+
+
+-- two general octonions:
+
+
+o : os := octon(o1,oi,oj,ok,oE,oI,oJ,oK)
+p : os := octon(p1,pi,pj,pk,pE,pI,pJ,pK)
+
+
+-- the norm of an octonion is defined as the sum of the squares of the 
+-- coefficients:
+
+
+norm o
+
+-- and the norm is multiplicative:
+ 
+norm(o*p)-norm(p*o)
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}