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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "This is octon3.mws" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "(Revised:
 October 27, 2001)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 235 "This Maple worksheet creates an octonion package called \+
'Octonion', version 4, and saves it in libname[1]  `Octonion.m` file. \+
It runs under Maple V R. 5 provided that Maple package CLIFFORD versio
n 5 has been loaded using with(Cliff5)." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 28 "To load the package, execute" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 ">with(Oct
onion);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
250 "The package requires that the CLIFFORD package be loaded first. I
f the CLIFFORD package is not loaded, an error message will appear ind
icating that it has to be loaded first. You can find the CLIFFORD pack
age at http://math.tntech.edu/rafal/cliff5/ ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Below you will find a des
cription of commands available in the 'Octonion' package." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Any comments and b
ug reports please send to:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 16 "Rafal Ablamowicz" }}{PARA 0 "" 0 "" {TEXT -1 25 "
Department of Mathematics" }}{PARA 0 "" 0 "" {TEXT -1 34 "Tennessee Te
chnological University" }}{PARA 0 "" 0 "" {TEXT -1 24 "Cookeville, TN \+
38501 USA" }}{PARA 0 "" 0 "" {TEXT -1 30 "e-mail: rablamowicz@tntech.e
du" }}{PARA 0 "" 0 "" {TEXT -1 24 "phone: USA(931) 372-3569" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Cookeville, Oct
ober 27, 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 
"" 0 "" {TEXT -1 90 "1. Name 'oversion' stores information about the c
urrent version of the OCTONION package.  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Typical use: oversion();  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 715 "Octonion[oversion]:= proc()\noptio
ns `Copyright 1995-2001 by Rafal Ablamowicz, Tennessee Technological U
niversity`;\nprint(`+++++++++++++++++++++++++++++++++++++++++++`);\npr
int(`'Octonion' - A Maple V (Rel. 5.1) Package for Computations with O
ctonions (version 4)`); \nprint(`Last revised: October 25, 2001`);\npr
int(`Copyright 1995-2002, by Rafal Ablamowicz, Tennessee Technological
 University`);\nprint(`Department of Mathematics, Box 5054`);\nprint(`
Tennessee Technological University, Cookeville, TN 38505`);\nprint(`ph
one: USA (931) 372-3569, fax: USA (931) 372-6353`);\nprint(`e-mail: ra
blamowicz@tntech.edu`);\nprint(`http://math.tntech.edu/rafal/cliff5/`)
;\nprint(`+++++++++++++++++++++++++++++++++++++++++++`);\nend:\n" }}
{PARA 0 "" 0 "" {TEXT -1 452 "2.  Defining octonion nonassociative mul
tiplication 'omul'.  Octonions are defined as the paravectors in Cl(0,
7) thus an octonion basis is \n[Id, e1, e2, e3, e4, e5, e6, e7].  It i
s stored under a global variable '_octbasis'. Basis for pure octonions
 [e1, e2, e3, e4, e5, e6, e7] is stored under a global variable 'pure_
octbasis'. Any element that belongs to this vector space is now of typ
e 'octonion'. The infix form of this multiplication is  `&o`.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 273 "NOTE: Oc
tonion mutliplication is defined as in P. Lounesto, \"CLICAL User Manu
al\", page 19. To display the default multiplication table, use proced
ure 'omultable()' described below. To define your own octonion multipl
ication table, use procedure 'def_omultable' (see below)." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Typical use:  omul
(Id + e1 + e2, e1 + e3); or (Id + 2*e1 + e2) &o (e2 + e3 + e1 + Id);" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "
Octonion[omul]:=proc(a1::octonion,a2::octonion) local expr,o1,o2,to1,t
o2,coeffsa,coeffsb,i,j,p;\nglobal _pureoctbasis,_octbasis,Soct,_octgen
,`&c`,`&cQ`,`&w`,`&q`,`&o`;\noptions `Copyright 1995-1992 by Rafal Abl
amowicz, Tennessee Technological University`,remember;\nif a1=0 or a2=
0 then RETURN(0) fi;\nif member(a1,\{`Id`,1\}) then RETURN(a2) fi;\nif
 member(a2,\{`Id`,1\}) then RETURN(a1) fi;\nif type(a1,\{`*`,`+`\}) or
 type(a2,\{`*`,`+`\}) then " }{TEXT -1 5 "\n    " }{MPLTEXT 1 0 641 "o
1:=collect(Cliff5[displayid](a1),_octbasis);\n  o2:=collect(Cliff5[dis
playid](a2),_octbasis);\n  coeffsa:=[coeffs(o1,_octbasis,'to1')];\n  c
oeffsb:=[coeffs(o2,_octbasis,'to2')];\n  to1:=[to1]:to2:=[to2]:\nexpr:
=\nconvert([seq(seq(coeffsa[i]*coeffsb[j]*'Octonion[omul]'(eval(to1[i]
),eval(to2[j])),i=1..nops(to1)),j=1..nops(to2))],`+`);\nif has(expr,'I
d') then expr:=collect(simplify(expr),_octbasis) else\n               \+
        expr:=collect(simplify(expr),_pureoctbasis) \nfi;\nRETURN(expr
);\nfi;\nRETURN(Cliff5[vectorpart](subs(Soct,args[1] &cQ args[2] &cQ _
octgen),0) +\n       Cliff5[vectorpart](subs(Soct,args[1] &cQ args[2] \+
&cQ _octgen),1));\nend:\n" }}{PARA 0 "" 0 "" {TEXT -1 37 "3. Octonioni
c conjugation 'o_conjug'." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 43 "Typical use: o_conjug(Id + e1 + e2 + 4*e5);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "Oc
tonion[o_conjug]:=proc(a1::octonion) local p;\noptions `Copyright 1995
-2002 by Rafal Ablamowicz, Tennessee Technological University`,remembe
r;\nRETURN(eval(a1 - 2*Cliff5[vectorpart](a1,1)));\nend:\n" }}{PARA 0 
"" 0 "" {TEXT -1 27 "4. Octonionic norm 'onorm'." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Typical use: onorm(Id - 2
*e1 + 3*e3);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 247 "Octonion[onorm]:=proc(a1::octonion) \noptions `Copyr
ight 1995-2002 by Rafal Ablamowicz, Tennessee Technological University
`,remember;\nif args[1] = 0 then RETURN(0) fi;\nRETURN(sqrt(Cliff5[sca
larpart](args[1] &o Octonion[o_conjug](args[1])))); \nend:\n" }}{PARA 
0 "" 0 "" {TEXT -1 29 "5. Octonionic inverse 'oinv'." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Typical use: oinv(Id - \+
2*e1 + e2);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 362 "Octonion[oinv]:=proc(a1::octonion) local N,ND; globa
l _octbasis;\noptions `Copyright 1995-2002 by Rafal Ablamowicz, Tennes
see Technological University`,remember;\nif args[1] <> 0 then \n   N:=
normal(args[1]); \n   ND:=denom(N)*N;\n   RETURN(expand(denom(N)*Octon
ion[o_conjug](ND)/(Octonion[onorm](ND))^2))\nelse ERROR(`octonion `, a
rgs[1], `has no inverse`) \nfi;\nend:\n" }}{PARA 0 "" 0 "" {TEXT -1 
58 "6. A commutator [a,b] = ab - ba of two octonions a and b.\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "Octonion[commutator]:=proc(o1::oct
onion,o2::octonion)\noptions `Copyright 1995-2002 by Rafal Ablamowicz,
 Tennessee Technological University`;\n RETURN(o1 &o o2 - o2 &o o1) \n
end:\n" }}{PARA 0 "" 0 "" {TEXT -1 69 "7. An associatotor [a,b,c] = (a
b)c - a(bc) of three octonions a,b,c.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 207 "Octonion[associator]:=proc(o1::octonion,o2::octonion,o3::octo
nion) \noptions `Copyright 1995-2002 by Rafal Ablamowicz, Tennessee Te
chnological University`;\n RETURN((o1 &o o2) &o o3 - o1 &o (o2 &o o3))
 \nend: \n" }}{PARA 0 "" 0 "" {TEXT -1 98 "8. Associative 3-form Phi(a
,b,c) = (1/2)*Re(a (b_bar c) - c (b_bar a)) where b_bar = o_conjug(b) \+
\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "Octonion[Phi]:=proc(o1::octo
nion,o2::octonion,o3::octonion) local expr:\noptions `Copyright 1995-2
002 by Rafal Ablamowicz, Tennessee Technological University`;\n   expr
:=(1/2)*(o1 &o (Octonion[o_conjug](o2) &o o3) - o3 &o (Octonion[o_conj
ug](o2) &o o1)):\n   RETURN(coeff(expr,Id)) \nend:\n" }}{PARA 0 "" 0 "
" {TEXT -1 44 "9. Real and imaginary parts of an octonion:\n" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 197 "Octonion[realpart]:=proc(o1::octonion) l
ocal o11;\noptions `Copyright 1995-2002 by Rafal Ablamowicz, Tennessee
 Technological University`;\n  o11:=Cliff5[displayid](o1):\n  RETURN(c
oeff(o11,Id)) \nend:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "Octonion
[purevectorpart]:=proc(o1::octonion) local o11;\noptions `Copyright 19
95-2002 by Rafal Ablamowicz, Tennessee Technological University`;\n  o
11:=Cliff5[displayid](o1):\n  RETURN(Cliff5[vectorpart](o11,1)) \nend:
\n" }}{PARA 0 "" 0 "" {TEXT -1 590 "10.  Procedure 'omultable' display
s current octonion multiplication table in a form of a 7 by 7 matrix w
here in the (i,j)-entry it displays the octonion product between the i
th and the jth basis element in the pure octonion basis. This means th
at the (i,j)-entry will equal omul(ei,ej), i,j = 1,...,7. The table is
 automatically created using a set of default Fano triples. The defaul
t Fano triples are stored in a global variable '_default_Fano_triples'
. User can enter his/her own set of Fano triples and use procedure 'de
f_omultable' to define his/her own octonion multiplication table.\n" }
}{PARA 0 "" 0 "" {TEXT -1 13 ">omultable();" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "Octonion[omultable]:=pro
c() local i,j,M; global omul,_pureoctbasis;\noptions `Copyright 1995-2
002 by Rafal Ablamowicz, Tennessee Technological University`;\nif not \+
evalb(op(4,eval(Octonion[omul]))=NULL) then\n   M:=linalg[matrix](7,7,
(i,j)->Octonion[omul](_pureoctbasis[i],_pureoctbasis[j]));\n   RETURN(
evalm(M));\nelse \n   RETURN(`Octonion multiplication table is not cur
rently defined. Use 'def_omultable' to define a new table.`) \nfi;\nen
d:\n" }}{PARA 0 "" 0 "" {TEXT -1 265 "11. Procedure 'def_omultable' al
lows the user to define the octonion multiplication table in terms of \+
seven Fano triples which define seven lines in the Fano plane. The def
ault Fano triples are: \n\n[1,3,7], [1,2,4], [1,5,6], [2,3,5], [2,6,7]
, [3,4,6], and [4,5,7]. \n" }}{PARA 0 "" 0 "" {TEXT -1 64 "They are st
ored under a global variable '_default_Fano_triples'." }}{PARA 0 "" 0 
"" {TEXT -1 355 "\nIf [i,j,k] is a Fano triple, where i,j,k = 1,..., 7
, then the octonion multiplication of [ei,ej,ek] is defined as follows
: \n\n(1) omul(ei, ej) = ek, omul(ej, ek) = ei, omul(ek, ei) = ej,\n\n
(2) omul(ej, ei) = -ek, omul(ek, ej) = -ei, omul(ei, ek) = -ej,\n\n(3)
 omul(ei, ei) = -Id, omul(ej, ej) = -Id, omul(ek, ek) = -Id.\n\nThe pr
ocedure can be used as follows:" }}{PARA 0 "" 0 "" {TEXT -1 24 "\n(1) \+
def_omultable();   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 363 "When used without any argument, it will define octonio
n multiplication using the default Fano triples.\n\n(2) def_omultable(
F);\n\nWhen used with one argument of type 'list(list)', that is, when
 F is a list consisting of seven three-element lists, each three-eleme
nt list consisting of three integers between 1 and 7, then octonion mu
ltiplication is defined as above." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1170 "Octonion[def_omultable]:=proc() \+
local L,x,i;\noptions `Copyright 1995-2002 by Rafal Ablamowicz, Tennes
see Technological University`;\n######################################
####\nif nargs=0 then RETURN(def_omultable(_default_Fano_triples,_defa
ult_squares)) fi;\n##########################################\nif nops
(args[1])<>7 or not type(args[1],Fano_triples) then \n   ERROR(`list e
ntered does not contain seven Fano triples as required`) \nfi;\nsubsop
(4=NULL,eval(Octonion[omul])): #erase current octonion multiplication \+
table\nfor i from 1 to 7 do\n   L:=[seq(e.x,x=args[1][i])]:\n   Octoni
on[omul](L[1],L[2]):=L[3]:Octonion[omul](L[2],L[1]):=-L[3]:\n   Octoni
on[omul](L[2],L[3]):=L[1]:Octonion[omul](L[3],L[2]):=-L[1]:\n   Octoni
on[omul](L[3],L[1]):=L[2]:Octonion[omul](L[1],L[3]):=-L[2]:\nod:\nif n
args=1 then \n   for i from 1 to 7 do Octonion[omul](e.i,e.i):=_defaul
t_squares[i] od;\n   RETURN()\nfi;\n##################################
########\nif nargs=2 then\n   if nops(args[2])<>7 then \n   ERROR(`sec
ond input,when used, must be a list of 7 elements which are squares of
 the pure octonion basis`) fi;\nfor i from 1 to 7 do Octonion[omul](e.
i,e.i):=args[2][i] od;\nRETURN()\nfi;\nend:\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 440 "12. Type `Fano_tripl
es` is a list F consisting of seven lists f_i, i=1..7, each of which m
ust contain three positive integers from the set \{1,2,3,4,5,6,7\} in \+
such a way that each integer is contained in exactly three lists. For \+
example, the default Fano triples are:\n\n[[1,3,7],[1,2,4],[1,5,6],[2,
3,5],[2,6,7],[3,4,6],[4,5,7]]\n\nHowever, the following is not a list \+
of Fano triples:\n\n[[1,3,7],[1,2,4],[1,5,6],[2,3,5],[2,6,7],[3,4,6],[
3,5,6]]\n" }}{PARA 0 "" 0 "" {TEXT -1 86 "because 3 is contained four \+
of the seven sublists (6 is also contained in 4 sublists)." }}{PARA 0 
"" 0 "" {TEXT -1 238 "\nValid Fano triples like [[1,3,7],[1,2,4],[1,5,
6],[2,3,5],[2,6,7],[3,4,6],[4,5,7]] may be used to define seven lines,
 each containing three points, such that each point is on three lines \+
and each line is on three points in the Fano plane." }}{PARA 0 "" 0 "
" {TEXT -1 236 "\nUser may define a different set of Fano triples in o
rder to define different ocotonionic multiplication table. For example
, the following is another set of valid Fano triples:\n\n[[1,2,5],[1,3
,4],[1,7,6],[2,3,7],[3,6,5],[2,4,6],[4,5,7]]\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 547 "Octonion[`type/Fano_triples`]:=proc(F::list(list)) l
ocal f,s,S,i,k:\noptions `Copyright 1995-2002 by Rafal Ablamowicz, Ten
nessee Technological University`;\nif nops(F)<>7 then RETURN(false) fi
;\nfor f in F do \n    if not type(f,list(posint)) or nops(f)<>3 then \+
RETURN(false) fi;\nod;\nS:=\{1,2,3,4,5,6,7\}:\ns:=`union`(seq(convert(
F[i],set),i=1..7));\ns:=s minus (s intersect S):\nif nops(s)<>0 then R
ETURN(false) fi;\nfor i in S do\n    k:=0:\n    for f in F do if membe
r(i,f) then k:=k+1 fi od;\n    if k<>3 then RETURN(false) fi;\nod;\nRE
TURN(true);     \nend:\n" }}{PARA 0 "" 0 "" {TEXT -1 20 "13. Type 'oct
onion'\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "Octonion[`type/octonio
n`]:=proc(a1::algebraic) local S, setoctbasis;\noptions `Copyright 199
5-2002 by Rafal Ablamowicz, Tennessee Technological University`;\nif t
ype(a1,numeric) then RETURN(true) fi;\nif not type(a1,\{clibasmon,clim
on,clipolynom\}) then RETURN(false) fi;\nS:=Cliff5[cliterms](args[1]);
\nsetoctbasis:=\{Id,e.(1..7)\}:\nRETURN(evalb(S minus (S intersect set
octbasis) = \{\}));\nend:\n" }}{PARA 0 "" 0 "" {TEXT -1 103 "14. Initi
alization file for the 'Octonion' package.  This file is loaded automa
tically when the command" }}{PARA 0 "" 0 "" {TEXT -1 25 "with(Octonion
); is given." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 676 "Octonion[init]:=proc() \nlocal x,y,i,j,E124,E235,E34
6,E457; \nglobal Foct,Soct,_octbasis,_pureoctbasis,_octgen,`&o`,def_om
ultable,\n_default_Fano_triples,_default_squares,_default_Clifford_pro
duct;\nif assigned(Cliff5)=false then \n   ERROR(`You must load packag
e 'Cliff5' before you can load package 'Octonion'`) \nfi;\n\n`&o`:=pro
c() Octonion[omul](args) end:\n\nunprotect(_octbasis):\n_octbasis:=[Id
,e.(1..7)];   #standard octonion basis as Maple global variable\n_pure
octbasis:=[e.(1..7)];  #pure octonion basis as Maple global variable\n
_default_Fano_triples:=[[1,3,7],[1,2,4],[1,5,6],[2,3,5],[2,6,7],[3,4,6
],[4,5,7]];\n_default_squares:=[-Id$7]:\n_default_Clifford_product:='c
mulNUM':\n" }}{PARA 0 "" 0 "" {TEXT -1 124 "(1) Defining octonion mutl
iplication which can be changed here, if necessary, or later in a work
sheet using def_omultable. \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "u
nprotect(Foct);\nunprotect(Soct);\nFoct:=proc(a1::posint,a2::posint) i
f args[1]=args[2] then RETURN(-1) else RETURN(0) fi end:\nSoct:=\{seq(
seq(B[i,j]=Foct(i,j),i=1..7),j=1..7)\}:   #regular octonion multiplica
tion as in Cl(0,7)\n" }}{PARA 0 "" 0 "" {TEXT -1 59 "E124:=subs(Soct,e
1 &cQ e2 &cQ e4);   #defined and computed\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "E124:=e1we2we4; #assigned\n" }}{PARA 0 "" 0 "" {TEXT 
-1 59 "E235:=subs(Soct,e2 &cQ e3 &cQ e5);   #defined and computed\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "E235:=e2we3we5;  #assigned\n" }}
{PARA 0 "" 0 "" {TEXT -1 59 "E346:=subs(Soct,e3 &cQ e4 &cQ e6);   #def
ined and computed\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "E346:=e3we4w
e6;  #assigned\n" }}{PARA 0 "" 0 "" {TEXT -1 35 "E457:=subs(Soct,e4 &c
Q e5 &cQ e7);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "E457:=e4we5we7; \+
 #assigned\n" }}{PARA 0 "" 0 "" {TEXT -1 115 "_octgen:=subs(Soct, subs
(Soct, (Id - E124) &cQ (Id - E235)) &cQ subs(Soct, (Id - E346) &cQ (Id
 - E457))); #defined\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "_octgen:
=e2we3we4we7-e1we3we7-e4we5we7-e1we2we5we7-e2we4we5we6-e1we5we6-e3we4w
e6+\n              e1we2we3we6-e2we6we7+e1we4we6we7-e3we5we6we7-e1we2w
e3we4we5we6we7-e2we3we5-\n              e1we3we4we5+Id-e1we2we4; #assi
gned\n" }}{PARA 0 "" 0 "" {TEXT -1 102 "Assigning a multiplication tab
le for the octonion basis.  This table has been computed using the abo
ve" }}{PARA 0 "" 0 "" {TEXT -1 51 "definitions of Foct, Soct, E124, E2
35, E346, E457.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "def_omultable(
_default_Fano_triples,_default_squares);" }}{PARA 0 "" 0 "" {TEXT -1 
33 "\n(2) Protecting procedure names.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 109 "protect('Soct','Foct','Squat','Fquat',omul,oinv,onorm,o_conju
g,_default_Fano_triples,_default_squares);\nend:\n" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 264 "#save Octonion, `c:/maplev51/lib/Octonion.m`;\n#sa
ve Octonion, `a:/Octonion.m`;\n#save Octonion, `c:/maplev51/clifford/w
ww4/Octonion.m`;\n#save Octonion, `f:/clifford/www4/Octonion.m`;\n#sav
e Octonion, `f:/www/rafal/cliff4/Octonion.m`;\n\nsavelib(Octonion,`Oct
onion.m`):" }}{PARA 0 "" 0 "" {TEXT -1 24 "Last revised: 12-27-2001" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 23 0" 22 }{VIEWOPTS 
1 1 0 1 1 1803 }