Function: Octonion[`type/Fano_triples`] - a list of lists used to define octonionic multiplication table

Calling Sequence:

type(F,Fano_triples);

Parameters:

F - a list of lists

Description:

A list of lists F is of type 'Fano_triples' if:
(1) the list F contains seven lists F1, F2, F3, F4, F5, F6, and F7;
(2) each of the seven lists F1, ... , F7 contains three integers from the set {1,2,3,4,5,6,7};
(3) each of the seven integers {1,2,3,4,5,6,7} appears in eactly three of the seven lists F1, ... , F7.

A default list of Fano triples is stored in a global list _default_Fano_triples.

A valid list of seven Fano triples may be used to label seven points and seven lines in the Fano plane F_2.

The set of integers {1,2,3,4,5,6,7} is used because we use {e1,e2,e3,e4,e5,e6,e7} for the pure octonion basis.

If [i,j,k] is one of the seven valid Fano triples F1, ... , F7 , then:
(1) omul(ei,ej) = ek, omul(ej,ek) = ei, omul(ek,ei) = ej;
(2) omul(ej,ei) = -ek, omul(ek,ej) = -ei, omul(ei,ek) = -ej;

The default multiplication table is initialized at the time when the 'Octonion' package is being loaded. It can also be re-defined by issuing the following command:

> def_omultable(_default_Fano_triples);

where _default_Fano_triples is a global list with default Fano triples. See Octonion[`type/Fano_triples`] for more information.

Use Octonion[omultable] to display currently defined multiplication table.

See Octonion[omul] for octonionic multiplication.

To display all enviromental variables used by 'CLIFFORD' and 'Octonion' packages, use Cliff5[CLIFFORD_ENV] .

Examples:

> restart:with(Cliff5):with(Octonion);

Warning, new definition for init

> _default_Fano_triples;#default Fano triples

For example, the first list implies the following about {e1,e3,e7}:

> 'omul(e1,e3)'=omul(e1,e3);'omul(e3,e7)'=omul(e3,e7);'omul(e7,e1)'=omul(e7,e1);

and

> 'omul(e3,e1)'=omul(e3,e1);'omul(e7,e3)'=omul(e7,e3);'omul(e1,e7)'=omul(e1,e7);

and so on.

However, the following is another valid list of Fano triples:

> new_Fano_triples:=[[6,2,5],[6,3,4],[6,7,1],[2,3,7],[3,1,5],[2,4,1],[4,5,7]];

> type(new_Fano_triples,Fano_triples);

while the following is not:

> another_Fano_triples:=
[[4,2,5],[6,3,4],[6,7,1],[2,3,7],[3,1,5],[2,4,1],[4,5,7]];

> type(another_Fano_triples,Fano_triples);

>

The reason is that '4' appears in four lists.

See Also: Octonion[def_omultable] , Octonion[omultable] , Octonion[omul]