jhypercomplex.numbers.cayley_dickson_algebras
Class OctonionLattice

java.lang.Object
  jhypercomplex.numbers.cayley_dickson_algebras.HypercomplexLattice
      jhypercomplex.numbers.cayley_dickson_algebras.OctonionLattice

public class OctonionLattice
extends HypercomplexLattice

Author:

Markus Maute (http://www.JHyperComplex.com)

Field Summary

Fields inherited from class jhypercomplex.numbers.cayley_dickson_algebras.HypercomplexLattice

CANONICAL

Constructor Summary

OctonionLattice()
Constructor

Method Summary

Octonion[]	getAllUnitElements()
Octonion	getBrandtTransformer(int nr) // TODO explain //
Octonion[]	getE7Roots()
java.util.TreeMap<java.lang.String,java.lang.String>	getMutliplicativeOrderDistribution()
Octonion[]	getOctavianUnits(int from, int to)
Octonion[]	getSO16Roots()
SevenPointsSTS	getSTS7()
Octonion	getUnitElement(int nr)
boolean	isUnitElement(Octonion o)
void	set128And112BasisHardCoded()
void	setCanoniclalBasis()
void	setOctavianUnits_Canonical() This construction of the unit integral octonions servers as a reference.
void	setOctavianUnits_D8Representation() Set 128 elements +/-l0+/-l1+/-l2+/-l3+/-l4+/-l5+/-l6+/-l7 with an odd number of "-"-signs
void	setOctavianUnits_FanoPlaneRepresentation(int class_nr, int nr) Representation based on one of the 30 different Fano planes (= STS(7)). 112 basis vectors are constructed from the seven lines of the Fano plane considering all 16 sign combinations. 112 basis vectors are constructed from the dual vectors. 16 basis vectors are +/- one element base vectors.
void	setOctavianUnits_SubalgebrasRepresentation() The basis is constructed from a STS(7) defined by the 7 (quaternionic) subalgebras of order 4.

Methods inherited from class jhypercomplex.numbers.cayley_dickson_algebras.HypercomplexLattice

getAllUnitsAsString, getBasis, getElements, getNumberOfUnitElement, isIntegralUnitElement, isUnitElement

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

OctonionLattice

public OctonionLattice()
                throws java.lang.Exception

Constructor

Throws:

java.lang.Exception

Method Detail

setOctavianUnits_Canonical

public void setOctavianUnits_Canonical()
                                throws java.lang.Exception

This construction of the unit integral octonions servers as a reference. It can be found in the excellent paper "Integral Cayley Numbers" by H. S. M. Coxeter.

Throws:

java.lang.Exception

setOctavianUnits_SubalgebrasRepresentation

public void setOctavianUnits_SubalgebrasRepresentation()
                                                throws java.lang.Exception

The basis is constructed from a STS(7) defined by the 7 (quaternionic) subalgebras of order 4.

Throws:

java.lang.Exception

setOctavianUnits_FanoPlaneRepresentation

public void setOctavianUnits_FanoPlaneRepresentation(int class_nr,
                                                     int nr)
                                              throws java.lang.Exception

Representation based on one of the 30 different Fano planes (= STS(7)). 112 basis vectors are constructed from the seven lines of the Fano plane considering all 16 sign combinations. 112 basis vectors are constructed from the dual vectors. 16 basis vectors are +/- one element base vectors.

Parameters:

class_nr - Class number of the Fano planee (1 or 2).

nr - Number of Fano plane within a class (1,...15).

Throws:

java.lang.Exception

setOctavianUnits_D8Representation

public void setOctavianUnits_D8Representation()
                                       throws java.lang.Exception

Set 128 elements +/-l0+/-l1+/-l2+/-l3+/-l4+/-l5+/-l6+/-l7 with an odd number of "-"-signs

Throws:

java.lang.Exception

setCanoniclalBasis

public void setCanoniclalBasis()
                        throws java.lang.Exception

Throws:

java.lang.Exception

set128And112BasisHardCoded

public void set128And112BasisHardCoded()
                                throws java.lang.Exception

Throws:

java.lang.Exception

getOctavianUnits

public Octonion[] getOctavianUnits(int from,
                                   int to)
                            throws java.lang.Exception

Throws:

java.lang.Exception

getMutliplicativeOrderDistribution

public java.util.TreeMap<java.lang.String,java.lang.String> getMutliplicativeOrderDistribution()
                                                                                        throws java.lang.Exception

Returns:

A map: Multiplicative order --> Number of occurences among the 240 integral octonions.

Throws:

java.lang.Exception

getSO16Roots

public Octonion[] getSO16Roots()
                        throws java.lang.Exception

Returns:

The 112 roots of SO(16), the maximal subgroup of E8.

Throws:

java.lang.Exception

getE7Roots

public Octonion[] getE7Roots()
                      throws java.lang.Exception

Throws:

java.lang.Exception

getSTS7

public SevenPointsSTS getSTS7()

getBrandtTransformer

public Octonion getBrandtTransformer(int nr)
                              throws java.lang.Exception

// TODO explain //

Parameters:

nr - Number of Brand transformer

Returns:

Throws:

java.lang.Exception

getAllUnitElements

public Octonion[] getAllUnitElements()

Specified by:

getAllUnitElements in class HypercomplexLattice

getUnitElement

public Octonion getUnitElement(int nr)
                        throws java.lang.Exception

Overrides:

getUnitElement in class HypercomplexLattice

Parameters:

nr - Number of the unit integral elements, counting starts with "1".

Returns:

Integral unit hypernumber.

Throws:

java.lang.Exception

isUnitElement

public boolean isUnitElement(Octonion o)
                      throws java.lang.Exception

Parameters:

o - Octonion.

Returns:

'true' if the octonion is an integral octonion 'false' else.

Throws:

java.lang.Exception