jhypercomplex.numbers.cayley_dickson_algebras
Class Octonion

java.lang.Object
  jhypercomplex.numbers.Hypernumber
      jhypercomplex.numbers.cayley_dickson_algebras.CayleyDicksonAlgebra
          jhypercomplex.numbers.cayley_dickson_algebras.Octonion

All Implemented Interfaces:

CommonMethods

Direct Known Subclasses:

OctonionLoop

public class Octonion
extends CayleyDicksonAlgebra

Author:

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

Field Summary

Fields inherited from class jhypercomplex.numbers.cayley_dickson_algebras.CayleyDicksonAlgebra

_2_N_ON, CLASSICAL, COMPLEX, OCTONION, QUATERNION, SEDENION, TRIGINTADUONION, TWISTED

Fields inherited from class jhypercomplex.numbers.Hypernumber

ARG_NUM, CIRCULAR, COMP_NUM, components, HYPERBOLIC, ZERO_PRECISSION

Constructor Summary

Octonion()

Octonion(Octonion o)

Octonion(java.lang.String components)

Method Summary

Octonion	add(Hypernumber o)
Octonion	calculateBCHWith(Octonion o, int from_order, int to_order)
Octonion	conjugate()
static void	doCayleyDicksonDoubling(java.lang.String signature)
static void	doCayleyDicksonDoubling(java.lang.String signatures, java.lang.String vector_basis)
static void	doCayleyDicksonDoubling(java.lang.String signatures, java.lang.String vector_basis, int type)
static void	doCayleyDicksonProcess(java.lang.String signature)
static java.lang.String	getAlgebraType() The normed multiplication table is compared with a multiplication table of i) a non-split octonion ii) a split octonion
static Octonion	getAntiAssociator(Hypernumber h1, Hypernumber h2, Hypernumber h3)
Octonion	getAntiCommutatorWith(Hypernumber h) Calculates the anti-commutator of the hypernumber h1 with another one h2 which is defined as: {h1, h2} = h1*h2 + h2*h1
static Octonion	getAssociationType(int nr, Octonion o1, Octonion o2, Octonion o3, Octonion o4)
static Octonion	getAssociationTypeCommutator(int nr, Octonion o1, Octonion o2, Octonion o3, Octonion o4)
static Octonion	getAssociator(Octonion o1, Octonion o2, Octonion o3) Calculates the associator of 3 octonions o1, o2, o3 which is defined as {o1, o2, o3} = (o1*o2)*o3 - o1*(o2*o3) The associator is non-trivial as the octonion algebra is non-associative.
java.lang.String[]	getBasis()
java.lang.String[]	getBiVectorComponents()
static java.lang.String[][]	getCayleyDicksonTable(java.lang.String[] signature, int type)
static java.lang.String[][]	getCayleyDicksonTable(java.lang.String[] signature, java.lang.String[] vector_basis, int type)
Octonion	getClone()
static Octonion	getCommutator(Octonion o1, Octonion o2)
static Octonion	getCommutatorJacobian(Octonion o1, Octonion o2, Octonion o3)
Octonion	getCommutatorWith(Hypernumber o) Analysis *
static Octonion	getDifference(Octonion o1, Octonion o2)
static SevenPointsSTS	getFanoPlane()
Octonion	getGrade(int grade)
static java.util.TreeMap<java.lang.Integer,java.lang.Integer>	getGrades()
java.lang.String	getGradesOccuring()
static Octonion	getJacobian(Octonion o1, Octonion o2, Octonion o3)
static MultiplicationTable	getMultiplicationTable()
static java.lang.String[][]	getMultiplicationTableAsArray()
java.lang.String[][]	getMultTable()
static java.lang.String	getMultTableAsHtml()
static Octonion	getNegated(Octonion o)
Hypernumber	getNewInstance()
static java.lang.String[][]	getNormedMultiplicationTable()
static java.lang.String	getNormedMultiplicationtableAsHtml()
int	getNumberOfComponents()
int	getNumberOfGrades()
static Octonion	getNumericalInverse(Hypernumber h)
static Octonion	getOuterProduct(Octonion o1, Octonion o2)
static Octonion	getProduct(Octonion o1, Octonion o2)
static Octonion	getProductWithScalar(Hypernumber h, java.lang.String scalar)
static Octonion	getScalarProduct(Octonion o1, Octonion o2)
static java.util.HashSet<java.lang.String>	getSubalgebras(int order)
static java.util.HashSet<int[]>	getSubalgebrasUnsigned(int order)
static Octonion	getSum(Octonion o1, Octonion o2)
java.util.Collection<java.lang.String>	getTriadSet(int nr)
java.lang.String[]	getTriVectorComponents()
java.lang.String[]	getVectorComponents()
static Octonion	getVectorProduct(Octonion o1, Octonion o2)
static boolean	isSixthRootOfUnity(Octonion o)
static boolean	isThirdRootOfUnity(Octonion o)
Octonion	multiplyWithScalar(java.lang.String scalar) Multiplication with a scalar.
Octonion	negate() Arithmetics *
Octonion	rightMultiplyWith(Octonion o2)
void	setBiVectorComponents(java.lang.String comps)
static void	setDefaultTable()
static void	setFanoPlane(int nr) Sets a multiplication table based on one of the 30 Fano planes according to the definition in the octonion multiplication table class.
static void	setMultiplicationTable(MultiplicationTable mult_table)
static void	setMultiplicationTable(java.lang.String[][] mult_table)
void	setOctonionMultTable(OctonionMultiplicationTable mult_table)
void	setTriVectorComponents(java.lang.String comps)
static void	setType(int nr) Sets the n-th multiplication table, according to the definition in the octonion multiplication table class.
void	setVectorComponents(java.lang.String comps) Components *
Octonion	subtract(Hypernumber o) Subtracts a hypernumber.

Methods inherited from class jhypercomplex.numbers.cayley_dickson_algebras.CayleyDicksonAlgebra

getLoopMultiplicationTable, isCayleyDicksonAlgebra, isLoop

Methods inherited from class jhypercomplex.numbers.Hypernumber

add, applyLeibnizRule, asString, asString, calculateBCHWith, collectTerms, compressComponents, dual, getAlternator1, getAlternator2, getAntiCommutator, getAssociationType, getAssociationType, getAssociationTypeAsString, getAssociationTypeAsString, getAssociationTypeCommutator, getAssociationTypeCommutator, getAssociationTypeCommutatorAsString, getAssociationTypeCommutatorAsString, getAssociator, getBasisElementsAsString, getCommuAssociator, getCommutator, getCommutator, getCommutatorJacobian, getCommutatorSaglian, getCommutatorWith, getComponent, getComponents, getComponentsAsString, getComponentsAsString, getComponentsOfGrade, getConjugate, getDiagonalProducts, getDifference, getDual, getExpressionAsList, getFirstHetztelPeresian, getFormattedMultTable, getGradedInnerProduct, getHermitianConjugate, getImaginaryPart, getInstance, getIntersection, getJacobian, getJordanAssociator, getLeftCovariantDerivative, getMultiplicativeOrder, getMultiplicativeOrder, getMultTableDiagonal, getNegated, getNextBitmap, getNormedMultiplicationTable, getNormSquared, getNormSquaredAsValue, getNumberOfClosedElements, getNumberOfClosedPairs, getNumberOfComponents, getNumberOfNonzeroComponents, getOuterProduct, getProduct, getPseudoScalarComponent, getRandomClosedPair, getRandomNonClosedPair, getScalarComponent, getScalarProduct, getSecondHetztelPeresian, getSignature, getSubalgebras, getSubalgebrasAsString, getSubalgebrasSignatures, getSum, getTrace, getTraceRespDeterminant, getVectorDerivative, getVectorProduct, isAutomorphism, isClosedAlgebra, isCommutative, isComponentZero, isEqual, isLinearlyDependent, isNumerical, isNumericalityDetected, isOrthogonal, isOrthogonalTo, isZero, resetComponents, rightMultiplyWith, rightMultiplyWith, setBasis, setComponent, setComponents, setComponents, setNumerical, setNumerical, setPseudoScalarComponent, setRandomComponents, setRandomComponents, setRandomComponents, setRandomComponentsAll, setScalarComponent, simplifyNumericalFactors, simplifyNumericalSummands, simplifyPlusMinus, square

Methods inherited from class java.lang.Object

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

Constructor Detail

Octonion

public Octonion()

Octonion

public Octonion(java.lang.String components)
         throws java.lang.Exception

Throws:

java.lang.Exception

Octonion

public Octonion(Octonion o)

Method Detail

getClone

public Octonion getClone()

getNewInstance

public Hypernumber getNewInstance()

doCayleyDicksonDoubling

public static void doCayleyDicksonDoubling(java.lang.String signatures,
                                           java.lang.String vector_basis,
                                           int type)
                                    throws java.lang.Exception

Throws:

java.lang.Exception

doCayleyDicksonDoubling

public static void doCayleyDicksonDoubling(java.lang.String signature)
                                    throws java.lang.Exception

Throws:

java.lang.Exception

doCayleyDicksonDoubling

public static void doCayleyDicksonDoubling(java.lang.String signatures,
                                           java.lang.String vector_basis)
                                    throws java.lang.Exception

Throws:

java.lang.Exception

getCayleyDicksonTable

public static java.lang.String[][] getCayleyDicksonTable(java.lang.String[] signature,
                                                         java.lang.String[] vector_basis,
                                                         int type)
                                                  throws java.lang.Exception

Throws:

java.lang.Exception

getCayleyDicksonTable

public static java.lang.String[][] getCayleyDicksonTable(java.lang.String[] signature,
                                                         int type)
                                                  throws java.lang.Exception

Throws:

java.lang.Exception

doCayleyDicksonProcess

public static void doCayleyDicksonProcess(java.lang.String signature)
                                   throws java.lang.Exception

Throws:

java.lang.Exception

getBasis

public java.lang.String[] getBasis()

Specified by:

getBasis in class Hypernumber

setMultiplicationTable

public static void setMultiplicationTable(java.lang.String[][] mult_table)

setMultiplicationTable

public static void setMultiplicationTable(MultiplicationTable mult_table)

setOctonionMultTable

public void setOctonionMultTable(OctonionMultiplicationTable mult_table)

getMultiplicationTableAsArray

public static java.lang.String[][] getMultiplicationTableAsArray()

getMultiplicationTable

public static MultiplicationTable getMultiplicationTable()

setDefaultTable

public static void setDefaultTable()

getMultTableAsHtml

public static java.lang.String getMultTableAsHtml()

getNormedMultiplicationTable

public static java.lang.String[][] getNormedMultiplicationTable()

getNormedMultiplicationtableAsHtml

public static java.lang.String getNormedMultiplicationtableAsHtml()

getMultTable

public java.lang.String[][] getMultTable()

setVectorComponents

public void setVectorComponents(java.lang.String comps)
                         throws java.lang.Exception

Components *

Throws:

java.lang.Exception

getVectorComponents

public java.lang.String[] getVectorComponents()
                                       throws java.lang.Exception

Throws:

java.lang.Exception

setBiVectorComponents

public void setBiVectorComponents(java.lang.String comps)
                           throws java.lang.Exception

Throws:

java.lang.Exception

getBiVectorComponents

public java.lang.String[] getBiVectorComponents()
                                         throws java.lang.Exception

Throws:

java.lang.Exception

setTriVectorComponents

public void setTriVectorComponents(java.lang.String comps)
                            throws java.lang.Exception

Throws:

java.lang.Exception

getTriVectorComponents

public java.lang.String[] getTriVectorComponents()
                                          throws java.lang.Exception

Throws:

java.lang.Exception

getNumberOfComponents

public int getNumberOfComponents()

negate

public Octonion negate()
                throws java.lang.Exception

Arithmetics *

Overrides:

negate in class Hypernumber

Returns:

Negated hypernumber h.

Throws:

java.lang.Exception

add

public Octonion add(Hypernumber o)

Overrides:

add in class Hypernumber

subtract

public Octonion subtract(Hypernumber o)
                  throws java.lang.Exception

Description copied from class: Hypernumber

Subtracts a hypernumber.

Overrides:

subtract in class Hypernumber

Parameters:

o - Hypernumber to be subtacted.

Returns:

Result of subtraction.

Throws:

java.lang.Exception

multiplyWithScalar

public Octonion multiplyWithScalar(java.lang.String scalar)
                            throws java.lang.Exception

Description copied from class: Hypernumber

Multiplication with a scalar.

Overrides:

multiplyWithScalar in class Hypernumber

Parameters:

scalar - Scalar.

Returns:

Hypernumber multiplied with a scalar.

Throws:

java.lang.Exception

conjugate

public Octonion conjugate()
                   throws java.lang.Exception

Overrides:

conjugate in class Hypernumber

Returns:

The conjugate of the hypernumber, i.e. reverses the signs of the imaginary hypercomplex components.

Throws:

java.lang.Exception

getScalarProduct

public static Octonion getScalarProduct(Octonion o1,
                                        Octonion o2)
                                 throws java.lang.Exception

Throws:

java.lang.Exception

getGradesOccuring

public java.lang.String getGradesOccuring()
                                   throws java.lang.Exception

Throws:

java.lang.Exception

rightMultiplyWith

public Octonion rightMultiplyWith(Octonion o2)
                           throws java.lang.Exception

Throws:

java.lang.Exception

getCommutatorWith

public Octonion getCommutatorWith(Hypernumber o)
                           throws java.lang.Exception

Description copied from class: Hypernumber

Analysis *

Overrides:

getCommutatorWith in class Hypernumber

Throws:

java.lang.Exception

getSum

public static Octonion getSum(Octonion o1,
                              Octonion o2)
                       throws java.lang.Exception

Throws:

java.lang.Exception

getDifference

public static Octonion getDifference(Octonion o1,
                                     Octonion o2)
                              throws java.lang.Exception

Throws:

java.lang.Exception

getNegated

public static Octonion getNegated(Octonion o)
                           throws java.lang.Exception

Throws:

java.lang.Exception

getProductWithScalar

public static Octonion getProductWithScalar(Hypernumber h,
                                            java.lang.String scalar)
                                     throws java.lang.Exception

Throws:

java.lang.Exception

getProduct

public static Octonion getProduct(Octonion o1,
                                  Octonion o2)
                           throws java.lang.Exception

Throws:

java.lang.Exception

getCommutator

public static Octonion getCommutator(Octonion o1,
                                     Octonion o2)
                              throws java.lang.Exception

Throws:

java.lang.Exception

isThirdRootOfUnity

public static boolean isThirdRootOfUnity(Octonion o)
                                  throws java.lang.Exception

Throws:

java.lang.Exception

isSixthRootOfUnity

public static boolean isSixthRootOfUnity(Octonion o)
                                  throws java.lang.Exception

Throws:

java.lang.Exception

getAntiCommutatorWith

public Octonion getAntiCommutatorWith(Hypernumber h)
                               throws java.lang.Exception

Description copied from class: Hypernumber

Calculates the anti-commutator of the hypernumber h1 with another one h2 which is defined as: {h1, h2} = h1*h2 + h2*h1

Overrides:

getAntiCommutatorWith in class Hypernumber

Parameters:

h - Second hypernumber.

Returns:

The anti-commutator of the hypernumber with another one.

Throws:

java.lang.Exception

getCommutatorJacobian

public static Octonion getCommutatorJacobian(Octonion o1,
                                             Octonion o2,
                                             Octonion o3)
                                      throws java.lang.Exception

Throws:

java.lang.Exception

getJacobian

public static Octonion getJacobian(Octonion o1,
                                   Octonion o2,
                                   Octonion o3)
                            throws java.lang.Exception

Throws:

java.lang.Exception

getAlgebraType

public static java.lang.String getAlgebraType()
                                       throws java.lang.Exception

The normed multiplication table is compared with a multiplication table of i) a non-split octonion ii) a split octonion

Returns:

"nonsplit" or // TODO update commenmt "split" or "unknown"

Throws:

java.lang.Exception

setType

public static void setType(int nr)
                    throws java.lang.Exception

Sets the n-th multiplication table, according to the definition in the octonion multiplication table class.

Parameters:

nr - Number of multiplication table in the range 1,...,480.

Throws:

java.lang.Exception

setFanoPlane

public static void setFanoPlane(int nr)
                         throws java.lang.Exception

Sets a multiplication table based on one of the 30 Fano planes according to the definition in the octonion multiplication table class.

Parameters:

nr - Number of Fano plane in the range 1...30.

Throws:

java.lang.Exception

getAssociator

public static Octonion getAssociator(Octonion o1,
                                     Octonion o2,
                                     Octonion o3)
                              throws java.lang.Exception

Calculates the associator of 3 octonions o1, o2, o3 which is defined as {o1, o2, o3} = (o1*o2)*o3 - o1*(o2*o3) The associator is non-trivial as the octonion algebra is non-associative.

Parameters:

o1 - Octonion 1

o2 - Octonion 2

o3 - Octonion 3

Returns:

The associator of the three octonions.

Throws:

java.lang.Exception

getAntiAssociator

public static Octonion getAntiAssociator(Hypernumber h1,
                                         Hypernumber h2,
                                         Hypernumber h3)
                                  throws java.lang.Exception

Throws:

java.lang.Exception

calculateBCHWith

public Octonion calculateBCHWith(Octonion o,
                                 int from_order,
                                 int to_order)
                          throws java.lang.Exception

Throws:

java.lang.Exception

getVectorProduct

public static Octonion getVectorProduct(Octonion o1,
                                        Octonion o2)
                                 throws java.lang.Exception

Throws:

java.lang.Exception

getNumberOfGrades

public int getNumberOfGrades()

getGrade

public Octonion getGrade(int grade)
                  throws java.lang.Exception

Throws:

java.lang.Exception

getNumericalInverse

public static Octonion getNumericalInverse(Hypernumber h)
                                    throws java.lang.Exception

Throws:

java.lang.Exception

getOuterProduct

public static Octonion getOuterProduct(Octonion o1,
                                       Octonion o2)
                                throws java.lang.Exception

Throws:

java.lang.Exception

getAssociationType

public static Octonion getAssociationType(int nr,
                                          Octonion o1,
                                          Octonion o2,
                                          Octonion o3,
                                          Octonion o4)
                                   throws java.lang.Exception

Throws:

java.lang.Exception

getAssociationTypeCommutator

public static Octonion getAssociationTypeCommutator(int nr,
                                                    Octonion o1,
                                                    Octonion o2,
                                                    Octonion o3,
                                                    Octonion o4)
                                             throws java.lang.Exception

Throws:

java.lang.Exception

getFanoPlane

public static SevenPointsSTS getFanoPlane()
                                   throws java.lang.Exception

Returns:

Fano plane which is equivalent to a Steiner triple system on 7 points.

Throws:

java.lang.Exception

getTriadSet

public java.util.Collection<java.lang.String> getTriadSet(int nr)
                                                   throws java.lang.Exception

Throws:

java.lang.Exception

getGrades

public static java.util.TreeMap<java.lang.Integer,java.lang.Integer> getGrades()
                                                                        throws java.lang.Exception

Returns:

Array with the grades.

Throws:

java.lang.Exception

getSubalgebras

public static java.util.HashSet<java.lang.String> getSubalgebras(int order)
                                                          throws java.lang.Exception

Throws:

java.lang.Exception

getSubalgebrasUnsigned

public static java.util.HashSet<int[]> getSubalgebrasUnsigned(int order)
                                                       throws java.lang.Exception

Throws:

java.lang.Exception