public class Octonion extends CayleyDicksonAlgebra
CD_LIKE0, CD_LIKE1, CD_LIKE10, CD_LIKE11, CD_LIKE12, CD_LIKE13, CD_LIKE14, CD_LIKE15, CD_LIKE16, CD_LIKE17, CD_LIKE18, CD_LIKE19, CD_LIKE2, CD_LIKE20, CD_LIKE21, CD_LIKE22, CD_LIKE23, CD_LIKE24, CD_LIKE25, CD_LIKE26, CD_LIKE27, CD_LIKE28, CD_LIKE29, CD_LIKE3, CD_LIKE30, CD_LIKE31, CD_LIKE4, CD_LIKE5, CD_LIKE6, CD_LIKE7, CD_LIKE8, CD_LIKE9, CLASSICAL, CLASSICAL2, COMPLEX, MAX_TYPE, MIN_TYPE, OCTONION, QUATERNION, SEDENION, TRIGINTADUONION, TWISTED, TWISTED2
ARG_NUM, CIRCULAR, COMP_NUM, components, HYPERBOLIC, ZERO_PRECISSION
Constructor and Description |
---|
Octonion() |
Octonion(Octonion o) |
Octonion(java.lang.String components) |
Modifier and Type | Method and Description |
---|---|
Octonion |
add(BinaryAlgebra 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 signature,
int type) |
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()
Checking for the 480 variants of multiplication tables is omitted.
|
static java.lang.String |
getAlgebraType(boolean check_480_variants)
The normed multiplication table is compared with the normed multiplication tables
obtained by the different Cayley-Dickson doubling formulas for all 8 possible
signature combinations.
|
static Octonion |
getAntiAssociator(BinaryAlgebra h1,
BinaryAlgebra h2,
BinaryAlgebra h3) |
static Octonion |
getAntiCommutator(Octonion o1,
Octonion o2) |
Octonion |
getAntiCommutatorWith(BinaryAlgebra h)
Calculates the anti-commutator of the element a1 of the algebra with another one a2
which is defined as:
{a1, a2} = a1*a2 + a2*a1
|
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) |
java.util.TreeMap<java.lang.String,java.lang.Boolean> |
getAssociationTypeIdentitiesCheckList(int order,
int rank,
int n_comps,
boolean commutator) |
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) |
static Octonion |
getCommutatorSaglian(Octonion o1,
Octonion o2,
Octonion o3,
Octonion o4) |
Octonion |
getCommutatorWith(BinaryAlgebra o) |
static Octonion |
getConjugate(Octonion o) |
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) |
java.lang.String[][] |
getMTab()
Multiplication table and basis *
|
static MultiplicationTable |
getMultiplicationTable() |
static java.lang.String[][] |
getMultiplicationTableAsArray() |
java.lang.String[][] |
getMultTable() |
static java.lang.String |
getMultTableAsHtml() |
static Octonion |
getNegated(Octonion o) |
Octonion |
getNewInstance() |
static java.lang.String[][] |
getNormedMultiplicationTable() |
static java.lang.String |
getNormedMultiplicationtableAsHtml() |
int |
getNumberOfComponents() |
int |
getNumberOfGrades() |
static Octonion |
getNumericalInverse(BinaryAlgebra h) |
static Octonion |
getOuterProduct(Octonion o1,
Octonion o2) |
static Octonion |
getProduct(Octonion o1,
Octonion o2) |
static Octonion |
getProductWithScalar(BinaryAlgebra h,
java.lang.String scalar) |
static Octonion |
getScalarProduct(Octonion o1,
Octonion o2) |
static java.lang.String |
getSignature() |
static SignTable |
getSignTable() |
static SignTable |
getSignTable(java.lang.String plus,
java.lang.String minus) |
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) |
static Octonion |
getTernaryInnerProduct(Octonion o1,
Octonion o2,
Octonion o3) |
java.util.Collection<java.lang.String> |
getTriadSet(int nr) |
java.lang.String[] |
getTriVectorComponents() |
java.lang.String[] |
getVectorComponents() |
static Octonion |
getVectorProduct(Octonion o1,
Octonion o2) |
boolean |
isCayleyDicksonAlgebra() |
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) |
static void |
setSignTable(SignTable sign_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(BinaryAlgebra o)
Subtracts an element from an algebra.
|
getCDTypeAsString
add, add, applyLeibnizRule, calculateBCHWith, getCommutatorWith, getComponentsOfGrade, getDifference, getGradedInnerProduct, getIntersection, getLeftCovariantDerivative, getNumberOfComponents, getOuterProduct, getProduct, getScalarProduct, getSum, getTernaryInnerProduct, isCommutative, isEqual, isOrthogonal, isOrthogonalTo, isProjectionOperator, rightMultiplyWith
asString, asString, collectTerms, compressComponents, dual, getBasisElementAsString, getBasisElementsAsString, getClosedPairsAsString, getComponent, getComponents, getComponentsAsString, getComponentsAsString, getConjugate, getDiagonalProducts, getDual, getExpressionAsList, getFormattedMultTable, getHermitianConjugate, getImaginaryPart, getInstance, getMultiplicativeOrder, getMultiplicativeOrder, getMultTableDiagonal, getNegated, getNextBitmap, getNonzeroComponentsNumbered, getNonzeroComponentsNumbered, getNormedMultiplicationTable, getNormSquared, getNormSquaredAsValue, getNumberOfClosedElements, getNumberOfClosedPairs, getNumberOfNonzeroComponents, getNumericalInverse, getProduct, getProductWithScalar, getPseudoScalarComponent, getRandomClosedPair, getRandomNonClosedPair, getScalarComponent, getSubalgebras, getSubalgebrasAsString, getSubalgebrasSignatures, getTrace, getTraceRespDeterminant, getVectorDerivative, getVectorProduct, isAutomorphism, isCayleyDicksonLoop, isComponentZero, isLinearlyDependent, isNumerical, isNumericalityDetected, isProductClosed, isZero, resetComponents, rightMultiplyWith, rightMultiplyWith, setBasis, setComponent, setComponentAsBasisNames, setComponentExclusively, setComponents, setComponents, setIndexedComponents, setNumerical, setNumerical, setPseudoScalarComponent, setRandomComponents, setRandomComponents, setRandomComponents, setRandomComponentsAll, setRepresentation, setScalarComponent, simplifyNumericalFactors, simplifyNumericalSummands, simplifyPlusMinus, square
public Octonion()
public Octonion(java.lang.String components) throws java.lang.Exception
java.lang.Exception
public Octonion(Octonion o)
public Octonion getClone()
getClone
in class BinaryAlgebra
public Octonion getNewInstance()
getNewInstance
in class BinaryProductStructure
public static void doCayleyDicksonDoubling(java.lang.String signatures, java.lang.String vector_basis, int type) throws java.lang.Exception
java.lang.Exception
public static void doCayleyDicksonDoubling(java.lang.String signature) throws java.lang.Exception
java.lang.Exception
public static void doCayleyDicksonDoubling(java.lang.String signature, int type) throws java.lang.Exception
java.lang.Exception
public static void doCayleyDicksonDoubling(java.lang.String signatures, java.lang.String vector_basis) throws java.lang.Exception
java.lang.Exception
public static java.lang.String[][] getCayleyDicksonTable(java.lang.String[] signature, java.lang.String[] vector_basis, int type) throws java.lang.Exception
java.lang.Exception
public static java.lang.String[][] getCayleyDicksonTable(java.lang.String[] signature, int type) throws java.lang.Exception
java.lang.Exception
public static void doCayleyDicksonProcess(java.lang.String signature) throws java.lang.Exception
java.lang.Exception
public java.lang.String[] getBasis()
getBasis
in class BinaryProductStructure
public static void setMultiplicationTable(java.lang.String[][] mult_table)
public static void setMultiplicationTable(MultiplicationTable mult_table)
public static void setSignTable(SignTable sign_table) throws java.lang.Exception
java.lang.Exception
public static SignTable getSignTable(java.lang.String plus, java.lang.String minus) throws java.lang.Exception
java.lang.Exception
public static SignTable getSignTable() throws java.lang.Exception
java.lang.Exception
public static java.lang.String[][] getMultiplicationTableAsArray()
public static MultiplicationTable getMultiplicationTable()
public java.lang.String[][] getMTab()
BinaryProductStructure
getMTab
in class BinaryProductStructure
public static void setDefaultTable()
public static java.lang.String getMultTableAsHtml()
public static java.lang.String[][] getNormedMultiplicationTable()
public static java.lang.String getNormedMultiplicationtableAsHtml()
public java.lang.String[][] getMultTable()
public void setVectorComponents(java.lang.String comps) throws java.lang.Exception
java.lang.Exception
public java.lang.String[] getVectorComponents() throws java.lang.Exception
java.lang.Exception
public void setBiVectorComponents(java.lang.String comps) throws java.lang.Exception
java.lang.Exception
public java.lang.String[] getBiVectorComponents() throws java.lang.Exception
java.lang.Exception
public void setTriVectorComponents(java.lang.String comps) throws java.lang.Exception
java.lang.Exception
public java.lang.String[] getTriVectorComponents() throws java.lang.Exception
java.lang.Exception
public int getNumberOfComponents()
getNumberOfComponents
in class BinaryProductStructure
public java.util.TreeMap<java.lang.String,java.lang.Boolean> getAssociationTypeIdentitiesCheckList(int order, int rank, int n_comps, boolean commutator) throws java.lang.Exception
java.lang.Exception
public Octonion negate() throws java.lang.Exception
negate
in class BinaryProductStructure
java.lang.Exception
public Octonion add(BinaryAlgebra o)
public Octonion subtract(BinaryAlgebra o) throws java.lang.Exception
BinaryAlgebra
subtract
in class BinaryAlgebra
o
- Algebraic element to be subtracted.java.lang.Exception
public Octonion multiplyWithScalar(java.lang.String scalar) throws java.lang.Exception
BinaryProductStructure
multiplyWithScalar
in class BinaryProductStructure
scalar
- Scalar.java.lang.Exception
public Octonion conjugate() throws java.lang.Exception
conjugate
in class BinaryProductStructure
java.lang.Exception
public static Octonion getScalarProduct(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public static Octonion getTernaryInnerProduct(Octonion o1, Octonion o2, Octonion o3) throws java.lang.Exception
java.lang.Exception
public java.lang.String getGradesOccuring() throws java.lang.Exception
java.lang.Exception
public Octonion rightMultiplyWith(Octonion o2) throws java.lang.Exception
java.lang.Exception
public Octonion getCommutatorWith(BinaryAlgebra o) throws java.lang.Exception
getCommutatorWith
in class BinaryAlgebra
java.lang.Exception
public static Octonion getSum(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public static Octonion getDifference(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public static Octonion getNegated(Octonion o) throws java.lang.Exception
java.lang.Exception
public static Octonion getProductWithScalar(BinaryAlgebra h, java.lang.String scalar) throws java.lang.Exception
java.lang.Exception
public static Octonion getProduct(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public static Octonion getCommutator(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public static Octonion getAntiCommutator(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public static Octonion getConjugate(Octonion o) throws java.lang.Exception
java.lang.Exception
public static boolean isThirdRootOfUnity(Octonion o) throws java.lang.Exception
java.lang.Exception
public static boolean isSixthRootOfUnity(Octonion o) throws java.lang.Exception
java.lang.Exception
public Octonion getAntiCommutatorWith(BinaryAlgebra h) throws java.lang.Exception
BinaryAlgebra
getAntiCommutatorWith
in class BinaryAlgebra
h
- Second element of the algebra.java.lang.Exception
public static Octonion getCommutatorJacobian(Octonion o1, Octonion o2, Octonion o3) throws java.lang.Exception
java.lang.Exception
public static Octonion getJacobian(Octonion o1, Octonion o2, Octonion o3) throws java.lang.Exception
java.lang.Exception
public static Octonion getCommutatorSaglian(Octonion o1, Octonion o2, Octonion o3, Octonion o4) throws java.lang.Exception
java.lang.Exception
public static java.lang.String getAlgebraType() throws java.lang.Exception
java.lang.Exception
public static java.lang.String getAlgebraType(boolean check_480_variants) throws java.lang.Exception
check_480_variants
- java.lang.Exception
public static void setType(int nr) throws java.lang.Exception
nr
- Number of multiplication table in the range 1,...,480.java.lang.Exception
public static void setFanoPlane(int nr) throws java.lang.Exception
nr
- Number of Fano plane in the range 1...30.java.lang.Exception
public static Octonion getAssociator(Octonion o1, Octonion o2, Octonion o3) throws java.lang.Exception
o1
- Octonion 1o2
- Octonion 2o3
- Octonion 3java.lang.Exception
public static Octonion getAntiAssociator(BinaryAlgebra h1, BinaryAlgebra h2, BinaryAlgebra h3) throws java.lang.Exception
java.lang.Exception
public Octonion calculateBCHWith(Octonion o, int from_order, int to_order) throws java.lang.Exception
java.lang.Exception
public static Octonion getVectorProduct(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public int getNumberOfGrades()
getNumberOfGrades
in class BinaryAlgebra
public Octonion getGrade(int grade) throws java.lang.Exception
getGrade
in class BinaryAlgebra
java.lang.Exception
public static Octonion getNumericalInverse(BinaryAlgebra h) throws java.lang.Exception
java.lang.Exception
public static Octonion getOuterProduct(Octonion o1, Octonion o2) throws java.lang.Exception
java.lang.Exception
public static Octonion getAssociationType(int nr, Octonion o1, Octonion o2, Octonion o3, Octonion o4) throws java.lang.Exception
java.lang.Exception
public static Octonion getAssociationTypeCommutator(int nr, Octonion o1, Octonion o2, Octonion o3, Octonion o4) throws java.lang.Exception
java.lang.Exception
public static SevenPointsSTS getFanoPlane() throws java.lang.Exception
java.lang.Exception
public java.util.Collection<java.lang.String> getTriadSet(int nr) throws java.lang.Exception
java.lang.Exception
public static java.util.TreeMap<java.lang.Integer,java.lang.Integer> getGrades() throws java.lang.Exception
java.lang.Exception
public static java.util.HashSet<java.lang.String> getSubalgebras(int order) throws java.lang.Exception
java.lang.Exception
public static java.util.HashSet<int[]> getSubalgebrasUnsigned(int order) throws java.lang.Exception
java.lang.Exception
public static java.lang.String getSignature()
public boolean isCayleyDicksonAlgebra() throws java.lang.Exception
isCayleyDicksonAlgebra
in class BinaryAlgebra
java.lang.Exception