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Geometric Numbers
Geometric Numbers
Representations of numbers as multivectors
Complex numbers as 0,1
Complex numbers as
0,1
Complex numbers as even subspace of 2
Complex numbers as even subspace of
2
Hyperbolic numbers as 1
Hyperbolic numbers as
1
Dual numbers as 0,0,1
Dual numbers as
0,0,1
Quaternions as 0,2
Quaternions as
0,2
Quaternions as 3
Quaternions as
3
BiQuaternions () in 3
BiQuaternions () in
3
In[299]:=
=GeometricAlgebra[3,"Format"->"()","FormatIndex"{{1}->"i",{2}->"j",{3}->"k",{2,3}->"i",{3,1}->"j",{1,2}->"k",{1,2,3}""},"Ordering"->{{},{1},{2},{3},{2,3},{3,1},{1,2},{1,2,3}}]
Out[299]=
()
In[300]:=
{,i,j,k,i,j,k,}=["OrderedBasis"]
Out[300]=
{1,i,j,k,i,j,k,}
In[250]:=
2
i
2
j
2
k
2
Out[250]=
True
In[251]:=
2
i
2
j
2
k
Out[251]=
True
In[252]:=
**#==#**&/@["OrderedBasis"]
Out[252]=
{True,True,True,True,True,True,True,True}
In[253]:=
BiQuaternion[q:{_,_,_,_}]:=Re[q].{,i,j,k}+Im[q].{,i,j,k}
In[207]:=
BiQuaternionC[q:{_,_,_,_}]:=q.{,i,j,k}
In[222]:=
RealBiQuaternion[q_]:=q[Map[NumberMultivector[#,]&]]["Flatten"]
In[266]:=
ComplexBiQuaternion[q_]:=With[{re=q[{{},{1},{2},{3}}],im=q[{{1,2,3},{2,3},{3,1},{1,2}}]},(re+Iim).{,i,j,k}]
In[225]:=
{x,y}=RandomComplex[{-1-I,1+I},{2,4}];
In[256]:=
ComplexBiQuaternion[BiQuaternion[x]]==BiQuaternionC[x]
Out[256]=
True
In[257]:=
RealBiQuaternion[BiQuaternionC[x]]==BiQuaternion[x]
Out[257]=
True
In[258]:=
RealBiQuaternion[BiQuaternionC[x]**BiQuaternionC[y]]==BiQuaternion[x]**BiQuaternion[y]
Out[258]=
True
In[301]:=
ComplexBiQuaternion[BiQuaternion[x]**BiQuaternion[y]]==ComplexBiQuaternion[RealBiQuaternion[BiQuaternionC[x]**BiQuaternionC[y]]]
Out[301]=
True
In[302]:=
MatrixMultivector[MultivectorMatrix@BiQuaternionC[x],]==BiQuaternionC[x]
Out[302]=
True
In[303]:=
MatrixMultivector[MultivectorMatrix@BiQuaternion[x],]==BiQuaternion[x]
Out[303]=
True
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