Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Geometric Numbers
Representations of numbers as multivectors
Complex numbers as 0,1
Complex numbers as
0,1
Complex numbers as +2
Complex numbers as
+
2
Hyperbolic (split-complex) numbers as 1
Hyperbolic (split-complex) numbers as
1
In[27]:=
=
[1,"Format""","FormatIndex"{{1}"η"}]
 | GeometricAlgebra |
Out[27]=
In[28]:=
η=["Pseudoscalar"]
Out[28]=
η
In[29]:=
2
η
Out[29]=
1
In[30]:=
r=t+xη
Out[30]=
t+xη
In[31]:=
*
r
Out[31]=
t+(-x)η
In[32]:=
†
r
Out[32]=
x+tη
In[33]:=
r
Out[33]=
2
t
2
x
2
t
2
x
In[34]:=
-1
r
Out[34]=
t
2
t
2
x
x
-+
2
t
2
x
In[1898]:=
Tr[r]
Out[1898]=
2t
In[1899]:=
2
r
Out[1899]=
2
t
2
x
In[1900]:=
Det[r]
Out[1900]=
2
t
2
x
Define boost (rotation by hyperbolic angle - rapidity):
ϕ
In[68]:=
K=Cosh[ϕ]-Sinh[ϕ]η;
Define boosted vector:
In[36]:=
rʹ=tʹ+xʹη;
In[37]:=
K⟑r
Out[37]=
tCosh[ϕ]-xSinh[ϕ]+(xCosh[ϕ]-tSinh[ϕ])η
Derive classic Lorentz transformation formulas:
In[38]:=
Solve[K⟑rrʹ,{tʹ,xʹ}]/.ϕArcTanh[v]//Simplify
Out[38]=
tʹ,xʹ
t-vx
1-
2
v
x-tv
1-
2
v
Dual numbers as 0,0,1
Dual numbers as
0,0,1
Quaternions as 0,2
Quaternions as
0,2
In[1924]:=
=
[{0,2},"Format""","FormatIndex"{{-2}"i",{-1}"j",{-2,-1}"k"}]
| GeometricAlgebra |
Out[1924]=
In[1925]:=
{,i,j,k}=MultivectorBasis[]
Out[1925]=
In[1926]:=
i⟑jk
Out[1926]=
True
In[1927]:=
j⟑ki
Out[1927]=
True
In[1928]:=
k⟑ij
Out[1928]=
True
In[1929]:=
2
i
2
j
2
k
Out[1929]=
True
Quaternions as +3
Quaternions as
+
3
In[80]:=
=
[3,"Format""","FormatIndex"{{1,2,3}""},"Ordering"{{},{1},{2},{3},{2,3},{3,1},{2,1},{1,2,3}}]
| GeometricAlgebra |
Out[80]=
In[83]:=
["OrderedBasis"]
Out[83]=
In[1931]:=
{,,,}=MultivectorBasis[,{0,2}];i=;j=-;k=-;
e
12
e
13
e
23
e
23
e
13
e
12
In[1935]:=
i⟑jk
Out[1935]=
True
In[1936]:=
j⟑ki
Out[1936]=
True
In[1937]:=
k⟑ij
Out[1937]=
True
In[1938]:=
2
i
2
j
2
k
Out[1938]=
True
Split-quaternions s as +2,1
Split-quaternions as
s
+
2,1
In[330]:=
=
| GeometricAlgebra |
Out[330]=
2,1
In[336]:=
{i,j,k}=["Basis",2]
Out[336]=
In[343]:=
2
i
Out[343]=
True
In[342]:=
2
j
2
k
Out[342]=
True
Bi-quaternions () as 3
Bi-quaternions () as
3
In[299]:=
=
[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}}]
| GeometricAlgebra |
Out[299]=
()
In[300]:=
{,i,j,k,i,j,k,}=["OrderedBasis"]
Out[300]=
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}
References