First, install Grassmann.jl for develpment(whenever the code has changed, it will be reflected):

# Enter any where you like to clone Grassmann.jl
cd ~/projects
git clone https://github.com/chakravala/Grassmann.jl
julia
# Enter Pkg REPL-mode
# By hitting ]
add PyCall
dev ~/projects/Grassmann.jl



In [1]:

    
using Reduce,Grassmann









    



Reduce (Free CSL version, revision 4534), 05-Apr-18 ...



In [2]:

    
ℝ^3 == V"+++" == VectorSpace(3)









    Out[2]:





true



In [3]:

    
V = ℝ'⊕ℝ^3









    Out[3]:





⟨-+++⟩



In [4]:

    
V'









    Out[4]:





⟨+---⟩'



In [5]:

    
W = V⊕V'









    Out[5]:





⟨-++++---⟩*



In [6]:

    
collect(V)









    Out[6]:





Grassmann.Algebra{⟨-+++⟩,16}(v, v₁, v₂, v₃, v₄, v₁₂, v₁₃, v₁₄, v₂₃, v₂₄, v₃₄, v₁₂₃, v₁₂₄, v₁₃₄, v₂₃₄, v₁₂₃₄)



In [7]:

    
collect(V')









    Out[7]:





Grassmann.Algebra{⟨+---⟩',16}(w, w¹, w², w³, w⁴, w¹², w¹³, w¹⁴, w²³, w²⁴, w³⁴, w¹²³, w¹²⁴, w¹³⁴, w²³⁴, w¹²³⁴)



In [8]:

    
collect(W)









    Out[8]:





Grassmann.Algebra{⟨-++++---⟩*,256}(v, v₁, v₂, v₃, v₄, w¹, w², w³, w⁴, v₁₂, v₁₃, v₁₄, v₁w¹, v₁w², v₁w³, v₁w⁴, v₂₃, v₂₄, v₂w¹, v₂w², v₂w³, v₂w⁴, v₃₄, v₃w¹, v₃w², v₃w³, v₃w⁴, v₄w¹, v₄w², v₄w³, v₄w⁴, w¹², w¹³, w¹⁴, w²³, w²⁴, w³⁴, v₁₂₃, v₁₂₄, v₁₂w¹, v₁₂w², v₁₂w³, v₁₂w⁴, v₁₃₄, v₁₃w¹, v₁₃w², v₁₃w³, v₁₃w⁴, v₁₄w¹, v₁₄w², v₁₄w³, v₁₄w⁴, v₁w¹², v₁w¹³, v₁w¹⁴, v₁w²³, v₁w²⁴, v₁w³⁴, v₂₃₄, v₂₃w¹, v₂₃w², v₂₃w³, v₂₃w⁴, v₂₄w¹, v₂₄w², v₂₄w³, v₂₄w⁴, v₂w¹², v₂w¹³, v₂w¹⁴, v₂w²³, v₂w²⁴, v₂w³⁴, v₃₄w¹, v₃₄w², v₃₄w³, v₃₄w⁴, v₃w¹², v₃w¹³, v₃w¹⁴, v₃w²³, v₃w²⁴, v₃w³⁴, v₄w¹², v₄w¹³, v₄w¹⁴, v₄w²³, v₄w²⁴, v₄w³⁴, w¹²³, w¹²⁴, w¹³⁴, w²³⁴, v₁₂₃₄, v₁₂₃w¹, v₁₂₃w², v₁₂₃w³, v₁₂₃w⁴, v₁₂₄w¹, v₁₂₄w², v₁₂₄w³, v₁₂₄w⁴, v₁₂w¹², v₁₂w¹³, v₁₂w¹⁴, v₁₂w²³, v₁₂w²⁴, v₁₂w³⁴, v₁₃₄w¹, v₁₃₄w², v₁₃₄w³, v₁₃₄w⁴, v₁₃w¹², v₁₃w¹³, v₁₃w¹⁴, v₁₃w²³, v₁₃w²⁴, v₁₃w³⁴, v₁₄w¹², v₁₄w¹³, v₁₄w¹⁴, v₁₄w²³, v₁₄w²⁴, v₁₄w³⁴, v₁w¹²³, v₁w¹²⁴, v₁w¹³⁴, v₁w²³⁴, v₂₃₄w¹, v₂₃₄w², v₂₃₄w³, v₂₃₄w⁴, v₂₃w¹², v₂₃w¹³, v₂₃w¹⁴, v₂₃w²³, v₂₃w²⁴, v₂₃w³⁴, v₂₄w¹², v₂₄w¹³, v₂₄w¹⁴, v₂₄w²³, v₂₄w²⁴, v₂₄w³⁴, v₂w¹²³, v₂w¹²⁴, v₂w¹³⁴, v₂w²³⁴, v₃₄w¹², v₃₄w¹³, v₃₄w¹⁴, v₃₄w²³, v₃₄w²⁴, v₃₄w³⁴, v₃w¹²³, v₃w¹²⁴, v₃w¹³⁴, v₃w²³⁴, v₄w¹²³, v₄w¹²⁴, v₄w¹³⁴, v₄w²³⁴, w¹²³⁴, v₁₂₃₄w¹, v₁₂₃₄w², v₁₂₃₄w³, v₁₂₃₄w⁴, v₁₂₃w¹², v₁₂₃w¹³, v₁₂₃w¹⁴, v₁₂₃w²³, v₁₂₃w²⁴, v₁₂₃w³⁴, v₁₂₄w¹², v₁₂₄w¹³, v₁₂₄w¹⁴, v₁₂₄w²³, v₁₂₄w²⁴, v₁₂₄w³⁴, v₁₂w¹²³, v₁₂w¹²⁴, v₁₂w¹³⁴, v₁₂w²³⁴, v₁₃₄w¹², v₁₃₄w¹³, v₁₃₄w¹⁴, v₁₃₄w²³, v₁₃₄w²⁴, v₁₃₄w³⁴, v₁₃w¹²³, v₁₃w¹²⁴, v₁₃w¹³⁴, v₁₃w²³⁴, v₁₄w¹²³, v₁₄w¹²⁴, v₁₄w¹³⁴, v₁₄w²³⁴, v₁w¹²³⁴, v₂₃₄w¹², v₂₃₄w¹³, v₂₃₄w¹⁴, v₂₃₄w²³, v₂₃₄w²⁴, v₂₃₄w³⁴, v₂₃w¹²³, v₂₃w¹²⁴, v₂₃w¹³⁴, v₂₃w²³⁴, v₂₄w¹²³, v₂₄w¹²⁴, v₂₄w¹³⁴, v₂₄w²³⁴, v₂w¹²³⁴, v₃₄w¹²³, v₃₄w¹²⁴, v₃₄w¹³⁴, v₃₄w²³⁴, v₃w¹²³⁴, v₄w¹²³⁴, v₁₂₃₄w¹², v₁₂₃₄w¹³, v₁₂₃₄w¹⁴, v₁₂₃₄w²³, v₁₂₃₄w²⁴, v₁₂₃₄w³⁴, v₁₂₃w¹²³, v₁₂₃w¹²⁴, v₁₂₃w¹³⁴, v₁₂₃w²³⁴, v₁₂₄w¹²³, v₁₂₄w¹²⁴, v₁₂₄w¹³⁴, v₁₂₄w²³⁴, v₁₂w¹²³⁴, v₁₃₄w¹²³, v₁₃₄w¹²⁴, v₁₃₄w¹³⁴, v₁₃₄w²³⁴, v₁₃w¹²³⁴, v₁₄w¹²³⁴, v₂₃₄w¹²³, v₂₃₄w¹²⁴, v₂₃₄w¹³⁴, v₂₃₄w²³⁴, v₂₃w¹²³⁴, v₂₄w¹²³⁴, v₃₄w¹²³⁴, v₁₂₃₄w¹²³, v₁₂₃₄w¹²⁴, v₁₂₃₄w¹³⁴, v₁₂₃₄w²³⁴, v₁₂₃w¹²³⁴, v₁₂₄w¹²³⁴, v₁₃₄w¹²³⁴, v₂₃₄w¹²³⁴, v₁₂₃₄w¹²³⁴)



In [9]:

    
@basis ℝ'⊕ℝ^3 # equivalent to basis"-+++"









    Out[9]:





(⟨-+++⟩, v, v₁, v₂, v₃, v₄, v₁₂, v₁₃, v₁₄, v₂₃, v₂₄, v₃₄, v₁₂₃, v₁₂₄, v₁₃₄, v₂₃₄, v₁₂₃₄)



In [10]:

    
V # Minkowski spacetime









    Out[10]:





⟨-+++⟩



In [11]:

    
typeof(V) # dispatch by vector space









    Out[11]:





VectorSpace{4,0,0x0000000000000001}



In [12]:

    
typeof(v13) # extensive type info









    Out[12]:





Basis{⟨-+++⟩,2,0x0000000000000005}



In [13]:

    
v13∧v2 # exterior tensor product









    Out[13]:





-1v₁₂₃



In [14]:

    
ans^2 # applies geometric product









    Out[14]:





1v



In [15]:

    
using BenchmarkTools



In [16]:

    
@btime h = 2v1+v3 # vector element









    



  39.255 ns (3 allocations: 80 bytes)






    Out[16]:





2v₁ + 0v₂ + 1v₃ + 0v₄



In [17]:

    
h = 2v1+v3









    Out[17]:





2v₁ + 0v₂ + 1v₃ + 0v₄



In [18]:

    
@btime h⋅h # inner product









    



  203.845 ns (1 allocation: 16 bytes)






    Out[18]:





-3v



In [19]:

    
@basis "++++" S b;



In [20]:

    
let k = (b1+b2)-b3
   for j ∈ 1:9
       k = k*(b234+b134)
       println(k)
end end









    



0 + 1v₁₄ + 1v₂₄ + 2v₃₄
0 - 2v₁ - 2v₂ + 2v₃
0 - 2v₁₄ - 2v₂₄ - 4v₃₄
0 + 4v₁ + 4v₂ - 4v₃
0 + 4v₁₄ + 4v₂₄ + 8v₃₄
0 - 8v₁ - 8v₂ + 8v₃
0 - 8v₁₄ - 8v₂₄ - 16v₃₄
0 + 16v₁ + 16v₂ - 16v₃
0 + 16v₁₄ + 16v₂₄ + 32v₃₄



In [21]:

    
G3 = Λ(3) # equivalent to Λ(V"+++"), Λ(ℝ^3), Λ.V3









    Out[21]:





Grassmann.Algebra{⟨+++⟩,8}(v, v₁, v₂, v₃, v₁₂, v₁₃, v₂₃, v₁₂₃)



In [22]:

    
G3.v13 * G3.v12









    Out[22]:





v₂₃



In [23]:

    
i,j,k = hyperplanes(ℝ^3)









    Out[23]:





3-element Array{SValue{⟨+++⟩,2,B,Int64} where B,1}:
 1v₂₃ 
 -1v₁₃
 1v₁₂



In [24]:

    
@btime i^2, j^2, k^2, i*j*k









    



  241.122 ns (5 allocations: 112 bytes)






    Out[24]:





(-1v, -1v, -1v, 1v)



In [25]:

    
@btime -(j+k) * (j+k)









    



  291.012 ns (8 allocations: 240 bytes)






    Out[25]:





2



In [26]:

    
@btime -(j+k) * i









    



  182.004 ns (6 allocations: 192 bytes)






    Out[26]:





0 - 1v₁₂ - 1v₁₃



In [27]:

    
basis"--"









    Out[27]:





(⟨--⟩, v, v₁, v₂, v₁₂)



In [28]:

    
v1^2, v2^2, v12^2, v1*v2*v12









    Out[28]:





(-1v, -1v, -1v, -1v)



In [29]:

    
Λ(62)









    Out[29]:





Grassmann.ExtendedAlgebra{⟨++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++⟩,4611686018427387904}(v, ..., v₁₂₃₄₅₆₇₈₉₀abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ)



In [30]:

    
Λ(62).v32a87Ng









    Out[30]:





-1v₂₃₇₈agN



In [31]:

    
@btime Λ(62)









    



  0.001 ns (0 allocations: 0 bytes)






    Out[31]:





Grassmann.ExtendedAlgebra{⟨++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++⟩,4611686018427387904}(v, ..., v₁₂₃₄₅₆₇₈₉₀abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ)



In [32]:

    
@btime Λ(22)









    



  197.717 ns (0 allocations: 0 bytes)






    Out[32]:





Grassmann.SparseAlgebra{⟨++++++++++++++++++++++⟩,4194304}(v, ..., v₁₂₃₄₅₆₇₈₉₀abcdefghijkl)



In [33]:

    
V = ℝ^22









    Out[33]:





⟨++++++++++++++++++++++⟩



In [34]:

    
Λ(V+V')









    Out[34]:





Grassmann.ExtendedAlgebra{⟨++++++++++++++++++++++----------------------⟩*,17592186044416}(v, ..., v₁₂₃₄₅₆₇₈₉₀abcdefghijklw¹²³⁴⁵⁶⁷⁸⁹⁰ABCDEFGHIJKL)



In [35]:

    
Λ(7) + Λ(7)'









    Out[35]:





Grassmann.SparseAlgebra{⟨+++++++-------⟩*,16384}(v, ..., v₁₂₃₄₅₆₇w¹²³⁴⁵⁶⁷)



In [36]:

    
Λ.V3









    Out[36]:





Grassmann.Algebra{⟨+++⟩,8}(v, v₁, v₂, v₃, v₁₂, v₁₃, v₂₃, v₁₂₃)



In [37]:

    
Λ.C3









    Out[37]:





Grassmann.Algebra{⟨---⟩',8}(w, w¹, w², w³, w¹², w¹³, w²³, w¹²³)



In [38]:

    
mixedbasis"2"









    Out[38]:





(⟨++--⟩*, v, v₁, v₂, w¹, w², v₁₂, v₁w¹, v₁w², v₂w¹, v₂w², w¹², v₁₂w¹, v₁₂w², v₁w¹², v₂w¹², v₁₂w¹²)



In [39]:

    
w1+2w2









    Out[39]:





1w¹ + 2w²



In [40]:

    
ans(v1+v2)









    Out[40]:





3v



In [41]:

    
ℒ = (v1+2v2)∧(3w1+4w2)









    Out[41]:





0v₁₂ + 3v₁w¹ + 4v₁w² + 6v₂w¹ + 8v₂w² + 0w¹²



In [42]:

    
ℒ(v1+v2)









    Out[42]:





7v₁ + 14v₂ + 0w¹ + 0w²



In [43]:

    
L = [1,2] * [3,4]'; L * [1,1]









    Out[43]:





2-element Array{Int64,1}:
  7
 14



In [44]:

    
generator = [8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0;
       2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0;
       2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0;
       2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0;
       2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0;
       2 0 2 0 2 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0 0;
       2 0 0 2 2 2 0 0 2 0 2 0 0 0 0 0 2 0 2 0 0 0 0 0;
       2 2 0 0 2 0 2 0 2 0 0 2 0 0 0 0 2 0 0 2 0 0 0 0;
       0 2 2 2 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0;
       0 0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0;
       0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0;
       -3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]









    Out[44]:





24×24 Array{Int64,2}:
  8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  0  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  0  0  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  0  0  0  0  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  0  0  0  0  0  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  0  0  0  0  0  0  0  4  0  0  0  0  0  0  0  0  0  0  0  0  0
  2  2  2  2  0  0  0  0  2  2  2  2  0  0  0  0  0  0  0  0  0  0  0  0
  4  0  0  0  0  0  0  0  0  0  0  0  4  0  0  0  0  0  0  0  0  0  0  0
  2  2  0  0  2  2  0  0  2  2  0  0  2  2  0  0  0  0  0  0  0  0  0  0
  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  0  0  0  0  0  0  0  0
  2  0  0  2  2  0  0  2  2  0  0  2  2  0  0  2  0  0  0  0  0  0  0  0
  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  4  0  0  0  0  0  0  0
  2  0  2  0  2  0  0  2  2  2  0  0  0  0  0  0  2  2  0  0  0  0  0  0
  2  0  0  2  2  2  0  0  2  0  2  0  0  0  0  0  2  0  2  0  0  0  0  0
  2  2  0  0  2  0  2  0  2  0  0  2  0  0  0  0  2  0  0  2  0  0  0  0
  0  2  2  2  2  0  0  0  2  0  0  0  2  0  0  0  2  0  0  0  2  0  0  0
  0  0  0  0  0  0  0  0  2  2  0  0  2  2  0  0  2  2  0  0  2  2  0  0
  0  0  0  0  0  0  0  0  2  0  2  0  2  0  2  0  2  0  2  0  2  0  2  0
 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1



In [45]:

    
const E24,W24 = Λ(24), ℝ^24+(ℝ^24)';



In [46]:

    
const Leech = SBlade{Float64,W24}(generator./sqrt(8));



In [47]:

    
typeof(Leech)









    Out[47]:





SBlade{Float64,⟨++++++++++++++++++++++++------------------------⟩*,2,1128}



In [48]:

    
ndims(vectorspace(Leech))









    Out[48]:





48



In [49]:

    
Leech(E24.v1)









    Out[49]:





2.82842712474619v₁ + 0.0v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.0w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN



In [50]:

    
Leech(E24.v2)









    Out[50]:





1.414213562373095v₁ + 1.414213562373095v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.0w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN



In [51]:

    
Leech(E24.v3)









    Out[51]:





1.414213562373095v₁ + 0.0v₂ + 0.0v₃ + 1.414213562373095v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 1.414213562373095w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN



In [52]:

    
Leech(E24.v1 + 2*E24.v2)









    Out[52]:





5.65685424949238v₁ + 2.82842712474619v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.0w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN



In [53]:

    
ans⋅ans









    Out[53]:





39.99999999999999v



In [54]:

    
Leech(E24.v2 + E24.v5)









    Out[54]:





2.82842712474619v₁ + 1.414213562373095v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 1.414213562373095va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.7071067811865475w² + 1.414213562373095w³ + 1.414213562373095w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN



In [55]:

    
ans⋅ans









    Out[55]:





7.499999999999998v



In [56]:

    
basis"2"









    Out[56]:





(⟨++⟩, v, v₁, v₂, v₁₂)



In [57]:

    
(:a*v1 + :b*v2) ⋅ (:c*v1 + :d*v2)









    Out[57]:





(a * c + b * d)v



In [58]:

    
(:a*v1 + :b*v2) ∧ (:c*v1 + :d*v2)









    Out[58]:





(a * d - b * c)v₁₂



In [59]:

    
(:a*v1 + :b*v2) * (:c*v1 + :d*v2)









    Out[59]:





a * c + b * d + (a * d - b * c)v₁₂



In [60]:

    
basis"4"









    Out[60]:





(⟨++++⟩, v, v₁, v₂, v₃, v₄, v₁₂, v₁₃, v₁₄, v₂₃, v₂₄, v₃₄, v₁₂₃, v₁₂₄, v₁₃₄, v₂₃₄, v₁₂₃₄)



In [61]:

    
P,Q = :px*v1 + :py*v2 + :pz* v3 + v4, :qx*v1 + :qy*v2 + :qz*v3 + v4









    Out[61]:





(pxv₁ + pyv₂ + pzv₃ + 1.0v₄, qxv₁ + qyv₂ + qzv₃ + 1.0v₄)



In [62]:

    
P∧Q









    Out[62]:





(px * qy - py * qx)v₁₂ + (px * qz - pz * qx)v₁₃ + (px - qx)v₁₄ + (py * qz - pz * qy)v₂₃ + (py - qy)v₂₄ + (pz - qz)v₃₄



In [63]:

    
R = :rx*v1 + :ry*v2 + :rz*v3 + v4









    Out[63]:





rxv₁ + ryv₂ + rzv₃ + 1.0v₄



In [64]:

    
P∧Q∧R









    Out[64]:





0.0 + ((px * qy - py * qx) * rz - ((px * qz - pz * qx) * ry - (py * qz - pz * qy) * rx))v₁₂₃ + (((px * qy - py * qx) + (py - qy) * rx) - (px - qx) * ry)v₁₂₄ + (((px * qz - pz * qx) + (pz - qz) * rx) - (px - qx) * rz)v₁₃₄ + (((py * qz - pz * qy) + (pz - qz) * ry) - (py - qy) * rz)v₂₃₄



In [ ]: