Generalized Magic Object  

General framework wherein all discussed item can be put ^{m}GMagic Object {f(x_{i})g()} m: The order of the object with: G = IC, IR, IQ, IZ, IN Object: Square, Cube, Tesseract .... f(): number generating function g(): 'magic condition' The term "GMagic Object" can in fact take any form, as well s the funvtions f() and g() 

Magic Vector Space 
^{m}IRMagic Hypercube {"no restrictions""magic condition"} The magic condition posed on a square of real numbers viewed as vectors in a vector space 

pMultiMagic Hypercube _{p}^{n}H_{m}  ^{m}INMagic Hypercube {H[_{j}i] ε {1..m^{n}} '_{i=0}∑^{m} (H[_{j}i])^{k}' = ^{k}t_{n}}  
Magic Vector Space ^{3}IRMagic_Square  
Viewing the order 3 magic squares as a vector space is quite interesting using matrices as a vector space was new to me, so the reader more familiar with this is invited to suggest other things 

Base  Like any vector space the space has a base the base below is normalised to 1.0 as the magic sum  
B_{0} 1.0 0.0 0.0 2/3 1/3 4/3 2/3 2/3 1/3 
B_{1} 0.0 1.0 0.0 1/3 1/3 1/3 2/3 1/3 2/3 
B_{2} 0.0 0.0 1.0 4/3 1/3 2/3 1/3 2/3 2/3 

Note the transposed matrices of this base fall in the space B_{0}^{t} = (1.0,2/3,2/3) B_{1}^{t} = (0.0,1/3,2/3) B_{2}^{t} = (0.0,4/3,1/3) 

inner product V . W 
A vector space define an innerproduct which is a number note: that the below is defined on the matrices 

V . W = V^{i,j}W_{i,j} = _{i,j=0}∑^{2} V_{i,j}W_{i,j}  
outer product Cross product V * W 
The crossproduct defines a third vector based on two others note: that the below is defined on the matrices 

(V * W)_{i,j} =
((V_{(i+1)%3,(j+1)%3}W_{(i+2)%3,(j+2)%3}V_{(i+2)%3,(j+1)%3}W_{(i+1)%3,(j+2)%3}) + (W_{(i+1)%3,(j+1)%3}V_{(i+2)%3,(j+2)%3}W_{(i+2)%3,(j+1)%3}V_{(i+1)%3,(j+2)%3})) / 2 

This crossproduct opposed to it's IR^{3} counterpart is symmetric V * W = W * V Antisymmetric trials failed to remain within ^{3}IRMagicSquares since the seperate parts aren't. 

Magic Module ^{3}IZMagic_Square  
The Vector Space is defined on a Field, The Module has a simular definition on a Ring so whereas the above numbers are on Fields like IR the below uses the field IZ 

Base  The following base is appropriate for ^{3}IZmagic_Square  
IZB_{0} 0 1 1 1 0 1 1 1 0 
IZB_{1} 1 1 0 1 0 1 0 1 1 
IZB_{2} 1 1 1 1 1 1 1 1 1 

The following base would be appropriate for ^{3}INmagic_Square  
INB_{0} 1 2 0 0 1 2 2 0 1 
INB_{1} 0 2 1 2 1 0 1 0 2 
INB_{2} 1 1 1 1 1 1 1 1 1 

Note; of course these bases can be expressed in the basis for ^{3}IRMagic_Square  
inner product V . W 
A vector space define an innerproduct which is a number note: that the below is defined on the matrices 

V . W = V^{i,j}W_{i,j} = _{i,j=0}∑^{2} V_{i,j}W_{i,j}  
outer product Cross product V * W 
The crossproduct defines a third vector based on two others note: that the below is defined on the matrices 

(V * W)_{i,j} =
3 * ((V_{(i+1)%3,(j+1)%3}W_{(i+2)%3,(j+2)%3}V_{(i+2)%3,(j+1)%3}W_{(i+1)%3,(j+2)%3}) + (W_{(i+1)%3,(j+1)%3}V_{(i+2)%3,(j+2)%3}W_{(i+2)%3,(j+1)%3}V_{(i+1)%3,(j+2)%3})) 

Note this cross product is 6 times the one in ^{3}IRMagic_Square to avoid fractional numbers in the resulting matrix, since the expresions doesn't stay in IN this defines only a crossproduct in ^{3}IZMagic_Square 