The Hypercube Notation  

The (magic) hypercube is defined with dimension n and order m  
nPoint [_{k}i ; k = 0..n1 ; (i % m) = 0..m1 ] Range Rule: [_{k}i] 
the nPoint denote all the points in a Hypercube  
note: Range Rule: when no condition are given both k and i are assumed to run full range i can have any value but is modulated once onto the range 

the nPoint can also be used to denote the value at the position in this case dimension and order can be denoted as well: ^{n}[_{k}i]_{m} 

nVector <_{k}i ; k = 0..n1 ; (i % m) = 0..m1 > Range Rule: <_{k}i> 
the nVector denotes vectors in the Hypercube  
note: Range Rule: when no condition are given both k and i are assumed to run full range i can have any value but is modulated once onto the range 

amount restrictor #k=l 
restrict the amount of k's to l  
note: Range Rule: the unspecified values i are 0 so pe.: [_{k}i ; #k=2] = [_{j}0 _{k}i ; #j=n2, #k=2 ] : some plane in the hypercube <_{k}1 ; #k=2> = <_{j}0 _{k}1 ; #j=n2, #k=2 > : some diagonal direction 

Pathfinder Pf_{p} = <_{k}θ>; p = _{k=0}∑^{n1}(_{k}θ+1) 3^{k} θ ε {1,0,1} ; k = 0..n1 
a special kind of nVector whith only entries 1 0 and 1, thus used to traverse the hypercubes ragonals 

note: Pf_{(3n1)/2k} = Pf_{(3n1)/2+k}; k = 0..(3n1)/2 Pf_{(3n1)/2+3k+l=0∑k1θl3l = <n1p0,n1k1,n1lθl> ; l=0..k1, #k=1, p=k+1..n1 ; θl = 1,0,1 : an ragonal r = 1+l=0∑k1 θl } 

monagonals <_{k}1 ;#k=1 > 
The k'th monagonal  
<_{0}1>: row <_{1}1>: column <_{2}1>: pilar 

axes [_{h}0]<_{k}1 ;#k=1 > 
The kth axis  
[_{h}0]<_{0}1>: xaxis [_{h}0]<_{1}1>: yaxis [_{h}0]<_{2}1>: zaxis [_{h}0]<_{3}1>: waxis 

Aspectial Variants  
the notations here are a bit experimental at this stage, but I think acceptable!  
Aspectial Variant ^{n}H_{m}^{~R ^perm([0.n1])} = [_{h}0] { _{perm[k]}(<_{k}θ_{k} ; #k=1 >) ; k = 0..n1 }; θ_{k} ε {1,1}; R = _{k=0}Σ^{n1} ((θ_{k} == 1) ? 2^{k} : 0) 

Note: Reflection and coordinate permutation do not commute conceptually it might be easier to do the coordinate permutation first and reflect afterwards the formula on the righthandside feels like doing the reflections first and permute the coordinates thereafter, hence this is indicated also on the left by the order of the indicators. I currently trust a studies of the commutator will reveal simple operations, one simply ends up with an other member of the set of n! 2^{n} aspectial variants. 

Normalized position  Hypercube in normalized position  
^{n}H_{m} = [_{h}0] { _{k}(<_{k}1 ; #k=1 >) ; k = 0..n1 }; [_{h}0] = min([_{k}1 ;#k=0..n]) ; [_{k}1 ; #k=1] < [_{k+1}1 ; #k=1] ; k=0..n2 

Formulated is the 0position with the set of all monagonal directions in normal order. The 0position holds the minimum value of all the hypercubes corners The next monagonal position holds values ascending with ascending monagonal number 

Reflected position  Hypercube in reflected position  
^{n}H_{m}^{~R} = [_{h}0] { _{k}(<_{k}θ_{k} ; #k=1 >) ; k = 0..n1 }; θ_{k} ε {1,1} R = _{k=0}Σ^{n1} ((θ_{k} == 1) ? 2^{k} : 0) 

Reflected axes reverse their direction indicated by '1' (or 'm1') The directions reflected are bitwise added into a single reflection number R. 

Coordinate permuted position 
Hypercube in transposed position  
^{n}H_{m}^{^perm([0.n1])} = [_{h}0] { _{perm[k]}(<_{k}1 ;#k=1>) ; k = 0..n1 } 

coordinate permutation rearanges the directions amongst each other 