The Magic Encyclopedia ™

The Hypercube Notation
Work in Progress!!!!
(by Aale de Winkel)

Working on the Hypercube I developed a special notation to deal with the vast amount of positions, vectors and related items. I call this system
"the HyperCube Notation".
In this Article I'll summarize the things related to this notation, and will serve as a future reference page for this notation
Note in principle everything is here integer based.
Though this article is written with the hypercube in mind, it also applies to the hyperbeam with minor trivial modification.

The Hypercube Notation
The (magic) hypercube is defined with dimension n and order m

n-Point
[_ki ; k = 0..n-1 ; (i % m) = 0..m-1 ]
Range Rule: [_ki] the n-Point 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 n-Point can also be used to denote the value at the position
in this case dimension and order can be denoted as well: ⁿ[_ki]_m

n-Vector
<_ki ; k = 0..n-1 ; (i % m) = 0..m-1 >
Range Rule: <_ki> the n-Vector 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.:
[_ki ; #k=2] = [_j0 _ki ; #j=n-2, #k=2 ] : some plane in the hypercube
<_k1 ; #k=2> = <_j0 _k1 ; #j=n-2, #k=2 > : some diagonal direction

Pathfinder
Pf_p = <_kθ>; p = _k=0∑^n-1(_kθ+1) 3^k
θ ε {-1,0,1} ; k = 0..n-1 a special kind of n-Vector whith only entries -1 0 and 1,
thus used to traverse the hypercubes r-agonals
note: Pf_(3n-1)/2-k = -Pf_(3n-1)/2+k; k = 0..(3n-1)/2
Pf_{(3n-1)/2+3k+_l=0∑^k-1θ_l3^l =
<_n-1-p0,_n-1-k1,_n-1-lθ_l> ;

l=0..k-1, #k=1, p=k+1..n-1 ; θ_l = -1,0,1 :

an r-agonal r = 1+_l=0∑^k-1 |θ_l|}

monagonals
<_k1 ;#k=1 > The k'th monagonal
<₀1>: row
<₁1>: column
<₂1>: pilar

axes
[_h0]<_k1 ;#k=1 > The kth axis
[_h0]<₀1>: x-axis
[_h0]<₁1>: y-axis
[_h0]<₂1>: z-axis
[_h0]<₃1>: w-axis

Aspectial Variants
the notations here are a bit experimental at this stage, but I think acceptable!
Aspectial Variant
ⁿH_m^{~R ^perm([0.n-1])} = [_h0] { _perm[k](<_kθ_k ; #k=1 >) ; k = 0..n-1 };
θ_k ε {-1,1}; R = _k=0Σ^n-1 ((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 right-hand-side 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! 2ⁿ aspectial variants.
Normalized position Hypercube in normalized position
ⁿH_m = [_h0] { _k(<_k1 ; #k=1 >) ; k = 0..n-1 };
[_h0] = min([_k-1 ;#k=0..n]) ;
[_k1 ; #k=1] < [_k+11 ; #k=1] ; k=0..n-2
Formulated is the 0-position with the set of all monagonal directions in normal order.
The 0-position 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
ⁿH_m^~R = [_h0] { _k(<_kθ_k ; #k=1 >) ; k = 0..n-1 };
θ_k ε {-1,1} R = _k=0Σ^n-1 ((θ_k == -1) ? 2^k : 0)
Reflected axes reverse their direction indicated by '-1' (or 'm-1')
The directions reflected are bitwise added into a single reflection number R.

Coordinate permuted
position Hypercube in transposed position
ⁿH_m^{^perm([0.n-1])} =
[_h0] { _perm[k](<_k1 ;#k=1>) ; k = 0..n-1 }
coordinate permutation rearanges the directions amongst each other

The Hypercube Notation
The (magic) hypercube is defined with dimension n and order m
n-Point [_ki ; k = 0..n-1 ; (i % m) = 0..m-1 ] Range Rule: [_ki]	the n-Point 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 n-Point can also be used to denote the value at the position in this case dimension and order can be denoted as well: ⁿ[_ki]_m
n-Vector <_ki ; k = 0..n-1 ; (i % m) = 0..m-1 > Range Rule: <_ki>	the n-Vector 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
amount restrictor #k=l	note: Range Rule: the unspecified values i are 0 so pe.: [_ki ; #k=2] = [_j0 _ki ; #j=n-2, #k=2 ] : some plane in the hypercube <_k1 ; #k=2> = <_j0 _k1 ; #j=n-2, #k=2 > : some diagonal direction
Pathfinder Pf_p = <_kθ>; p = _k=0∑^n-1(_kθ+1) 3^k θ ε {-1,0,1} ; k = 0..n-1	a special kind of n-Vector whith only entries -1 0 and 1, thus used to traverse the hypercubes r-agonals
	note: Pf_(3n-1)/2-k = -Pf_(3n-1)/2+k; k = 0..(3n-1)/2 Pf_{(3n-1)/2+3k+_l=0∑^k-1θ_l3^l = <_n-1-p0,_n-1-k1,_n-1-lθ_l> ; l=0..k-1, #k=1, p=k+1..n-1 ; θ_l = -1,0,1 : an r-agonal r = 1+_l=0∑^k-1 \|θ_l\|}
monagonals <_k1 ;#k=1 >	The k'th monagonal
monagonals <_k1 ;#k=1 >	<₀1>: row <₁1>: column <₂1>: pilar
axes [_h0]<_k1 ;#k=1 >	The kth axis
axes [_h0]<_k1 ;#k=1 >	[_h0]<₀1>: x-axis [_h0]<₁1>: y-axis [_h0]<₂1>: z-axis [_h0]<₃1>: w-axis
Aspectial Variants
the notations here are a bit experimental at this stage, but I think acceptable!
Aspectial Variant ⁿH_m^{~R ^perm([0.n-1])} = [_h0] { _perm[k](<_kθ_k ; #k=1 >) ; k = 0..n-1 }; θ_k ε {-1,1}; R = _k=0Σ^n-1 ((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 right-hand-side 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! 2ⁿ aspectial variants.
Normalized position	Hypercube in normalized position
	ⁿH_m = [_h0] { _k(<_k1 ; #k=1 >) ; k = 0..n-1 }; [_h0] = min([_k-1 ;#k=0..n]) ; [_k1 ; #k=1] < [_k+11 ; #k=1] ; k=0..n-2
	Formulated is the 0-position with the set of all monagonal directions in normal order. The 0-position 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
	ⁿH_m^~R = [_h0] { _k(<_kθ_k ; #k=1 >) ; k = 0..n-1 }; θ_k ε {-1,1} R = _k=0Σ^n-1 ((θ_k == -1) ? 2^k : 0)
	Reflected axes reverse their direction indicated by '-1' (or 'm-1') The directions reflected are bitwise added into a single reflection number R.
Coordinate permuted position	Hypercube in transposed position
	ⁿH_m^{^perm([0.n-1])} = [_h0] { _perm[k](<_k1 ;#k=1>) ; k = 0..n-1 }
	coordinate permutation rearanges the directions amongst each other