The Magic Encyclopedia ™

Paths through the Hypercube
{note: investigative article}
(by Aale de Winkel)

This article deals with the various Paths through the Hypercube, and is merely there to define the notations to describe several types of paths, using the notations reached at in the bent-hyperagonal article.
Unless explicitely described the notation implicitely supposes a move to an adjecent position before path continuation
pe: m/2 {<1,1>,<1,-1>} = {m/2<1,1>,<1,0>,m/2<1,-1>}
Further the article presupposes modular space (or wrap around), so paths runs through the hypercubes borders onto the modular equivalent positions

Paths through the Hypercube
John R. Hendricks's "pathfinders" where defined to run through hyper-r-agonals, below defines a notation for
these and other definable paths through a n-dimensional hypercube, an n-Point can be used to fix the starting
point of a given path of any type, so this article is only concidering the path-form.

1-agonal m {<0_r1_s>} (r = 0.. m-1; r != s)

main n-agonal m {<1_r>} (r = 0.. m-1)

sub n-agnal m {<1_r-1_s>} (r = 0.. m-1; s = 0.. m-1; r != s)

bent hyper-n-agonal m/2 {<v_i; i = 0..n-1>,<w_i; i = 0..n-1>} ; with: <v> * <w> = <0>
both n-Vectors elements -1 and 1 only; henche these vectors are orthogonal

knight jump paths m {<v_r>} (r = 0.. m-1; v_r != 0)

bent knight jump paths m/2 {<v_i; i = 0..n-1>,<w_i; i = 0..n-1>}
both n-Vectors no 0 elements (vectors need not be orthogonal)

other paths {<v_r>_i} (r = 0.. m-1;)
(path supposed to be non-circular in less than m steps)
The number of vectors between the curly brackets is not yet defined (probably m), as suggested
above repeated vectors can be indicated by preapending the number of repetitions, this number
is multiplied by the "length-indicator" preappended before the curly brackets. (the mentioned
non-circular condition just avoids double counting the same cell element in the same path).

Paths through the Hypercube
John R. Hendricks's "pathfinders" where defined to run through hyper-r-agonals, below defines a notation for these and other definable paths through a n-dimensional hypercube, an n-Point can be used to fix the starting point of a given path of any type, so this article is only concidering the path-form.
1-agonal	m {<0_r1_s>} (r = 0.. m-1; r != s)
main n-agonal	m {<1_r>} (r = 0.. m-1)
sub n-agnal	m {<1_r-1_s>} (r = 0.. m-1; s = 0.. m-1; r != s)
bent hyper-n-agonal	m/2 {<v_i; i = 0..n-1>,<w_i; i = 0..n-1>} ; with: <v> * <w> = <0> both n-Vectors elements -1 and 1 only; henche these vectors are orthogonal
knight jump paths	m {<v_r>} (r = 0.. m-1; v_r != 0)
bent knight jump paths	m/2 {<v_i; i = 0..n-1>,<w_i; i = 0..n-1>} both n-Vectors no 0 elements (vectors need not be orthogonal)
other paths	{<v_r>_i} (r = 0.. m-1;) (path supposed to be non-circular in less than m steps)
other paths	The number of vectors between the curly brackets is not yet defined (probably m), as suggested above repeated vectors can be indicated by preapending the number of repetitions, this number is multiplied by the "length-indicator" preappended before the curly brackets. (the mentioned non-circular condition just avoids double counting the same cell element in the same path).