ragonals  

In order to define a line in hyperspace one needs to define a point and a directional vector. Effectively the 'r' is already clear when the directional vector is defined 

ragonal  a ragonal is the line between the corners of the rdimensional hypercube  
a hypercube of ndimensions has subhypercubes with rdimensions from r = 2 (square) till r = n (the hypercube itself. Consistently the monagonal (r = 1) is defined as a line parallel to the hypercube axes (regularly denoted by terms as "row", "column", "pilar" etc. The monagonal makes a convenient shorthand and is used througout this site. 

pan ragonal  the broken lines parallel to the ragonal  
althhough tecnically the monagonals are all whole (or "unbroken") the notion of a pan 1agonal or pan monagnal if often usefull, the regular (ie unbroken) ragonals are considered to be included when pan ragonals are concidered, if not one ought to explicetely state that fact. 

The Lexicon states that there are 2^{r1} (^{n}_{r}) m^{n1}
ragonal lines within a hypercube. the explanation of which I provide below 

Summing lines amount: 2^{r1} (^{n}_{r}) m^{n1} 
[ _{j}0 _{k}p _{l}q ] < _{j}1 _{k}θ _{l}0 > ;
k > j ; θ ε {1,1} ; p,q ε [0,..,m1] amounts: 1 j ; r1 k's ; nr l's 

For an ragonal pathfinder vector r positions need to be randomly selected from the possible n which can be done in (^{n}_{r}) ways, to avoid double counting the first of these selected can be set to the value 1 while the other r1 of these selected can be either 1 or +1 the other nr positions of the pathfinder vector ought to be set to 0. This argument defines the above denoted pathfinder vector and explains the factor 2^{r1} (^{n}_{r}) The summation line can start anywhere within the hypercube, so it is sufficient to start with the positions where the j'th coordinate is 0, all other positions need to be run through fully to catch all (broken) ragonal path's which needs to be summed over to obtain the hypercubes mainqualification. This explains the factor m^{n1} 

Pathfinder number Pf_{p} 
Pf_{p} with: p = _{k=0}∑^{n1} (_{k}i + 1) 3^{k} <==> <_{k}i> ; i ε {1,0,1}  
The correspondence above associates a number with any pathfinder vector in an isomorphic manner (ie it is a 1 on 1 correspondence). Given a pathfinder number it is easy to write that number in base 3 and subtract 1 from all the thus obtained digits. Counting all 1 and +1 digits one obtains the r This fact can be easily utilized in itterative procedures where the Pf_{p} can run from 0 to 3^{n}1 (the possible values) 

The procedure outlined by the above Pf_{p} note all ragonal pathfinders can be found The "summing lines" remark that half of these (those with +1 as the first nonzero entry) are enough to catch all possible lines to be summed over, it also suggest a starting point for the summing line. All other position coordinates need to be fully iterated to catch all (broken) ragonals. Values 0 and m1 for these entries might be enough to catch all nonbroken ragonals (with the exception of 1agonals where all values (0 .. m1) are needed) 

Subhypercube ^{s}S_{o}^{(P,V0..Vs1)} 
^{s}S_{o}^{(P,V0..Vs1)} with:
P = [_{k}i] ; V_{q} = <_{l}θ>_{q} ; q = 0..s1 } ;
l_{q} != l_{q'} ; s the subhypercubes dimension ; o its order 

To define a sdimensional subhypercube one needs to define the (0)position and s mutually exclusive pathfinders, this means that the nonzero elements of the s ragonal vectors do not share positions (meaning of the inequality) The subhypercube 1agonals are along the defining vectors, while the other vectors need to be itterated through each of the subhypercube orders possibilities, the hypercubes sagonals are given by the sums of all vectors (with only +1 or 1 multiplicaton factors) It is quite possible to include steps in the various directions, the subhypercubes order might even be higher then the hypercube, allowing numbers to be used more the once in a given sum (no experience yet with this) simularly one can of course define subhyperbeams (using various orders) 
samples within cube (so n == 3)  

Pf_{0}: 0 = 000_{3} <==> <1,1,1> Pf_{1}: 1 = 001_{3} <==> <1,1, 0> Pf_{2}: 2 = 002_{3} <==> <1,1, 1> Pf_{3}: 3 = 010_{3} <==> <1, 0,1> Pf_{4}: 4 = 011_{3} <==> <1, 0, 0> Pf_{5}: 5 = 012_{3} <==> <1, 0, 1> Pf_{6}: 6 = 020_{3} <==> <1, 1,1> Pf_{7}: 7 = 021_{3} <==> <1, 1, 0> Pf_{8}: 8 = 022_{3} <==> <1, 1, 1> 
Pf_{9}: 9 = 100_{3} <==> < 0,1,1> Pf_{10}: 10 = 101_{3} <==> < 0,1, 0> Pf_{11}: 11 = 102_{3} <==> < 0,1, 1> Pf_{12}: 12 = 110_{3} <==> < 0, 0,1> Pf_{13}: 13 = 111_{3} <==> < 0, 0, 0> Pf_{14}: 14 = 112_{3} <==> < 0, 0, 1> Pf_{15}: 15 = 120_{3} <==> < 0, 1,1> Pf_{16}: 16 = 121_{3} <==> < 0, 1, 0> Pf_{17}: 17 = 122_{3} <==> < 0, 1, 1> 
Pf_{18}: 18 = 200_{3} <==> < 1,1,1> Pf_{19}: 19 = 201_{3} <==> < 1,1, 0> Pf_{20}: 20 = 202_{3} <==> < 1,1, 1> Pf_{21}: 21 = 210_{3} <==> < 1, 0,1> Pf_{22}: 22 = 211_{3} <==> < 1, 0, 0> Pf_{23}: 23 = 212_{3} <==> < 1, 0, 1> Pf_{24}: 24 = 220_{3} <==> < 1, 1,1> Pf_{25}: 25 = 221_{3} <==> < 1, 1, 0> Pf_{26}: 26 = 222_{3} <==> < 1, 1, 1> 
^{3}C_{m}^{([0,0,0]<1,0,0><0,1,0><0,0,1>)} : the cube itself ^{2}S_{m}^{([0,0,0]<1,0,0><0,1,0>)} : the cubes xy square (front) ^{2}S_{m}^{([0,0,0]<1,0,0><0,0,1>)} : the cubes xz square (top) ^{2}S_{m}^{([0,0,0]<0,1,0><0,0,1>)} : the cubes yz square (left) ^{2}S_{m}^{([0,0,m1]<1,0,0><0,1,0>)} : the cubes xy square (back) ^{2}S_{m}^{([0,m1,0]<1,0,0><0,0,1>)} : the cubes xz square (bottom) ^{2}S_{m}^{([m1,0,0]<0,1,0><0,0,1>)} : the cubes yz square (right) ^{2}S_{m}^{([0,0,0]<1,0,0><0,1,1>)} : the cubes xyz oblique square ^{2}S_{m}^{([0,0,0]<0,1,0><1,0,1>)} : the cubes yxz oblique square ^{2}S_{m}^{([0,0,0]<0,0,1><1,1,0>)} : the cubes zxy oblique square ^{2}S_{m}^{([0,0,m1]<1,0,0><0,1,1>)} : the cubes xyz' oblique square ^{2}S_{m}^{([0,0,m1]<0,1,0><1,0,1>)} : the cubes yxz' oblique square ^{2}S_{m}^{([0,m1,0]<0,0,1><1,1,0>)} : the cubes zxy' oblique square (' used to indicate the use of the subdiagonal) 