The Magic Encyclopedia ™

r-Agonals
{note: investigative article}
(by Aale de Winkel)

The general term of r-agonal for all the straight lines in a hypercube allows for the possibility to define matters in a more systematic way. The expression below defines in a single expression all r-agonals in any dimension, thereby illustrating the power of the notation I introduced. Also based upon the expression a countig argument is given for the amount of r-agonals within a n-dimensional hypercube
This author introduced the term 1-agonal to stand for the terms like row, column, pillar etc. (though reported recently <John R Hendricks> used the term as early as 1962). Others use terms like i-rows and orthogonals as general terms, 1-agonal (or monagonal) however is consistent with 2-agonal (or diagonal), 3-agonal (ie triagonal), and allows for more practical use in general statements.

r-agonals
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

r-agonal a r-agonal is the line between the corners of the r-dimensional hypercube
a hypercube of n-dimensions has subhypercubes with r-dimensions 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 r-agonal the broken lines parallel to the r-agonal
althhough tecnically the monagonals are all whole (or "unbroken") the notion of a
pan 1-agonal or pan monagnal if often usefull, the regular (ie unbroken) r-agonals
are considered to be included when pan r-agonals are concidered, if not one ought
to explicetely state that fact.

The Lexicon states that there are 2^r-1 (ⁿ_r) m^n-1 r-agonal lines within a hypercube.
the explanation of which I provide below

Summing lines
amount:
2^r-1 (ⁿ_r) m^n-1 [ _j0 _kp _lq ] < _j1 _kθ _l0 > ; k > j ; θ ε {-1,1} ; p,q ε [0,..,m-1]
amounts: 1 j ; r-1 k's ; n-r l's
For an r-agonal 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 r-1 of these selected can be either -1 or +1 the other
n-r positions of the pathfinder vector ought to be set to 0. This argument defines the above
denoted pathfinder vector and explains the factor 2^r-1 (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) r-agonal path's which needs to be summed over to obtain the hypercubes
main-qualification. This explains the factor m^n-1

Pathfinder number
Pf_p Pf_p with: p = _k=0∑^n-1 (_ki + 1) 3^k <==> <_ki> ; 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 3ⁿ-1 (the possible values)

The procedure outlined by the above Pf_p note all r-agonal 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-)
r-agonals. Values 0 and m-1 for these entries might be enough to catch all non-broken r-agonals
(with the exception of 1-agonals where all values (0 .. m-1) are needed)

Subhypercube
^sS_o^{(P,V₀..V_s-1)} ^sS_o^{(P,V₀..V_s-1)} with: P = [_ki] ; V_q = <_lθ>_q ; q = 0..s-1 } ; l_q != l_q' ;
s the subhypercubes dimension ; o its order
To define a s-dimensional subhypercube one needs to define the (0)-position
and s mutually exclusive pathfinders, this means that the non-zero elements
of the s r-agonal vectors do not share positions (meaning of the inequality)
The subhypercube 1-agonals are along the defining vectors, while the other
vectors need to be itterated through each of the subhypercube orders
possibilities, the hypercubes s-agonals 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)
Pf₀: 0 = 000₃ <==> <-1,-1,-1>
Pf₁: 1 = 001₃ <==> <-1,-1, 0>
Pf₂: 2 = 002₃ <==> <-1,-1, 1>
Pf₃: 3 = 010₃ <==> <-1, 0,-1>
Pf₄: 4 = 011₃ <==> <-1, 0, 0>
Pf₅: 5 = 012₃ <==> <-1, 0, 1>
Pf₆: 6 = 020₃ <==> <-1, 1,-1>
Pf₇: 7 = 021₃ <==> <-1, 1, 0>
Pf₈: 8 = 022₃ <==> <-1, 1, 1> Pf₉: 9 = 100₃ <==> < 0,-1,-1>
Pf₁₀: 10 = 101₃ <==> < 0,-1, 0>
Pf₁₁: 11 = 102₃ <==> < 0,-1, 1>
Pf₁₂: 12 = 110₃ <==> < 0, 0,-1>
Pf₁₃: 13 = 111₃ <==> < 0, 0, 0>
Pf₁₄: 14 = 112₃ <==> < 0, 0, 1>
Pf₁₅: 15 = 120₃ <==> < 0, 1,-1>
Pf₁₆: 16 = 121₃ <==> < 0, 1, 0>
Pf₁₇: 17 = 122₃ <==> < 0, 1, 1> Pf₁₈: 18 = 200₃ <==> < 1,-1,-1>
Pf₁₉: 19 = 201₃ <==> < 1,-1, 0>
Pf₂₀: 20 = 202₃ <==> < 1,-1, 1>
Pf₂₁: 21 = 210₃ <==> < 1, 0,-1>
Pf₂₂: 22 = 211₃ <==> < 1, 0, 0>
Pf₂₃: 23 = 212₃ <==> < 1, 0, 1>
Pf₂₄: 24 = 220₃ <==> < 1, 1,-1>
Pf₂₅: 25 = 221₃ <==> < 1, 1, 0>
Pf₂₆: 26 = 222₃ <==> < 1, 1, 1>
³C_m^{([0,0,0]<1,0,0><0,1,0><0,0,1>)} : the cube itself
²S_m^{([0,0,0]<1,0,0><0,1,0>)} : the cubes x-y square (front)
²S_m^{([0,0,0]<1,0,0><0,0,1>)} : the cubes x-z square (top)
²S_m^{([0,0,0]<0,1,0><0,0,1>)} : the cubes y-z square (left)
²S_m^{([0,0,m-1]<1,0,0><0,1,0>)} : the cubes x-y square (back)
²S_m^{([0,m-1,0]<1,0,0><0,0,1>)} : the cubes x-z square (bottom)
²S_m^{([m-1,0,0]<0,1,0><0,0,1>)} : the cubes y-z square (right)
²S_m^{([0,0,0]<1,0,0><0,1,1>)} : the cubes x-yz oblique square
²S_m^{([0,0,0]<0,1,0><1,0,1>)} : the cubes y-xz oblique square
²S_m^{([0,0,0]<0,0,1><1,1,0>)} : the cubes z-xy oblique square
²S_m^{([0,0,m-1]<1,0,0><0,1,-1>)} : the cubes x-yz' oblique square
²S_m^{([0,0,m-1]<0,1,0><1,0,-1>)} : the cubes y-xz' oblique square
²S_m^{([0,m-1,0]<0,0,1><1,-1,0>)} : the cubes z-xy' oblique square
(' used to indicate the use of the subdiagonal)

r-agonals
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
r-agonal	a r-agonal is the line between the corners of the r-dimensional hypercube
r-agonal	a hypercube of n-dimensions has subhypercubes with r-dimensions 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 r-agonal	the broken lines parallel to the r-agonal
pan r-agonal	althhough tecnically the monagonals are all whole (or "unbroken") the notion of a pan 1-agonal or pan monagnal if often usefull, the regular (ie unbroken) r-agonals are considered to be included when pan r-agonals are concidered, if not one ought to explicetely state that fact.
The Lexicon states that there are 2^r-1 (ⁿ_r) m^n-1 r-agonal lines within a hypercube. the explanation of which I provide below
Summing lines amount: 2^r-1 (ⁿ_r) m^n-1	[ _j0 _kp _lq ] < _j1 _kθ _l0 > ; k > j ; θ ε {-1,1} ; p,q ε [0,..,m-1] amounts: 1 j ; r-1 k's ; n-r l's
Summing lines amount: 2^r-1 (ⁿ_r) m^n-1	For an r-agonal 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 r-1 of these selected can be either -1 or +1 the other n-r positions of the pathfinder vector ought to be set to 0. This argument defines the above denoted pathfinder vector and explains the factor 2^r-1 (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) r-agonal path's which needs to be summed over to obtain the hypercubes main-qualification. This explains the factor m^n-1
Pathfinder number Pf_p	Pf_p with: p = _k=0∑^n-1 (_ki + 1) 3^k <==> <_ki> ; i ε {-1,0,1}
Pathfinder number Pf_p	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 3ⁿ-1 (the possible values)
The procedure outlined by the above Pf_p note all r-agonal 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-) r-agonals. Values 0 and m-1 for these entries might be enough to catch all non-broken r-agonals (with the exception of 1-agonals where all values (0 .. m-1) are needed)
Subhypercube ^sS_o^{(P,V₀..V_s-1)}	^sS_o^{(P,V₀..V_s-1)} with: P = [_ki] ; V_q = <_lθ>_q ; q = 0..s-1 } ; l_q != l_q' ; s the subhypercubes dimension ; o its order
Subhypercube ^sS_o^{(P,V₀..V_s-1)}	To define a s-dimensional subhypercube one needs to define the (0)-position and s mutually exclusive pathfinders, this means that the non-zero elements of the s r-agonal vectors do not share positions (meaning of the inequality) The subhypercube 1-agonals are along the defining vectors, while the other vectors need to be itterated through each of the subhypercube orders possibilities, the hypercubes s-agonals 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)
Pf₀: 0 = 000₃ <==> <-1,-1,-1> Pf₁: 1 = 001₃ <==> <-1,-1, 0> Pf₂: 2 = 002₃ <==> <-1,-1, 1> Pf₃: 3 = 010₃ <==> <-1, 0,-1> Pf₄: 4 = 011₃ <==> <-1, 0, 0> Pf₅: 5 = 012₃ <==> <-1, 0, 1> Pf₆: 6 = 020₃ <==> <-1, 1,-1> Pf₇: 7 = 021₃ <==> <-1, 1, 0> Pf₈: 8 = 022₃ <==> <-1, 1, 1>	Pf₉: 9 = 100₃ <==> < 0,-1,-1> Pf₁₀: 10 = 101₃ <==> < 0,-1, 0> Pf₁₁: 11 = 102₃ <==> < 0,-1, 1> Pf₁₂: 12 = 110₃ <==> < 0, 0,-1> Pf₁₃: 13 = 111₃ <==> < 0, 0, 0> Pf₁₄: 14 = 112₃ <==> < 0, 0, 1> Pf₁₅: 15 = 120₃ <==> < 0, 1,-1> Pf₁₆: 16 = 121₃ <==> < 0, 1, 0> Pf₁₇: 17 = 122₃ <==> < 0, 1, 1>	Pf₁₈: 18 = 200₃ <==> < 1,-1,-1> Pf₁₉: 19 = 201₃ <==> < 1,-1, 0> Pf₂₀: 20 = 202₃ <==> < 1,-1, 1> Pf₂₁: 21 = 210₃ <==> < 1, 0,-1> Pf₂₂: 22 = 211₃ <==> < 1, 0, 0> Pf₂₃: 23 = 212₃ <==> < 1, 0, 1> Pf₂₄: 24 = 220₃ <==> < 1, 1,-1> Pf₂₅: 25 = 221₃ <==> < 1, 1, 0> Pf₂₆: 26 = 222₃ <==> < 1, 1, 1>
³C_m^{([0,0,0]<1,0,0><0,1,0><0,0,1>)} : the cube itself ²S_m^{([0,0,0]<1,0,0><0,1,0>)} : the cubes x-y square (front) ²S_m^{([0,0,0]<1,0,0><0,0,1>)} : the cubes x-z square (top) ²S_m^{([0,0,0]<0,1,0><0,0,1>)} : the cubes y-z square (left) ²S_m^{([0,0,m-1]<1,0,0><0,1,0>)} : the cubes x-y square (back) ²S_m^{([0,m-1,0]<1,0,0><0,0,1>)} : the cubes x-z square (bottom) ²S_m^{([m-1,0,0]<0,1,0><0,0,1>)} : the cubes y-z square (right) ²S_m^{([0,0,0]<1,0,0><0,1,1>)} : the cubes x-yz oblique square ²S_m^{([0,0,0]<0,1,0><1,0,1>)} : the cubes y-xz oblique square ²S_m^{([0,0,0]<0,0,1><1,1,0>)} : the cubes z-xy oblique square ²S_m^{([0,0,m-1]<1,0,0><0,1,-1>)} : the cubes x-yz' oblique square ²S_m^{([0,0,m-1]<0,1,0><1,0,-1>)} : the cubes y-xz' oblique square ²S_m^{([0,m-1,0]<0,0,1><1,-1,0>)} : the cubes z-xy' oblique square (' used to indicate the use of the subdiagonal)