I but recently got to understand what <Gil Lamb> calls
Dynamic Numbering. His spreadsheets merely arrowed one number toward another showing me the underlying "latin-structures".
What he really wanted to show is where the succesive numbers of the "Generating Squares" are placed, changing "Generating
Squares" and placing the succesive numbers the same way another square is obtained. After I got this picture clear it dawned
me that this was the same thing as using a given magic square as an index into the "Generating Square", a view which makes
"Dynamic Numbering" real easy to implement, since one merely needs to move the "Magic Squares" to the "analitic numberrange"
and transform each number into coordinates onto the Generating square.
The below formalises this point of view. Currently concentrated around the square, but the method generalises obviously onto
hypercubes of any dimension. After implementation of this I'll augment this (and related) texts accordingly
Dynamic Numbering
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Generating Squares
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(i=0∏n piNqi)[x,y] =
i=0∑n k=0∏i pkqk {
([x % k=0∏i pk] \ k=0∏i-1 pk) +
pn ([y % k=0∏i qk] \ k=0∏i-1 qk) };
i=0∏n pi = i=0∏n qi = m
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Examples up till 4 rectangles:
(pNq * rNs)[x,y] = (x%r) + r (y%s) + rs [(x\r) + p (y\s)]
(tNu * pNq * rNs)[x,y] =
(x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] + pqrs [(x\pr) + t (y\qs)]
(vNw * tNu * pNq * rNs)[x,y] =
(x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] +
pqrs [((x%prt)\pr) + t ((y%qsu)\qs)] + pqrstu [(x\prt + v (y\qsu)]
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Coordinate Squares
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MS[x,y] => [ MS[x,y] % m, MS[x,y] \ m ].
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Generated Squares
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S[x,y] => GS[ MS[x,y] % m, MS[x,y] \ m ].
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With the above point of view "Dynamic Numbering" moves into the realm of the more basic isomorphisms on a magic square.
As another method which uses "Generating Squares" is my own Pan-Transform for the double even orders, as it turns out:
Pan(GS) = GS(Pan(Nm)) meaning that the pan-transform applied on a generating square gives the same result as
applying "Dynamic Numbering" with the same "Generating Square" on the pan-transformed "Normal Square" Nm.
It is my current understanding that any construction method for a (Magic) square one obtains say Con(Nm),
applying dynamic numbering to this square (with GS) is the same as doing the construction itself on GS ie:
{GS} Con(Nm) = Con(GS)
Application of "Dynamic Numbering" however is not limited to using only "Generating Squares" but can be done with any square.
Doing this for order 8 on my "Pan(N8)" introduced me to a 26 {compact complete} order 8 squares
{GS}k Pan(N8) of which I only reckognized the 10 (k=1) squares (note {GS}k S, means the
application of Dynamic Numbering k times with the same GS). A lot more squares can be obtained by application of different
GS's one after another
Dynamic Numbering can of course also be applied to Generating Squares, these squares seem to share the {pandiagonal}
feature, but don't have there monagonal sums in incremental order.
Dynamic Numbering forms a powerfull tool combining a lot of squares to each other. The three {compact complete} square
families I already knew as:
Pan(N4), Pan(N2 N2) and Pan(2N1 2N4)
with dynamic numbering also can be called:
Pan(N4), {N2 N2}Pan(N4) and
{2N1 2N4} Pan(N4)
which dispenses with 2 counts of performing the construction, Dynamic Numbering merely changes the number source.