Historical Notes on Generalized Magic Objects



In an email someone asked whether I knew the dimension of an order n magic square with real numbers inside. Since to me the concept of dimension is that of spatial extent and thus the dimension of a square would be 2, a cube 3, a tesseract 4 etc. However what he meant was the amount of base vectors which would span the space when viewed as a vector- space, with no limitations on the involved numbers



however since these kind of vector spaces omit the basic requirement of regular magic squares for the participating squares entries (the numbers) for being unique within a given square, this let me to the idea of putting in some number generating function, besides that one might also consider some variations on the 'magic-condition' (as one also has in regular magic squares, where one has 'semi-magic' (off diagonal sums) and 'semi magic' squares (block-wise of sums)) this leads to the general magic objects defined as:

nG-Magic Object {f()|g()}
n: The order of the object
with: G = IC, IR, IQ, IZ, IN
Object: Square, Cube, Tesseract ....
f(): number generating function
g(): 'magic condition'

See the item 3IR-Magic_Squares for examples on third order magic vector spaces,
and the item 'f() and g()' for remarks on the numbergenerating function f()
as well as on the magic condition g()!



Yet unclear how these objects will evolve, but it is I think worth investigating
If you want to participate in further investigation of the subject, one can do this through email, or create a page of your own (Email me the URL, I'll then have that URL in onto my pages)



for the moment my email contact (who wants to be anonymous) and I worked out the
3IR-Magic_Squares
as the third order 'matrix' vector space, some quite interesting quations hoever remain open for the moment, since I nor mentioned email contact worked with 'matrix'-vector spaces



The regular magic squares can be defined in terms of the above as
nIN-Magic Squares {f(i) = i (i = 1..n2)|'sums' = magic number}
is the main item discusses in my Magic squares and cubes



The p MultiMagic squares can be defined in terms of the above as
nIN-Magic Squares {f(i) = ik (i = 1..n2; k = 1..p)|'sums' = magic number}
(the above defines p seperate magic squares)
in the above examples 'sums' means of course all sums horizontally, vertically,
and along both diagonals