Quadrant Magic

A discussion I had with Harvey Heinz which ultimately led to the definition of Isomorphic-like Magic Stars (magic stars based on properties of magic squares) several sub magic figures of a magic square where defined, since the focus of the discussion lie in the use of the squares as a basis for stars the features where defined centered around the four subcenters (see below) of the square, Harvey coined the name "Quadrant Magic" for these kind of features. The text below defines these kind of feature, and focusses on the four quadrant. It is quite possible to define the figures elswhere in the square.

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Quadrant Magic features (definition)

A Quadrant Magic feature of a square of order n is 4 figures of n numbers each around one of the 4 subcenters of a square summing to the squares magic number.

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Subcenters (defined)

Subcenters for squares of orders 4k+1 are most naturally defined by the coordinates
(k,k), (n-1-k,k), (k,n-1-k) and (n-1-k,n-1-k)
note that the quadrants here are 2k+1 by 2k+1 squares, which means that the central row and column elements are common to two quadrant (the squares center common to all the quadrants.
(Quadrant magicness was first defined in these types of squares)
for orders 4k - 1 subcenters might be defined as the centers of the quadrants above and below the central row and aside the central column, thus given by the coordinates
(k-1,k-1), (n-k,k-1), (k-1,n-k) and (n-k,n-k)
In order to define Quadrant Magic features, elements of the central row and column have to be worked into the figures (which makes the squares subcenter decentral to the feature)
(currently this is only defined, it is not yet tested for)
Even orders have no nutural subcenters for there quadrants, Quadrant Magic features might be definable for these orders in a somewhat different manner (future upload)

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Quadrant Magic features (orders 4k+1)

The following table defines the quadrant magic features along with there name and some remarks

NOTE: The table below show the original defined patterns, which lead to an extensive studies by Harvey results he obtained (with help of my program) can be found in his: Quadrant Magic also more on this on this site in future uploads. (see next section)

definitions of quadrant magic features (around the upper left subcenter) simular definitions around the other three subcenters
the "plusmagic" feature
				(0,k)					used by harvey to obtain complete magic Plus_Stars tested for by program
				(1,k)
				-----
	(k,0)	(k,1)	-----	(k,k)	-----	(k,2k-1)	((k,2k)
				-----
				(2k-1,k)
				(2k,k)
the "crosmagic" feature
	(0,0)						(0,2k)		not usable to obtain Isolike stars tested for by program
		(1,1)				(1,2k-1)
			....		....
				(k,k)
			....		....
		(2k-1,1)				(2k-1,2k-1)
	(2k,0)						(2k,2k)
the "diammagic" feature
				(0,k)					usable to obtain Diam_Stars tested for by program
			(1,k)		(1,k+1)
		....				....
	(k-1,1)						(k-1,2k)
(k,0)				(k,k)				(k,2k)
	(k+1,1)						(k+1,2k-1)
		....				....
			(2k-1,k)		(2k-1,k+1)
				(2k,k)
the "sringmagic" feature (l=k/2)
		(l,l)	(l,l+1)	......	(l,3l-1)	(l,3l)			not usable to obtain Iso_Stars tested for by program this ring is closed for orders 8m+1 for orders 8m+5 the central elements in the ring need to be left out
		(l+1,l)				(l+1,3l)
		----		(k,k)		----
		(3l-1,l)				(3l-1,3l)
		(3l,l)	(3l,l+1)	......	(3l,3l-1)	(3l,3l)
the "lringmagic" feature
		(0,0)	(0,2)	......	(0,2k-2)	(0,2k)			not usable to obtain Iso_Stars tested for by program the lring is twice an sring
		(2,0)				(2,2k)
		----		(k,k)		----
		(2k-2,0)				(2k-2,2k)
		(2k,0)	(2k,2)	......	(2k,2k-2)	(2k,2k)
the "tcrosmagic" feature (l=k/2)
		(l,l)	(l,l+1)		(l,3l-1)	(l,3l)			NOT tested for by program (this is defined for order 17(?)) (definable for orders 4 m2 + 1)
		(l+1,l)	(l+1,l+1)		(l+1,3l-1)	(l+1,3l)
				(k,k)
		(3l-1,l)	(3l-1,l+1)		(3l-1,3l-1)	(3l-1,3l)
		(3l,l)	(3l,l+1)		(3l,3l-1)	(3l,3l)

note that the defined features above are tested around all the four subcenters and when all four test ok the square is said to be (thus-) magic.
note also that for the smallest of these orders (order 5) diam-magic and plus-magic are the same, also lring and cros refer to the same numbers, while sring and tcros is non-existent.
preparing this page I noticed that many other patterns might be definable for a given order It is far too much to list all posible patterns (but the above list might grow in future uploads.