Examples including Digital Roots

This Pattern works just as well with decimals. The value was incremented by 0.271003 yielding these results. ( 0.271003 = 100/369)

• Not just Integers works with decimals.
• Columns Rows and Diagonals add up to 100.
• Averages around the center square.
• Center Square is 11.111 reduces to 5
• Top value 21.951 reduces to 9.
• All of the above holds true with the same pattern and different fill sequences.

This Pattern is another application of the single fill pattern on a 27x27. 

• Rows Columns Diagonals all equal sums.
• The number 1 is at the bottom row center , 729 at the top row center.
• The average values of the square around the center square are equal.
• It is interesting to observe the patterning of the of the digital roots of the numbers as well.

• Even when the offset fill is applied the pattern still effectively balances the matrix demonstrating just how fractal this pattern is.

• The core matrix is the individual matrix which can be a 3x3 , 9x9, 27x27 etc. using any of the fill patterns strictly adhering to the central pattern.  Copies of the core matrix when aligned comprise a layer. 
• The core matrix above is a 9x9 matrix. The 9x9 matrix is positioned in 9 layers on the Z Axis. 
• A 3x3 matrix will have 3 layers on the Z-Axis, a 9x9 has 9 layers , an 81x81 has 81 layers  etc.
• Each successive layer must be transposed by 1 position on the x- axis or on the y axis.
• The shifting of the layers is always performed  in the same direction and each layer is shifted progressively by 1 position per layer.
• There is no limit to the maximum size of the x, y, or z axis. (The z-axis is based on the core matrix size, the core matrix size is not limited.)
• The core matrices are aligned side by side and end to end and extend  indefinitely. The area or size of the core matrix can be focused anywhere,  the sums in the x, y and z axis in that area of focus will  be equivalent just like the Magic Square.
• If the outside edges of a column or row were brought together (producing a round tube ) the sums on the X, Y , and Z axis surrounding that connecting line would be seamless as no line would exist there at all. 
• An Excel Spreadsheet with a sample 9x9x9 Magic cube is located in the Download section.

Not only can the magic squares be nested around themselves , they can be inserted within a frame work of a magic square of a different order. The 5x5's nested within a 4x4 frame work as an example. Not only do the columns, rows and diagonals sum equally , but the average about  the center  also applies.