# The Magic Square 3x3 and 9x9 matrixes X3 up to Infinity

“If you only knew the magnificence of the 3, 6 and 9, then you would have the key to the universe.”

― **Nikola Tesla**

“If you only knew the magnificence of the 3, 6 and 9, then you would have the key to the universe.”

― **Nikola Tesla**

Creating a base 3 magic square with equivalency in rows, columns, diagonals, and area.

A simple geometric pattern is completely unlimited in size and application.

A simple example, the 3x3 matrix is perhaps a very old, important, and powerful pattern hidden in plain sight. The numbers themselves reveal the relative position and sequence populating the matrix. These positions indicate the numbers within a 3x3 matrix, the position of a 3x3 matrix within a 9x9 matrix, the position of a 9x9 matrix within a 27x27 matrix continuing onward.

The 9x9 matrix . Composed of 9 - 3x3 matrix. Following the simple pattern in the 3x3 the 9x9 can be populated many different ways.

Just a random sampling of other fill patterns used in a 9x9 matrix.

The 27x27 composed of 9 - 9x9's with 9 - 3x3's each occupying 9 positions.

- 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.

This 9x9x9 Magic Cube - 9 Magic squares of 1 to 81 stacked to produce 9 columns of equivalent sums of 369. Only 1 layer in the cube can have equivalent diagonals. On this one I arranged it to be Layer 5.

The average value of the squares surrounding the center square is the value of the center square.

An Excel spreadsheet with this Magic cube is located in the down load section.

The resulting fill in the 3x3's are incremented x9 in the 1-3, 4-6, and 7-9 positions.

The arrangement of the numbers within the patterns for each different size Magic Squares are completely different. For example the numerical arrangement of the numbers within a 3x3 is completely different from a 4x4 or 5x5 or 6x6 etc..

In this 3x3 in a 4x4 in a 5x5 60x60 Magic Square containing 3 different magic square patterns of 3 different sizes combined to produce equal sums in rows, columns, and diagonals. The average value of the squares surrounding the center are also equal, even while these completely different patterns are mirrored rotated and inverted.

The average value surrounding the center is equal to the average value of the rows or columns. Notice the center square of each 3x3 is the average value of that square. In the 12x12 which is composed of 4 - 3x3's the average of the 4 values in the center of the 12x12 is the average of value of the 12x12. In the average of the squares around the center, some of the averaged squares around the center run through the center or inside edges of the 3x3 squares. It's not just whole squares , the values within the squares are positioned precisely by the patterns to produce equal averages of the squares around the center. Below is an Excel Worksheet containing the information shown.

This Magic Cube was populated sequentially from 1 to 729 staring with 1 on layer 1 and the 2 on layer 2 etc. using a 9x9 Magic Square . This is just another variation of how the pattern may be applied. The fundamental pattern is a Basic Magic Square 3x3 pattern applied across the 9 layers. Each layer is shifted by 1 position permitting the columns to sum equally. The layers were configured to place the value 365 at the center of the middle layer5 . The diagonals on Layer 5 are equal to sums of the rows and columns of layer 5. The average value of the squares around the center is the center square value 365. The sums of the layers are offset by 9 because there are 9 layers.

The repetition will start at a single value and increase up to a maximum value and then repeat the incremented numbers a fixed number of times until it begins descending to a single value again.

The scaler determines the maximum minimum and incremental value within the scaled Magic Square.

This is typical of how the Scaling works and its affects.

Scaling doesn't affect the balance of the rows and columns sums nor the average about the center.

This scaling is applied to the each cell of the final nested Magic Square. If the final Magic square is and 18x18 in a 9x9 then the scaling is applied to the values in the 18x18 as well as the 9x9.

This 9x9x9 Magic Cube is 9 Magic Squares of 1 to 81 stacked to produce a 9x9x9 Magic Cube with Columns , Rows , and Diagonals having Sum values of 369. Layer 5 has the diagonals with sums of 369 also. The average value of the cubic squares surrounding the center square is the value of the center square. Magic Squares and Magic Cubes do not have a maximum size limitations.

A 6561x6561 Magic Square at 1 square inch per number the dimensional size is greater than 1-1/2 football fields square with 1,625,702,400 different solutions. That is for a x9 type fill and the number is tripled to 4,877,107,200 solutions when you include 2 methods for a x3 type fill.

The 6561 is an 8th order matrix (3 raised to the 8th power). The number of permutations for 8 using 8 is 40,320 combinations. Since the row and column positions can be cojoined independently while strictly adhering to the simple pattern for each, there are 40,320 combinations for the rows and 40,320 combinations for columns which render 1,625,702,400 different patterns (solutions)in a x9 type fill.

This is a 60x60 Magic Square comprised of a 3x3 in a 4x4 in a 5x5 Magic Square. Keep in mind the 3x3 and the 4x4 and the 5x5 each have a different pattern. These patterns combine with perfect precision even when the patterns are rotated or transposed and/or the values within the patterns are scaled, the row , column, and diagonal sums are equal and the average value of the squares around the center are equal as well. It is easy to assume the sums of the rows ,columns and diagonals are equal, but the average around the center seems an absolute wonder.

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