**THE TORUS: HOW A MAGIC SQUARE BECOMES A CIRCLE**

Magic squares in the Luo Shu format describe a torus. The definition of a magic square in the Luo Shu format can be found here: LINK

The 9x9 magic square in the Luo Shu format can be transcribed so all the numbers are reduced to their Pythagorean root number.

For instance, 37 = 3 + 7 = 10 = 1 + 0 = 1

This creates a new magic square using only the numbers one thru nine but still maintain some features of a Luo Shu magic square, i.e., the constant of the square is the center number times the size of the square or 5 x 9. And the total sum of all the numbers in the square is the size of the square, squared, times the center number or 81 x 5.

There are some interesting features to this new magic square:

1. The numbers in common to the two squares demonstrate how a magic square in the Luo Shu format is constructed. Follow the weave of numbers begining with the "1" below the center cell and one can see how the numbers wrap around the square in continuum.

2. The number five is the center number of the square as well as one of the major diagonals. The number five is the center number of the 3x3 Magic Square known as the Luo Shu.

3. Beginning in the upper left corner cell, the numerical pattern in each column, row, and one diagonal direction is: 1, 6, 2, 7, 3, 8, 4, 9, 5, 1 - similar to the He Tu pattern.

4. The magic square forms a three dimensional torus. That is, the numbers continue in order from one edge to the opposite edge.

5. The diagonal in one direction consists of one integer only, and the diagonal in the opposite direction is always a continuous progression of numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9....

6. The 9x9 Magic Square and the 27x27 Magic Square in the Luo Shu format can have their numbers transcribed into their Pythagorean root numbers and still maintain a magic square as well as define a torus. The next magic square that could do this would be the 45x45 Magic Square or any order magic square whose root number = 9.

7. At the heart of every magic square in the Luo Shu format lies a Pythagorean triplet. In the 9x9 Magic Square it is the triplet of 9, 40, and 41.

This new magic square has a modified Pythagorean triplet: 3x3, 4 and 5, very close to the 3 - 4 - 5 classic Pythagorean triplet found in the 3x3 Luo Shu Magic Square.

**THE CIRCLE AND THE SQUARE**

The symbolism of the circle and the square has millenniums of tradition. The gnomon, the sun, the calendar, the four cardinal directions, the right angle triangle and magic squares in the Luo Shu format are part of the ingredients of the symbolism of the circle and the square.

This so called "sacred geometry" was used as a model for city layouts, temple design, and in places of religious / political importance in many cultures including early China (Shang dynasty), southeast Asia, India, Iran, Armenia, Turkey and elsewhere.

The long tradition of this symbolism continued throughout history and influenced church design and architecture of the Byzantine era and the early Italian Renaissance.