A Right Angle Triangle in Every Magic Square in the Luo Shu Format


A RIGHT ANGLE TRIANGLE IN EVERY LO SHU MAGIC SQUARE


This section will introduce the concept of the Lo Shu as a model for odd magic squares of any size constructed in this unique format.  The Lo Shu featues many mathematical relationships but the most important is the right angle triangle theorem.

All magic squares in the Luo Shu format are the relationship between two consecutive whole integers.  The very first magic square is of the order three, a relationship between 1 and 2, squared. 

                                                    22  -  12  =  3                   12  +  22   =  5               

                                                 left of center cell                 center cell

A Luo Shu Magic Square can be constructed of any size to infinity, and each would have its own distinguishing  Pythagorean triad of numbers.  

A New Interpretation of the Lo Shu

   Y     =      CENTER CELL OF ANY LUO SHU MAGIC SQUARE,  X2  +  1 /  2
X     =     THE CELL TO THE LEFT OF THE CENTER CELL, THE SIZE  OF THE LUO SHU
Y - 1 =     THE CELL DIRECTLY ABOVE  X 



 

CYCLE OF LUO SHU        PYTHAGOREAN TRIAD            PYTHAGOREAN THEOREM
or MAGIC SQUARE       X         Y  -  1          Y               X2   +  (Y  -  1)2  =    Y2   

1                    3             4              5              9    +       16    =      25

2                    5            12            13            25    +     144    =     169 

3                    7            24            25            49    +     576    =     625

4                    9            40            41            81    +   1,600   =   1,681

5                   11           60            61           121    +   3,600   =   3,721

6                   13           84            85           169    +   7,056   =   8,225

7                   15          112          113           225    +  12,544  =  12,769

28                  63          336           343          679       25,336       26,215

 

The totals of the seven columns generate seven numbers that are all divisable by seven.
The even numbers are divisible by 28 (the sum of the first seven numbers).
The sum total of these 49 numbers is:

28   +   63  +   336   +   343   +   679   +   25,336   +   26,215   =   53,000




A NEW DEFINTION OF A MAGIC SQUARE IN THE LUO SHU FORMAT

The characteristics of any odd magic square constructed in this form, the Luo Shu format, will have these features:

 

1.   X will represent the order of the square.

2.   Y will represent the center number and center cell of the square

3.     X, Y, and Y-1 will always represent a Pythagorean Triad.

4.   A Luo Shu Magic Square is based on the squares of consecutive numbers:

the greater of which will be in the upper right corner cell of the square and
the lesser will be one cell to the left of the lower right corner cell.

5.   The lower right corner cell will represent the sum of the numbers 1 thru X. 

6.   The cell above the center cell will represent the highest number in the square, or X2.

7.   The sum of any pair of numbers equidistant from the center cell along any axis will equal  X2 + 1.  

8.    The cell below the center cell will always be 1, and a square of any size may be constructed beginning here.

 9.   All Magic Squares in the Luo Shu format will have a cruciform or cross shape of odd numbers along the horizontal and vertical axis.

10.  All Magic Squares in the Luo Shu format will generate a torus geometric figure.

11.  The progression of magic squares in the Luo Shu format describe fractal growth.

12.  All Magic Squares in the Luo Shu format will have the same characteristic tracing of the Knight's Tour (see example 11x11 Luo Shu Magic Square).

 

For the proof written in Mathematical terms, see Aale de Winkel's web site: http://www.magichypercubes.com/Encyclopedia/index.html

Click on "L" for Luo Shu to find the article with the proof 

The interpretations of a new definition of Magic Squares in the Luo Shu format and the resulting formulas to describe a Luo Shu Magic Square are the intellectual property of the author.  This material can not be reproduced with out the express permission of the author.  All the material contained in this web site is protected by copyright law.