Sundial, Type B

Sundial, Type B
MYSTERY SOLVED

Sunday, December 8, 2024

9X9 MAGIC SQUARE

The feature that makes this magic square most unusual is another magic square appears within its framework.  That demands further inspection.

I found this square on Jollymaths.com

Reducing the numbers of this 9x9 magic square to their Pythagorean root number generates a square of repetitive Luo Shu magic squares.  The entire square becomes nine individual 3x3 magic squares.


Notice there are nine consecutive numbers in each grouping with the first nine numbers located in the first row middle column grouping.  


The second grouping of consecutive nine numbers, ten thru 18 can be found in the lower right grouping of numbers.  The third set of consecutive nine numbers, nineteen thru twenty-seven, can be discovered in the first column middle row group.  If each group of nine numbers is assigned its position in a 3x3 grid according to numerical order, then the Luo Shu magic square shows up once again.

Wednesday, December 4, 2024

THE 21X21 MAGIC SQUARE and HE TU

If all the numbers in the 21x21 magic square are reduced to their Pythagorean root number, the He Tu pairing emerges.  In addition, the 3x3 magic square appears along the major diagonal that runs from the upper right to the lower left corner cells.  The He Tu pairing can be found in the 3x3, the 9x9, the 15x15, the 21x21 and the 27x27 magic squares.




The magic constant of this square is (21)(221) = 4,641; every row, column, and major diagonal sum to this number.  Note the cross of odd numbers that run through the horizontal and vertical axis of the square.


Every number one is paired with every number six and so on for two-six, three-seven and four-nine.  The pattern that emerges is different for each square the pairing occurs in.



This post demonstrates that the He Tu pairing is to be found in the Luo Shu magic squares.

Note:  There are some unpaired numbers along the borders of the magic square.  Keeping in mind the magic square is the two dimensional image of a torus, the numbers that are unpaired actually have their paired mate located at the opposite border in the same row or column so in three dimensions the pairing is complete.  Five is unpaired as it represents the central pivot.  

As stated above, the pairing can be found in several squares, beginning with the 3x3 M.S. and skipping every six orders so the next magic square that the He Tu will be discovered is the 9x9 M.S.

This proves that the He Tu pairing originates from the Luo Shu format of magic squares.  This assertion has been made previously by Lars Berglund in his book, The Secret of the Luo Shu, Numerology in Chinese Art and Architecture (1990).  Berglund puts forward his theory but offers no evidence, now we have the evidence.

Thursday, November 28, 2024

La Sagrada Familia

FROM THE PAPER  "THE SECRET OF THE MAGIC SQUARE, NUMEROLOGY IN ART AND ARCHITECTURE
(copyright pending 2024)

La Sagrada Familia, Barcelona, Spain

The magnificent church, La Sagrada Familia, was started in 1892 by architect Antoni Gaudi and is presently still under construction although a completion date of 2026 has recently been announced.  Two significant features relevant to this paper are the appearance of a magic square albeit an even magic square of order 4, and the coded use of numbers reflecting the Pythagorean Theorem. 


Image 19.  A 4x4 magic square at the entrance of La Sagrada Familia with a magic constant of 33.

This is not the traditional magic square of order 4 as that square would have 34 as the magic constant.  This square uses two 10s and two 14s in order to have the magic constant equal 33, a reference to the age of Christ when he died.  A traditional 4x4 magic square would have the numbers one thru sixteen and there are 880 different arrangements of numbers to achieve this end.                        


Stained glass image of quincuncial composition in La Sagrada Familia using the classic quincunx and comparing to jewelry from the Lazaro Galdiano Museum, Madrid.


Stained glass window exhibiting the Pythagorean Theorem in code, La Sagrada Familia.

Counting the number of petals in the large circle, one comes up with 16 or 42.

Counting the number of circles including the large circle (but not the real small circles) one comes up with 25 or 52.

Counting the tomb shaped figures on one half of the window, one comes up with 9 or 32.

32 + 42 = 52

Could it be a coincidence that Gaudi uses a magic square, makes a coded reference to the Pythagorean Theorem, and uses quincuncial composition?  The magic square represents the establishment of order on earth by connecting with Heaven thru the use of mathematics; repetitive concepts familiar to early Christian art and architecture. 

La Sagrada Familia, outside of church.
Another example of Gaudi's use of square numbers:  there are 81 (92) circular windows in the above picture, the last grouping has 9 windows and both 81 and 9 are powers of three (the triad). 

for PDF of the complete paper (36 pages), 
email:  robert.luoshu@gmail.com
$7.00

The Secret of the Magic Square, Numerology in Art and Architecture
25 images and 15 tables, discusses in detail the magic square and its relevance in early China and the West as a motif in art and architecture.

©2025 pending