PORTABLE SUNDIAL, TYPE B

That the Luo Shu appears on Chinese sundials from the seventeenth century has never been documented until now.

This type of sundial did not appear much before 1600 AD.  Many replications of this instrument can be found at several museums.  The Type B sundial, or equatorial dial, became popular in Europe during the seventeenth century.  

One can clearly see the numbers of the Luo Shu in the second ring adjacent to the ring with the eight trigrams.  Even numbers are in black, odd numbers are in red.  It is perfectly appropriate for the Luo Shu to be used as a motif to mark astronomical instruments that are related to time measurement and space directionality.  

Instruments such as the portable sundial help humankind to connect with Heaven.  The Luo Shu has a legacy of being related to the calendar, the Pythagorean Theorem, the carpenter's square and the gnomon.  

From Science and Civilisation in China, p. i of every volume:

The carpenter's square is no ordinary tool, but the gnomon for measuring the lengths of the sun's solstitial shadows. 

The above pictures were taken from Time, Science, and Society in China and the West edited by J.T. Fraser, et al (1986).  Another example can be found in Science and Civilisation in China, Vol. 3, p. 308, Figure 134.  Please note the editor offers explanations of the markings in each of the rings around the compass but makes no mention of the Luo Shu being present in the second ring.  This is the first time this observation has been noted.

Symbols such as the Luo Shu and carpenter's square are frequently used in art and architecture to identify things of political or religious significance such as temples, churches, art, architecture, and astronomical instruments.  

My paper, The Secret of the Luo Shu, Numerology in Art and Architecture explains in detail the use of the Luo Shu and related symbols as an artistic motif used by many cultures for over two thousand years.  

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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.

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.