Clarity,
Office 17622,
PO Box 6945,
London.
W1A 6US
United Kingdom
Phone/ Voicemail:
+44 (0)20 3287 3053 (UK)
+1 (561) 459-4758 (US).
MAGIC SQAURES AND TORIC RELATIONSHIPS
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. There are some interesting features to this new magic square:
1. Five is the center number of the square as five is the center number of the numbers one thru nine. Five is also the center of the 3x3 Magic Square known as the Luo Shu.
2. The diagonals in one direction consists of one integer only, and the diagonals in the opposite direction is always a continuous progression of numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9....
3. The magic square forms a three dimensional torus. That is, the numbers continue in order from one edge to the opposite edge allowing the two dimensional square to be folded once into a tube shape then a second time to form a torus.
4. The numerical pattern in the column, row, and one diagonal direction is: 1, 6, 2, 7, 3, 8, 4, 9, 5, 1…
5. 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 able to do this would be the 45x45 magic square.
The significance of all this is that the torus shape has important industrial uses.
When the numbers of magic squares are used to construct a torus shape, the numbers have an ideal harmonic balance that may create a synergy that offers the least resistance to electricity.
The harmonics may be related to the solfeggio scale (in red), an ancient musical scale that had powerful implications not understood by present society. There are six frequencies separated by an interval of 111.
Beginning with 63 and doing the “111 expansion” one arrives at:
63If we refer to the 27x27 Magic Square in the Luo Shu format (see attachment below) and follow the progression of the above numbers: 63, 174, 285, 396, 507, 618, to finally 729 – the last number of the square – a pattern is established. If we continue the pattern in the 27x27 magic square, we will pick up the other numbers of the solfeggio harmonics (in red):
174
285
396
507
618
729 = 27x27
840
951
84
195
306
417
528
639
Furthermore, these numbers, 84 thru 639 are linked on the diagonal to the numbers 111, 222, 333, 444, 555, 666.
And the numbers 174, 285, and 396 are linked on the diagonal to the numbers 147, 258, and 369 – the numbers of the Enneagram.
All of these twenty four numbers listed break down to the Pythagorean root number of 3, 6, or 9.
Theory proposal: when the numbers of a large magic square in the Luo Shu format are transcribed into their Pythagorean root numbers a magic square is generated that is in perfect harmonic balance that also (can) describe a torus and this may have powerful toric implications.
Last edited by robertluoshu; September 12th, 2010 at 11:45 PM.
Hi Robert,
Apropos Magic Squares, the other day I came across this book in Google Books:
Magic squares and cubes
By William Symes Andrews, Paul Carus, Lorraine Screven Frierson, Charles Albert Browne
Perhaps you've seen it already. Cool stuff in that old book.
Hey Luis,
Yes, I have the Andrews book, published in 1919. I liked the paperback so much that I wanted to invest in a hardcover version and paid $90 over the internet for a xeroxed copy of the paperback with a hard bookbinding. Copyright law does not apply to books of this era.
On page 150, CA Browne published the first known 27x27 Magic Square. However, his square is not in the Luo Shu format BUT his square does feature 365 as the center number of the square. Browne's article, Magic Squares and Pythagorean Numbers, makes several interesting points:
- The use of the number 729 in Plato's Republic
- The importance of the number 729 in the Pythagorean system
- Odd numbers were superior to even numbers
- The reference of the 27x27 Magic Square and the calendar
- Pythagorean harmonics (the octaves and numerical ratios)
Similar to the topics that we have discussed here on this thread. It is impressive that Browne could build such a large square without the aid of a computer, every time he changed a number he would have to re add the row and column affected, 27 numbers in each.
I have included two attachments in my previous post that show:
- the 27x27 Magic Square in the Luo Shu format with the numbers of the solfeggio frequencies highlighted in a specific pattern
- the 27x27 Magic Square reduced to its Pythagorean numbers one through nine
Last edited by robertluoshu; September 10th, 2010 at 11:30 PM.
pocossin (September 10th, 2010)
Hi Robert.
Thanks for your mail and question. Yes this does have a lot to do with the work between myself and Lee Burton. The pattern you find of 369369 also has an opposite of 639639 in which all values in The Well and Binary Opposites of the Bagua are found in the values 3 and 6, but in the 1 to 8 of The Well and Bagua the other values given in the contracted integers = 2 to 7 and 4 to 5 as I have already explained in the previous postings.
The central value of the Well = 9 and all expansions in the 8 directions all = 9s and run as follows:
75318642>9<75318642>9< = 2 and 7s
51627384>9<51627384>9< = 4 and 5s
------------------------------
36936936>9<36936936>9
------------------------------
51627384>9<
12345678>9<
---------------
63963963>9< so we have gone full circle as 123456789 + 6s = 2 and 7s.
---------------
The way these values can be tied to what is now termed Solfeggio is in the 111 sequences that start with 63 + 111 =
174
285
396
417
528
639
741
852
963 - but this pattern is also reflected in all + 111 counts as
123
234
345
456 etc, so it's inherent in all integers in the fact that the central value of The Well and Lo Shu give the central value of 5 = 10s (1) and 9 = 9s, but all values can be included =
Central value =
852 = C5 =10
396 = C9 = 9
741 = C4 = 8
285 = C8 = 7
639 = C3 = 6
174 = C7 = 5
528 = C2 = 4
963 = C6 = 3
417 = C1 = 2
852 = C5 = 10 = 1. So the contracted integer runs are total interchangeable with each other, and also the value of C = the value found in all expansions as in the Lo Shu C5 = 11111s in all crossover or reversals of the values, or n the Well C9 = 9999s (this is present in every C = value). This generation of values C = is to infinity but only need be taken to 9 in the values of the Magic Squares that are manufactured in the 9 Stack or which appear in Rodin's Torus.
Lee gives all the values in his work and so both math comply.
It's the value of 4 and 5 that appears in the Ho Tu as 5>162738495>1 gives the 5 directions of 1 to 6, 2 to 7, 3 to 8 4 to 9 and 5 to 10 as the 5 Elements and that each of these relationships found in the Trigrams of Ch'ien and Kun in the value's of Heaven and Earth produce a 1 to 8 in the Trigram sequence that produces the 8 x 8 Magic Squares - as in the reduction all the values that give the values of Time contract back 3, 6 and 9s. The average value of each of the Outer Squares/Directions/Elements = 2160 so a total of 8640 appears = to 240 + 120 = 360 = the values of the 360 day-year = 4320 light and 4320 dark or the values in the 9 x 9 square of the Lo Shu that are said to set the gods and demons in motion, and in the addition of the values 10 + 5 the Great Cycle of 12,960, 25,920, and = 51840 is attained. Interestingly the values of the 9 x 9 give the working's of the Ho Tu in each integer expansion of the Trigrams 1 to 8 = so then 1 can be + 9 = 10 so the expansion is Metric. In the values of the Trigrams all = 6s so 1 = 6 to 89 = 534 = 90 = 540 so 5 or 45 = 270 (same as values found at Angkor Wat and to those that can be applied to the GPs as 540 + 216).
The value of 729 is the main value being mentioned in many recent posting on this by Lee and now some other people's investigations. So please read both my own and Lee's work for the full pic.
I hope this is of assistance - sincerely - Paul.
This I like very much. The numeric pattern of 1, 6, 2, 7, 3, 8, 4, 9, 5.... is related to the Ho Tu and the Luo Shu torus.
Question on the 111 expansion beginning with 63:
From 396 to 417 is not 111 but 21. Why is there this interruption then a continuance of an expansion of 111?
I find it interesting that even with this numerical interruption above, there is no interruption of this pattern in the 27x27 Magic Square in the Luo Shu format.
Last edited by robertluoshu; September 11th, 2010 at 04:58 PM.
Hi Robert indeed when first seeing the Solfeggio site this question as to why 21 appeared between 396 and 417 was my own first question ???? The working's given in the Book of Changes only follows sequences with center 1 = 417 so all values to infinity from central 1 = 222 etc in the cross-over of the Magic Square as +
963
417
852 so all are 1627384951 as in the Ho Tu, in which all the Squares 8 x 8 in the 5 Elements generate the values of Time and the Measure of the Cycles of Change.
The way you now laid out the math concurs with my own working's and with Lee's, as each central value produces a set value as C1 = 2s, C2 = 4s, C3 = 6s and so on as already described in previous postings and in The Sum of Things.
The only part that that I would question as a base is in the workings of the 8 Spoke wheel and not a division of 9 being the base as 9s is the product or the central value of The Well. If you are dealing with the Binary Trigrams 1 to 8, and all other values within the Trigram Pairings then as also pointed out the integer relationships are 1(9)8 or 8(9)1, 2(9)7 and 7(9)2, 3(9)6 and 6(9)3, 4(9)5 and 5(9)4. In the listing of the Ho Tu it gives 4 to 5 as 1627384951 which already seem to have now looked at. You are not only dealing with integers but also their link to the binary structure of the Changes, but it's all now so close to a mutual math that I see no problems in them coming together.
The well has a central of 9 and gives 999s in all directions of cross-over integers within the squares to infinity, and the Lo Shu from central 5 gives 1111s as 10s to infinity. But in the Torus or Torii every value produces 9 from 1 to 8, and only in finding the values of 1 to 8, and all the Opposite Binary pairings = 9 are the Central relationships confirmed as each C value gives sequences from 111 to 999. Lee's work will also confirm this for you in respect of the contract values, but you must undo the Trigrams and Hexagrams to also gain the expanded.
I feel we now close enough in the math to say that the work is understood and that in the values of the addition of the extra 73rd part of 120 hours to 360 in the division of 8640/72 the value of 8760 is found in the 365.
So indeed Robert I leave all this with you and look toward a future in which the math of The Changes is now being revealed. My regards and all the best - Paul.
robertluoshu (September 17th, 2010)
Hi Robert.
Shame this thread was left in limbo as we were getting so close to a mutual on these workings.
I can now add that Scot C Nelson who has worked with Marko Rodin on the math of the Torus appears to be putting forth the same math, a change in what they term Vortex math to now become I Ching Math. The only difference between us all is the 8 Spoke with a central 9 that manufactures the all values linked between The Well and Ba gua - thereby giving us a integer to binary interchange in the 1 to 8 = 9s. Scot has used the contracted integers as I put forward and seen the manufacture of the Central Value to the cross-overs or reversals in the Magic Squares of 111, 222 333 to 999 expanded to infinity as a balanced perfect Magic Square in both number and binary.
I noticed you also used the 9 division whereas myself and Lee would only see 1 to 9 as a product of 1 to 8 in both integer and binary images of trigrams. The 1 to 9 appearing in the product of the Ho Tu's values as they create the Lo Shu. I give this in the Outer Direction's or Element's as 1234 to 6789 with 5 pairing to 10 in the 9 x 9 Magic Square that in turn gives us the value of 10s and 15 (all generated from the 8 x 8 Magic Squares or circle of 64 Hexagrams in each Quarter plus the Central 5th Element). The Ho Tu using 1627384951 which is linked to the obvious expansion of the contracted integers 5 and 4 = 516273849-51etc also to infinity.
All this math is so close, and so I hope we can continue this thread to gain a mutuality on how everything works. More has been added to Lee Burton's site and to my own book's new edition of 'The Sum of Things' now available from ABE Books. So as Rodin is now considering the same workings I hope we can bring our math to a conclusion and allow the old to meet the new.
My sincere regards - Paul.
Hello mi name is Romina and i live in Uruguay. Sorry! my english is horrible! I am maths teacher. I am studyin the King Wen sequence. I know that there is an extensive work that made R.S. Cook but my observations are about the simmetry and sums. I would like to discuss with someone interesting in this work. Thanks. Romina.
P.D. Si contestan en español mejor!!! Gracias.
Qué tal, Romina,
Bienvenida. Gracias por escribir y preguntar. Te cuento que también soy de Uruguay, aunque transplantado hace 25 años a los EEUU. Hay otro uruguayo por acá, Sergio, y tenemos a Charly que es de Argentina.
Si querés saber más sobre el trabajo de Cook, digerido y en español, no tenés más que ir a lo del amigo Raúl Jurovietzky y el sitio Saiching de Argentina y empezá por acá:
http://www.saiching.org/novedades.htm#junio2008
Saludos,
Gracias por su respuesta. Me alegro de encontrar uruguayos por acá. Tal vez no me expresé bien pero mi idea no es discutir el trabajo de Cook, sino discutir un trabajo que yo elaboré sobre la disposición de hexagramas del Rey Wen. En el trabajo que he elaborado he realizado observaciones sobre simetría y sumas que me han sorprendido muchísimo. Quería saber si en esta comunidad hay alguien interesado en discutir eso. He visitado la página de la sociedad argentina de i ching y fue a partir de un artículo de Raúl Jurovietzky que comencé a realizar el trabajo. También me he comunicado con él y le he mandado el trabajo, estoy esperando su respuesta. Pero tal vez encontraba por aquí a otras personas interesadas en esto. Gracias nuevamente. Saludos. Romina.
Clarity,
Office 17622,
PO Box 6945,
London.
W1A 6US
United Kingdom
Phone/ Voicemail:
+44 (0)20 3287 3053 (UK)
+1 (561) 459-4758 (US).
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