This blog is about the Cycle of Nine implemented in the Digital Root or Modulus 9-Function. The Digital Root generates many Patterns that were used in Ancient Architectures.
One of the most important Digital Root Patterns is the Vedic Square. It is the Digital Root of the Multiplication Table of the numbers 1 to 9.
This Table contains the Harmonics of the Numbers 1 to 9. These Harmonics are highly related to the Harmonic Pattern behind the Cycles in our Universe.
The first part of this Blog is about the Digital Root. It contains the patterns that are behind the Cycle of Nine.
This part is very technical but it makes it possible to show that there is a deep structure behind the Modulus-9.
This pattern has to do with just two numbers, 2 and 3. They generate the Spirals of Expansion and Compression of our Universe.
2 and 3 and their Sum 5 are also the Numbers behind the Harmonics of our Universe.
The Second Part is about the Vedic Square. It is called the Vedic Square because this Square is one of the most important tools in Ancient Vedic Mathematics.
Vedic Mathematics was used in many Ancient Cultures (China, Egypt, Greece) with different names. The Chinese art of Feng Shui was called Vaastu Shastra in India.
Pythagoras, trained in Egypt (Heliopolis), used the same principles and used the same Patterns the Ancient Vedic Scientists were using.
The last part is about the Game of Chess. This game is just like many other Ancient Games a Simulator of the Game of the Universe.
This blog contains many links to other Blogs and Resources on the Internet. These references make it possible to dig deeper into this fascinating subject.
About the Digital Root
When you divide a number X by a number N the Remainder of the division is called X Modulus N. 22 mod 7 = 1 because 22 = 3×7 + 1.
The Modulus-function N maps the Set of the Natural Numbers to the Numbers 0, 1, 2, ….,N-1.
One of the most famous and ancient Modulus-functions is called the Digital Root. The Digital Root is the Modulus 9 function.
Because 10 mod 9 = 1 every Power of 10 has a Modulus 9 of 1. Therefore (a10**X+ b10**Y+…) mod 9 = a + b, the Sum of the Digits of the Number. 62 mod 9 = 6+2 = 8.
Digital Roots have been recorded for thousands of years, formalized by Pythagoras in 530BC and even earlier in Indian Vedic Mathematics (Vaastu Shastra).
Digital Roots are used in Numerology. In Numerology Numbers have a Meaning.
In Gematria Letters and Words are transformed into Numbers which have a meaning.
In Ancient Languages like Hebrew Letters are also Numbers. Numerologists believe that Words with the same Digital Root have the same Meaning.
The numbers 0 to 9 of the Digital Root are the Points of the Tetraktys of Pythagoras.
The Modulus 9 pattern contains 2 number groups (3,6, 9) and (1,2,4, 5,7,8).
Later we will see that the last group contains 2 subgroups (1,4,7) and (2,5,8). Together with (3,6,9) we can map these 3 Triangels on the Modulus 9 Circle.
4 is the Middle of 1+7=8, 5 is the Middle of 2+8=10 =1 and 6 is the Middle of 3+9=12=3. 5 is also the Middle of the Middle.
The group (1,2,4,5,7,8 ) is called the Ring Z/9 in Mathematics. Z/9 is isomorphic with the Sequence 2**N mod 9 where N is positive and negative. The sequence 1,2,4,8,16(7),32(5),64(1),128 (2),256 (4),… repeats itself until infinity.
This Sequence is the Expansion and Compression Pattern of the Number 2.
The Ring Z/9 is part of the Tetraktys and forms a Hexagram. This Hexagram is a 2D-projection of the Cube of Space. When we combine the (3,6,9)-pattern with the Hexagon a (4×4) Triangle is created.
The number 2 is the Container, the Cube, inside the Tetraktys. That is the Reason why the Second letter in the Hebrew Alphabet Beth means Vessel or Container.
(3,6,9) is a Triangular Cycle that repeats itself until Infinity. The Number 3, the Trinity, is the Mover of the Container of 2. This Rotation moves With and Against the Clock.
This is the reason why the 3th Letter of the Hebrew Alphabet, Gimel, means Camel. The Camel of Gimel carries the Water into the 2 Containers of Beth.
The Number-2-pattern contains 3 Binary Groups (called Polar Pairs) with a Sum of Nine (1,8), (2,7), (4,5). The Number-3-Pattern contains 2 Polar Pairs (3,6) and (0,9). The Polar Pairs represent the Lines of the Tetraktys.
(0,9) maps unto Itself and represents The Beginning and The End, The Now. (0,9) is a Point and a Line.
The Polar Pairs of the Z/9 create a Cyclic Pattern that contains two Squares, (1,2,4,0) and (5,7,8,0). Both of them Share the Zero, The Void.
The Sum of the Opposite Numbers of the Z/9, (4,8 = 12=3), (1,5 =6 ), (2,7=9) of the Tetraktys shows the 3,6,9-pattern again.
There are 8 Ternary Groups ((1,5,9), (1,6,8), (2,6,7), (2,5,8),(2,4,9),(3,4,8),(3,5,7),(4,5,6)) with a Sum of 15. This Ternary Group represents Triangels. All of them are part of the famous Lo Shu 3×3 Magic Square.
When we use the number 3 as a generator 3 Triangles are created (1,4,7), (2,5,8) and (3,6,9).
The 3 Triangles move With and Against the Clock ((1,4,7) and (7,4,1)).
It takes 3 rotations to get every Triangle back to its original position. (1,4,7) becomes (7,1,4) and (4,7,1). This means that there are 6 permutations of every Triangle.
Every addition of two Triangels produces another Triangle. An Example: (1,4,7) + (2,5,8) = (3,9,6).
When we create a Matrix to find all the combinations a new group of 9 transformations ((1,1,1),(2,2,2),(3,3,3),(4,4,4),(5,5,5),(6,6,6),(7,7,7),(8,8,8),(9,9,9)) appears. They are the Triangels that are a Line and a Point. An Example: (1,4,7) + (1,1,1)=(2,5,8).
There are now (18 +9=27) x27 = 729=3**6 = 9**3 possibilities.
The same 27×27 Matrix appears when we Multiply the 3 Triangels. An Example: (5,8,2)x(5,8,2)= (25,64,4)= (7,1,4) and (3,6,9)x(5,8,2)=(15,48,18)= (6,3,9).
Another interesting patterns becomes visible when we look at the Opposite Numbers of 3 Triangels (1,5), (2,6), (3,7),(4,9) en (5,9) in the Picture above. They recreate the Triangels. An Example: (5+9=5, 2+6=8 ,8+3=11=2).
About the Digital Root of the Golden Mean
The 27×27 Matrix pattern also emerges out of 24 repeating numbers (1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9) of the Digital Root of the Fibonacci Sequence (The Golden Ratio).
When we group the Golden Ratio pattern in 2’s (2×12) the Polar Pairs appear. The 12 pattern has a Sum of 108 = 0 Modulus 9. 108 and 24 are related to the Gayatri Mantra.
1 1 2 3 5 8 4 3 7 1 8 9
8 8 7 6 4 1 5 6 2 8 1 9
When we group the pattern of 24 numbers (3×8) of the Golden Ration into Trinities the Triangle Pattern appears again.
1 2 3 4 -3 -2 -1 -4 (Pattern-number)
1 1 2 3 (7) 5 8 4 3 (2)
7 1 8 9 (7) 8 8 7 6 (2)
4 1 5 6 (7) 2 8 1 9 (2)
The Pattern of the Pattern is (1,2,3,4,-3,-2,-1,-4). The last part of the Pattern (-3,-2,-1,-4) can be transformed into the first part (1,2,3,4) by adding 4.
The Digital Sum of the first 3×4 numbers is 7 and the Digital Sum of the last 3×4 numbers is 2.
When we rearange the 24 cycle in 6 groups of 4 digits another pattern shows itself: (1,4,8,5), (1,3,8,6), (2,7,2,7),(3,1,6,8), (5,8,4,1), (8,9,1,9).
When we combine all the different rotations of the 3 Triangels a Cyclic Flow Pattern appears that looks like the Jitterbug of Buckminster Fuller.
The Jitterbug is a 3D projection of the 4D 24-Cell (again 24!) also called the Hyperdiamond.
The 24-cell is self-dual and is the regular polytope with no analogue among the five Platonic solids of 3-space.
About the Vedic Square
One of the Simple Structures of Numbers that contains a lot of patterns is the Vedic Square. The Vedic Square was called the Eight Mansions in China. The Vedic Square is the Digital Root of the Multiplication Table of the numbers 1 to 9.
The Multiplication Table is a subset of the 27×27 Matrix of the 3 Triangels.
The Multiplication Table contains the Harmonics of the Numbers 1 to 9.The Vedic Square was used to build the Pyramids, create the Chinese I Ching, the Game of Chess, Dante Alighieri used it to structure his trilogy La Divina Commedia, the Sistine Chapel was build and the frescoes and symbols were arranged according to its concepts and the first chapter of Genesis was written and imbued with its numerous concepts graphic images.
Scholars and Artists discovered that the various lines of the Vedic Square could be used to direct a design. By selecting a line of numbers, and using a constant angle of rotation, various designs could be produced. These designs are visible in abstract Islamic Art.
The Vedic Square is a Symmetrical Structure because AxB=BxA. This is called the Associative Property of Multiplication. The Square is a combination of Two Triangels and contains 45 distinctive numbers.
The Vedic Square repeats itself until infinity when you extend the Square to a NxN Square.
The number pattern of the diagonal of the Vedic Square, 1,4,9,7,7,9,4,1,9, is the Digital Root Pattern of the Square Roots. This patterns repeats itself until Infinity.
The Vedic Square contains the 5 Polar Pairs, the 8 Lo Shu Ternary Groups and the 3 Trinity Patterns ((1,4,7), (2,5,8), (3,6,9). It also contains the Star of David, The Zodiac, the Tree of Life and many other Mystic Patterns.
It is possible to transform the Vedic Square to the Lo Shu Magic Square.
The patterns of the Vedic Square Rotate. The End of a Horizontal and a Vertical Pattern connects with the Beginning of the Pattern. This means that the Vedic Square is a Torus.
This Torus is called the Rodin Torus. The Rodin Torus is a Coil that produces a Uniform Electro-Magnetic Field.
The 3-6-9 and 6-3-9 Cycle in the Vedic Square can be thought of as Clockwise and Counter-Clockwise, or as Electricity and Magnetism. They are transport-channels.
The ((3,6,9),(6,3,9)-Matrix divides the Vedic Square in 9 2×2 Squares.
The 9 2×2 Squares have a Sum of 9,18 and 27 which is 1×9,2×9 and 3×9. If we leave out the (3,6,9)-Matrix and divide by 9, a 3×3 matrix results with 1,2,3 on the Outside and a Cross of 2’s in the middle. This 3×3 matrix shows the Expansion of the 2 into the (1,2,3).
The Rows and Colums of Ring Z/9 add up to 45. The Rows and Colums of the Number 3-Pattern add up to 54 which is a Mirror of 45. The (4,5)-pattern generates the Star of David and the Zodiac.
About Indian Vastu Science
The Game of Chess originated in India. It was passed on to the medieval West through the intermediary of the Persians and the Arabs.
The form of the Chess-Board corresponds to the Vastu-Mandala, the 9×9 diagram which also constitutes the basic lay-out of a temple or a city.
Hindu mythology has it that Vaastu Purusha was born of Lord Shiva’s sweat when he fought the deadly demon Andhakasura.
Vaastu Purusha himself became uncontrollable and destructive and the heavenly gods finally subjugated him and brought him down on earth with face down, with his face in the Northeast and his feet in the Southwest.
45 deities stayed there, 32 of them in the outer enclosure and 13 of them in the inner enclosure holding him in place at various points or locations on his body.
32 =64/2 and the Number of the 32 Paths of Wisdom of the oldest book of Hebrew Mysticms the Sepher Yesirah (the Book of Formation or Book of Creation, ספר יצירה).
64 is the Number of the I Tjing. 45 (5×9) is the Sum of the Lo Shu Magic Square and the Number of the Vedic Square.
All these Mystic Structures come from the same Source and are different Views on the same Pattern, the Tetraktys, the Triangular Numbers created by the Meru Prastara or Sri Yantra also known as the Pascal Triangle.
The Vastu Mandala is an expansion of a Point (the Bindu) into the Line(2), The Trinity (3) and the Rotating (With the Clock and Against the Clock) and Expanding Square (4), represented by the Symbol of the Swastika. The Swastika is a Fractal Generating Pattern.
Every Point is a generator from which the Swastika-pattern generates a new Swastika. The 2×2 Square is transformed by the Swastika Pattern into the 4×4 and the 8×8 Square.
As you can see the Vastu Jain Symbol is an Indian Version of the Tetraktys of Pythagoras.
The Swastika contains the Four Points of the last line of the Tetraktys that are related to the Tethahedron.
About the Game of Chess
The Chess-Board symbolizes the Unfolding of Space by the Number-2-pattern and it synthesizes the Complementary Cycles of Sun and Moon.
The number 64, the sum of the Black & White (Yin/Yang) Squares on the Chess-board, is a divisor of the number 25920 (25920/64=405, 25920/9= 2880/9=320/5=72), which measures the Precession of the Equinoxes.
The Polar Pairs in the Modulo 9 Pattern are expressions of the Planets.
(1,8), the Castles, relates to the Planet Mars.
(2,7), the Bishops, relates to the Planet Venus. Venus is the Ruler of the Heart and the (2,7) is situated in the Middle of the Vedic Square.
When viewed from the Earth, the Planet Venus inscribes a near perfect five-pointed star (pentagram) around the sun every eight years. The points of a five-pointed star (pentagram) touch the circle of a pentacle every 72 degrees. Likewise, many in Islam expect 72 virgins in heaven.
A full 360 degrees of procession takes 25,920 years, which is also seventy-two (72) 360-year cycles.
(3,6), the Knights relates to the Planet of the Messenger, Mercury. Mercury is Hermes, the Messenger God, with winged sandals. The moves of the Knights create a pattern that looks like the Swastika.
The (3,6)-number-lines are Transport-Channels (Gimel) as you can see in the Vedic Square and the Rodin Torus. The planet Mercury traces a Hexagram during its movement around the Zodiac.
(0,9) is the Planet Jupiter, the Ruler of Modulus 9 who determinates the Rules of the Game. (0,9) is the Beginning and the End of the Game and is the cause of the Rotation of the Swastika related to the (3,6.9)-pattern.
The numbers 4 and 5 are the Moon (Queen) and the Sun (King). The Moon moves the quickest of all the planets, so does the Queen on the chessboard.
The Number 5 of the King is the Center of the 3×3 Lo Shu Magic Square and the Center of the Tetraktys.
The 8 Pawns represent the number 2 and are connected to the Planet Saturn, the 2nd Son of the Central Sun and the Trinity (1+ 2 = 3). The Pawns start to move with 2 steps and later move 1 step. The 2 is the Center (The Son of the Sun) of the Trinity.
The 2 is also the Generator of the Expansion Pattern and the Polar Companion of the 7, the Center of the 3D-version of the Square, the Cube of Space.
The Pawn (2, Saturn) promotes into a Queen (Moon, 2×2) when he has reached the Other Side.
LINKS
About the Multiplication Table of 9
About the Tetraktys and the Lo Shu
About the Harmonics of the Universe
About Harmonics and Entrainment
About the Jitterbug of Buckminster Fuller
A Simulation of the Jitterbug Pattern (CUBIC WONDER)
Between 1969 and 1975 I was a student at the University of Leiden.