## Posts Tagged ‘Sum’

Monday, December 19th, 2011

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.

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 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.

Lo Shu Magic Square

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.

The 3 Triangles of (1,4,7), (2,5,8) and (3,6,9) copied from the linked Website.

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).

this solution gives the densest
lattice packing of spheres in 24 dimensio

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).

The pattern of the Golden Mean copied from the Linked Website

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.

The 24-cell also called the Hyperdiamond

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 Sistine Chapel is designed with the Vedic Square. Click on this picture to see a 3D version.

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.

La Divina Comedia of Dante with the Tower of Babel on the background. This Ziggurat is a Geometric Structure highly related to the Vedic Square.

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 Patterns of Islamic Art are created out of the Vedic Square

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.

Vedic Square

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 Rodin Modulus 9 Number Torus

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).

Patterns in the Vedic Square

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.

The Vastu-Mandala

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 Jain Symbol is a version of the Tetraktys

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.

The Borobudur represents Mountain Meru, Pascal's Triangle.

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.

About the Multiplication Table of 9

About Genesis and the Vedic Square

About the Divina Comedia and the Vedic Square

About the Sistine Chapel and the Vedic Square

About the Tetraktys and the Lo Shu

About the Harmonics of the Universe

About the Jitterbug of Buckminster Fuller

A Simulation of the Jitterbug Pattern (CUBIC WONDER)

About Plato and the Sri Yantra

About Vastu Science and the Borobudur

### About Mystical Number Theory and Pascal’s Triangle

Friday, December 2nd, 2011

The first part of this Blog is about the Triangular numbers, related to the Number 3, the Holy Trinity.

The second part shows that Pascal’s Triangle (called Meru’s Mountain in Mystics), the Binomial Expansion,  contains every Possible Mystical Number Pattern (including the Triangular Numbers) you can Imagine.

Pascal Triangle also shows that our Universe is a combinatorial miracle. It explores every possibility, is always in balance, expands and moves back to the beginning which is and was the Void, the Empty Set, the merge of Every Paradox, that is Possible.

The Sēpher Yəṣîrâh (Book of Formation or Book of Creation, ספר יצירה) is the oldest book on Jewish Mysticism. The Sefer Yetzirah describes how the universe was created by the “God of Israel” through 32 Wondrous Ways of Wisdom.

The Number 32 is the Sum of the 10 Sephirot and the 22 Letters of the Hebrew Alphabet.

The Sephirot is related to the  Tetraktys of Pythagoras. The Tetraktys embodies the Four main Greek Cyclical (PlatonicMusical Harmonies: the Fourth (4:3), the Ffth (3:2), the Octave (2:1) and the Double Octave (1:4).

1+2+3+4 = 10. 10 is the 4th Triangular Number. The Nth triangular number is the Sum of the numbers 1 -> N. This Sum is equal to 1/2N(N+1).

Between the 10 Sephirot run 22 Channels or Paths which connect them.

The Sephirot are the Points of the Tetraktys. The Hebrew Letters are the Lines between the Points. The Lines of the  Sephirot and the Tetraktys create a Cube (6) at the Top and a Tetrahedron (4) at the Bottom.

The Letters of the Hebrew  Alphabet are divided in the 3 Mother Letters (אמש, the Trinity), the Seven Doubles (The Planets) and the Twelve Simples (the Zodiac).

The 22 letters of the Hebrew Alphabet are a combination of the Trinity, the 7 Planets and 12 Signs of the Zodiac.

When you analyse the Sepher Yeshirah the Cube of Space (the Kaaba) appears out of the Hebrew Alphabet. The Kaaba is related to the Seventh Planet, Saturn.

The 3 Axis of the Cube of Space are the Trinity, the 6 (2×3) Faces of the Cube stand for the Planets with the 7th Saturn, the Son of the Central Sun (3+1 (Center)+3) in the Center and the 12 (4×3) Boundary Lines of the Cube represent the 12 Signs of the Zodiac.

As you can see the Number Three, the Triangle,  plays an important role. It is the First Structure that is Closed in Itself and is therefore Topological related to the Circle. The Circle (and the Triangle) is able to rotate With and Against the Clock. This property is called Spin in Physics.

It is very important to realize that Everything Rotates in our Universe around a Central Object that rotates around another Central Object. The Central Object Gives Time, determinates the Rythm or Harmonics,  of the Rotation Structure.

The Trinity rotates around the Void. The 7 Chakra’s of the Human rotate around the 4th Chakra, the Heart Chakra, ((3+(1)+3)=7).

The Planets rotate around the Sun and the Sun rotates around the Central Black Hole. The arrow of Sagitarius points to this Black Hole.

On a Six Sided Dice the Sum of all the Numbers is Seven (1+6,2+5,3+4). The Sum of the Six Numbers is 3 X 7 = 21. If we add the Center (Saturn) the Number 22 appears.

22/7 is a good approximation of the number π. π relates the Square (and the Cube) to the Circle.

The Cube of Space symbolizes  the Playing Board of the Game of Life. On the Playing Board we have a Free Choice to move into the many Paths that are available. Every Path has its own Probability and this Probability can be calculated. If we don’t know what to do we could throw a Dice.

The Cube of Space contains the same six lines that exist in the I Ching. Four of the lines are of equal length, the other two, the diagonals, are longer. For this reason symmetry cannot be statically produced and the Dance (of Shiva) results.

The Circle represents the Cycles of Time of the Matrix of the Demiurg. Behind all the Probabilities of all the Possible Paths lies a Hidden Order.

A Hexagram, represented by the Star of David,  is a Two-Dimensional (Orthographic) projection of a Cube. A Symmetric Projection of the Cube creates a Cross.

A Hexagram is a Two Dimensional Cube

One of the many meanings of the first word in the Bible “Bereshit“,  is “They (Elohim) created Six” which means that in Six Stages of  the Time Cycle the Cube of Space (or the Hexagram) was populated. On the Seventh Day the Center was filled.

The book of Genesis does not describe the creation of the Trinity (They, Elohim, 1+2+3, 1x2x3) itself. This stage was later covered in the Zohar.

In my blog “About the Sum of Things” it is shown that Six Stages are part of an Expansion Pattern governed by the Powers of Two. After 2**6 (64) Expansions (or Compressions) the Same Fractal Pattern repeats itself on a higher level.

64 is the Number of the I Tjing and the Game of Chess. The number 32 of the Sepher Yeshirah is 64/2 and is a Contraction of the I Tjing.

The I Tjing is a contraction of the oldest Divination System in the Word called FA. FA is still used all over the world by the followers of the oldest wisdom-system created by the YOrubA in Africa. The Yoruba lived at the place where the ancient Paradise was situated.

The Fourth Day (Sun (4), Moon (5))

The Tetraktys contains the Numbers 1, 3, 6 and 10. These numbers are called Triangular Numbers.

The number 21 is also a Triangular Number because it is the Sum of  the Sixth Level of the Tetraktys,  the Numbers 1 to 6.

The Fifth Level of the Tetractys is related to the Number 15 (1+2+3+4+5). This number connects the Tetractys and the Sephirot to the 3×3 Lo Shu Magic Square also called the Seal of Saturn.

The nth Triangle number T(n) is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n.  T(n)=n(n+1)/2.

The Triangular Numbers contain the Perfect Numbers. A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Six (1+2+3=1x2x3) is the first Perfect Number and 28 (1+2+4+7+14) is the next.

The Sum of two Triangular Numbers is a Square

The Sum of two adjacent Triangular Numbers T(n) +  T(n+1) is a Square Number because Two Triangels can be combined in a Square. 1+3=2**2 and 3+6=3**2.

There are many relationships between the Triangular Numbers. These relationships were the focus of the research of the Mystical Group of the Mathematikoi of Pythagoras.

6 (Bereshit, the Cube, the Hexagram) + the 22 Letters of the Hebrew Alphabet = 28, the Next Perfect Number (1+2+3+4+5+6+7).

28 is like the numbers 6 and 15 also a Hexagonal Number. As you can see in the picture below 28 is the fourth Hexagonal Number. As we have seen before a Hexagon is a Projection of a Cube so 28 represents a Cube in a Cube in a Cube. A Cube in a Cube is called a Tessarect or a HyperCube.

28 is a Hexagonal Number

The Number 15 is a Cube in a Cube called a Tessarect or a HyperCube

The first sentence in Genesis (“In the beginning Elohim created Heaven and Eearth“) contains 7 words and 28 letters. This indicates that the Creation Process was already in the 7th stage of the Tetraktys and in its 2nd Fractal Expansion,  the Birth of the Material Universe.

The sum of the entire verse is the 73rd Triangular Number. The prime Numbers 37 and 73 are geometrically related. They form the third and the fourth term in the sequence of Star Numbers (1, 13, 37, 73, 121).

Hexagon/Star pairs are closely related to Triangular numbers. Their product is always a Triangle, and they can be symmetrically generated from a Pair of Triangles.

Star Numbers are a Combination of Two Triangular Numbers

The Symbol of the Heart Chakra contains Two Triangles.

When a number represents a Geometric Structure it is called a Figurative Number.

Every possible figurative number is generated by the Triangle of Pascal.

The Fractal Sierpinsky Triangle is the Triangle of Pascal Modulo 2.

The Triangle of Pascal was known long before Pascal (re)discovered it.

It was known in Ancient India as the Meru Prastara and in China as  the Yang Hui.  Meru Prastara relates the triangel to a Mystical Mountain called Mount Meru. Mount Meru is also implemented in the Sri Yantra.

The Triangle shows the Coefficients of the Function F(X,Y))= (X+Y)**n. If n=0 F(X,Y)=1 and if n=1 F(X,Y)=X+Y so the Coeffcients are (1,1).

Pascals Triangle is a 2-Dimensional System based on the Polynomal (X+Y)**N. It is always possible to generalize this structure to Higher Dimensional Levels. 3 Variables ((X+Y+X)**N) generate The Pascal Pyramid and n variables (X+Y+Z+….)**N  generate The Pascal Simplex.

The rows of the Pascal’s Triangle add up to the power of 2 of the row. So the sum of row 0 is 2**0 and  the sum of row 1 is 2**1 =2.

The Sum of the  rows of the higher n-dimensional versions of the Triangle is n**N where n is the Amount of Variables and N the level of expansion. So the Sum of Pascal’s Pyramid (3 variables X,Y,Z) is 3**N.

Triangle of Pascal

The most interesting property of the Triangle is visible in the Diagonals.

The First Diagonal contains only 1′s. The Ones represent Unique Objects. They are the Points in the Tetraktys.

The Second Diagonal contains the natural numbers. These Numbers are used to Count Objects that are The Same. The Natural Numbers are the Lines that connect the Points. The Natural Numbers are the Sum of the previous Ones.

The Third Diagonal contains the triangular numbers. The Triangular Numbers are the Sum of the previous Natural Numbers.

This pattern repeats itself all the time.

The Yang Hui is an ancient Chinese version of the Triangle of Pascal. This Triangle contains Nine (3x3) Levels.

The Fourth Diagonal contains the tetrahedral numbers (Pyramid Numbers) and the Fifth Diagonal, the pentatope numbers.

Fermat stated that Every Positive Integer is a Sum of at most three Triangular numbers, four Square numbers, ﬁve Pentagonal numbers, and n n-polygonal numbers.

The Tetrahedron with basic length 4 (summing up to 20) can be looked at as the 3-Dimensional analogue of the Tetraktys.

A Tetrahedral Number represents a 3D-version of the Tetraktys.

The Diagonals of the Triangle of Pascal contain every Possible 2-Dimensional Figurative Number (and Structure).

These Numbers are Projections of Higher Dimensional Numbers and Higher Dimensional Structures.

The Higher Dimensional Versions of the Triangle (the Pascal Pyramid, The Pascal Simplex) contain these structures.

The Rows of the Triangle Sum to the Powers of Two (2 Dimensions). These Powers control the Levels of Expansion.

Every 7th step the Fractal Pattern of the Triangle repeats itself on a higher Level.

The Figurative Numbers are the Geometric Shapes that are created by the Lines of the Natural Numbers that are connecting the Points of the One.

Pascal’s Triangle also contains the numbers of the Fibonacci Sequence (“The Golden Spiral“).

When we take the Modulo 9 (the Digital Root of Pythagoras) of the Numbers of Fibonacci a repeating patterns of 24 steps shows itself that can be represented by a Star Tetrahedron or Stella Octangula. The Star Tetrahedron is a Three Dimensional Star of David.

the Fibonacci Numbers as a Cube.

The StarTetrahedron, shows the Pattern behind the Sequence of Fibonacci.

Every Figurative Number N is the Sum of the Figurative Numbers N-1.  Every Geometric Shape is a combination of all the Previous Geometric Shapes.

This means that Every Geometric Shape is in the end The Sum of the Sum of the Sum of  …. Triangels, Trinities (Elohim) or Triangular Numbers and therefore an Extension of the Tetractys of Pythagoras.

The Expansion of the Whole is a (Fractal)  Combination of Combinations.

The Triangle of Pascal is related to the so called Binomial Theorem which is used in Combinatorics and Probability Theory to describe the Amount of Combinations of a Set of  Objects.

The rows of the Triangle of Pascal also shows the Bell Shaped Pattern of the Normal Distribution.

The Probability Distribution of the Triangle of Pascal converges to the Normal Distribution because of the Central Limit Theorem. Every Row has a Mean of N/2 and a Variance of (N**1/2)/2 which means that with every new row the Mean and the Variance become Bigger and Bigger.

The Triangle of Pascal and therefore the Figurative Numbers describe Everything that is Possible but every Expansion of the Triangle is less Likely to Occur.

The Triangle of Pascal Mod 3 represents the Tetraktys in the Tetraktys in the .....

Because of the Fractal Expansion/Contraction Pattern The Cube of  Space, related to the Element Earth,  explains Everything there is to Know on Our Level of Existence, Mother Earth.

The interesting part of the Figurative Numbers is that they representent Visual Patterns with which we can Reason.

We don’t need complex formulas because we can See what is Possible.

The interesting part of the Triangle of Pascal is that we can See that the Complex Figurative Structures are created out of a very Simple Structure, the Triangle.

If we want to understand our Reality we have to begin with looking at the Beginning and not start somewhere in the Middle.

If we look at the Fractal Expansion Pattern of the Triangle we See that Every new Stage is an Expansion Out of the Middle.

The Expansion of the Human, the Next Step in our Evolution,  is therefore an Expansion Out of the Heart, the Balance of Father Sky and Mother Earth.

Life is not only about Me and the Other.

Life is also about the Relationship between Me and the Other.

If we don’t Collaborate the Next stage in our Evolution will never happen.

The Content of the Sepher Yesirah

About the Cube of Space and Psychology

About the Sepher Yesirah and the I Tjing

A correspondence table of the Cube of Space

About the Sri Yantra and Plato

About the Lo Shu and the I Tjing

All kind of strange relationships between Triangular Numbers

A website about Mystical Number Theory

About the Golden Spiral and Plato

About Pascal’s Triangle and the Normal Distribution

A complete course in elementary Number Theory

About the Psychology of the Cube of Space

About the Tetraktys and the Zodiac

About the Process Theory of Paul Young

Mysteries of the Equilateral Triangle

About Visual Patterns in Number Theory

About Pascal’s Triangle and Cell Division

### About the Sum of Things

Thursday, September 22nd, 2011

The Lo Shu is a Chinese Model for Time and was used to design Cities, Temples, Cycles and Calendars.

The Lo Shu is a 3×3 Magic Square created by Fu Xi, the Founder of Chinese Civilization. Fu Xi lived around the time of the Great Flood.

The Magic Square was carried by a Turtle. The Turtle is the symbol of the Constellation Orion. Many Ancient Cultures claim that our Ancestors came from this constellation. The Square of Lo Shu is also referred to as the Magic Square of Saturn or Chronos (Time).

The Duality of Fu Xi and his Sister, The Snake of the Kundalini contains 3.5 (7/2) Twists. 2x3.5 = 7. 7 = (-1,-2,-3,0,1,2,3).

According to Chinese Creation Myth  the world started with 無極 (wuji: nothingness), the Tao. Out of the Tao, the Egg of the Zero (0) and the Supreme Pure One (1) emerged. In the next step the Trinity (3), the Three Pure Ones, was created.

The Tao produced One; One produced Two; Two produced Three; Three produced All things (Lao Tsu, Tao Te Ching, 42).

The Sri Yantra contains 43 Triangels and is generated by the Bronze Mean.

The Trinity is represented by a Triangle. A combination of Two Triangels is a Square. The combination of Two (2) Triangels is also the Symbol of the Heart Chakra, the merge of Heaven, the Upper Triangel and Earth, the Lower Triangel.

The most complex Yantra (“machine”), the Sri Yantra, contains 43 Triangels. The Sri Yantra is determinated by the Bronze Mean. The Bronze Mean is a generalization of the Sequence of Fibonacci based on the Trinity. The Bronze Mean generates Quasi-Crystals and Penrose Tilings.

The Square is a combination of two Triangels. It is represented by the Star of David, the Symbol of the Heart Chakra.

The Lo Shu Magic Square not only shows the Connection between the Four Forces that came out of the Two Forces, Expansion and Compression, it also shows the connection between Heaven and Earth.

In the Magic Square of Lu Shu the Even Numbers (2,4,6,8 = 2 x (1,2,3,4 = 2×2)) are Black(Yang, Male). The Odd Numbers (1,3,5,7,9 = 3×3) are White (Yin, Female). The Sum of all the Numbers is 15 (3×5).

The Even Numbers are situated at the Corners of the Square.

When the numbers of each row are multiplied (8x1x6; 3x5x7; 4x9x2) they together total to 225 ((3×5)x(3×5)) — as do those of the columns (8x3x4; 1x5x9; 6x7x2).

The Magic Square is dominated by two numbers (2**1, 2**2, 2**3)  and (3, 2×3, 3**2), It contains four Prime Numbers (1,3,5,7). It also contains the first Perfect Number 6 (1+2+3 =1x2x3). 6 is also a Harmonic Divisor Number and has a Harmonic Divisor 2. The Sixth Day in the week is the day of Saturn, the God of Time.

The sequence 1,2,4,8, 1, 3,6,9, 27 is called the Tetrad of the Pythagorians and was, according to Plato, used by the Demiurg, the Creator of our Universe, to create the Soul. The Soul is a mixture of the Same (1), the Different (2), and Existence (3). The Tetrad was used to define the Harmony of the Spheres.

When you connect the numbers of the Lo Shu a pattern appears that is called the Seal of Saturn or the Dance of Yu.

The Seal of Saturn or the Dance of Yu

The Sum of Magic Squares is a Magic Square and the product of a Magic Square M with a number a is again a Magic Square aM. Both Rules can be used to construct a Magic Square out of simple Binary Squares called Magic Carpets. One of these Magic Carpets is the Identity Matrix with One on the Diagonal.

Every Magic Square is related by Rotation and Mirroring to Eight other Magic Squares. This means that the central Lo Shu Magic Square with the Five in the Middle is related to Eight other Magic Squares with the other Numbers in the Middle. One of them contains the Nine and is called the Well.

The Lo Shu and the Well

The Sum of numbers of the Well is 18 (2×9) and the sum of all opposites across the 8 Spoke Wheel = 9 (9+0, 8+1, 7+2, 6+3, 5+4). 2×9 means that there are Two Complementary Magic Squares. One Magic Square is Rotating with the Clock and the other Against the Clock.

The Well is an Expansion and Contraction Cycle based on Powers of 2.

The Nine is the most important Number in the Magic Square. The Nine comes back in many ancient mythologies like the Egyptian Pesedjet and the Greek Ennead.

When you add the separate numbers of a number (54 = 5+ 4 = 9) the result is the Remainder when you divide the number by 9.  This number is called the Modulo 9 number.

The Powers of 2 (= Expansion) Mod 9 creates a pattern (1,2,4,8,(16)7,(32)5,(64)1,(128)2,…). This pattern is reversed when we divide by 2 (=Compression) (1/2 (0,5 =5), 1/4 (0,25 = 7), 1/8 (0,125 =8), 1/16 (0,625 = 4), 2,1).

The Expansion and Compression Pattern shows that the same pattern repeats itself when the Magic Square is expanded or compressed Six (6) Times. The 7th Step is the Same as the First Step.

After 2**6=64 (8×8) divisons of the cell a Specialized Cell is created, the stem cell. The same happens in Music.  The Game of Chess (Chiu King in China) with its 8×8 = 64 = 2**6 Playing Board contains every possible situation of the Game of Life.

The Two-Pattern does not contain the numbers 3,6 and 9. The number 9 is repeated every time when the pattern is expanded and compressed which means that Nine is the same as the Identity, the One (1+9 = 1). 3 and 6 Oscillate (1x2x3=6 and 1+2+3 =6) and  therefore represent the Duality (3+6 = 9, = the One).

If we combine the Expansion and Compression patterns a Torus (a Rotating Circle, The Wheel of Time) appears. The Zero (the Void) is in the Center and Contains the Vortex. The Vortex represents another Cycle (Black/White Hole, 3,6,9, the Wheel of Order) in which every structure/pattern is destroyed to start All Over Again.

Lo Shu Torus

If we subtract the numbers in the Square by the number of the Center, 5, a highly Symmetric Pattern emerges. This pattern shows that there are Two Mirror Universes both containing four Forces.

When we transform this pattern from Modulus 9 to Modulus 3 (the Trinity) we see that the number 4 equals 1 and represents a new beginning of the Cycle.

 4 9 2 -5 -1 4 -3 3 5 7 -5 -2 0 2 8 1 6 -5 3 -4 1

The Magic Square with the Zero in the Middle is surrounded by two Sequences of 123(4) with different Signs. The Signs are arranged in a Cyclic Pattern (- -, + -, + +, – +) that contains the Four Permutations of – and +. The Sum of the ++ and the — is -3 and +3, the Trinity. The Sum of +- en -+ is the One.

The Cyclic Pattern of + and – represents the Prisoners Dilemma that is solved by the Strategy of Tit for Tat. This strategy is applied by the Trickster, the God of Paradox, a concept of God in which God creates games that create games.

When you use the basic Addition and Multiplcation Rules of the Magic Square many important numbers and patterns  appear that are related to Ancient CalendersSymbols and Cycles.

The Lo Shu Magic Square appeared a few years ago when I wanted to know more about Acupuncture.  Acupuncture is related to the Sheng Cycle and the Sheng Cycle is related to the Lo Shu. A Magic Square is the perfect way to visualize the Constraints that control the Four Forces in our Universe.

Soon after this discovery I found out the Lo Shu is related to the Trinity and the Bronze Mean which is an abstraction of the Fibonacci Sequence. The Bronze Mean generates the Sri Yantra and so called Quasi Crystals. Quasi Crystals explain Acupuncture.

A few weeks ago I started a discussion with Kim Veltman. Kim Veltman is the director of the Maastricht McLuhan Institute. He is researching Ancient Architecture and wrote a very interesting document called “Alphabets, Elements and Cosmologies”.

Ancient Alphabets are related to Numbers and Tones so Ancient Texts also represent Numerical Patterns and Music. You want be surprised to read that the Lo Shu showed itself in every Text Kim Veltman has analysed.

His document started a new inquiry into the pattern of the Lo Shu. During this research I detected the book “the Sum of Things” by Paul Martyn-Smith. Paul Martin-Smith worked together with Lee Burton.

This blog contains just a litte bit of the tremendous amount of information Paul Martin-Smith & Lee Burton have found about the essence of the I Tjing which is contained in just one simple 3×3 Magic Square.

If you want to know more the only thing you can do is to buy this book and start reading.

The Calculations  behind the Number of Things

The Relationship between the Lo Shu and the I Tjing

About Saturn, the Son of the Sun

How to construct Magic Squares out of Binary Squares (Magic Carpets)

The Sistine Chapel (Rome) and the Seal of Saturn

About the Relationship between the Lo Shu and Music

About the History of the Lo Shu

About the Wheel of Eternal Order and the Wheel of Time

About the Relationship between the Lo Shu and Chinese Alchemy

About Plato, Pythagoras and the Lo Shu

About the Lo Shu Magic Square and the Chinese Domino Game