Posts Tagged ‘Pascal Triangle’

About Intelligent Design

Tuesday, April 17th, 2012

Thermodynamics, the Science of Heat, became an important science in the 19th century when physicists started to improve the Steam-Machine. This blog is about a recently discovered law of Thermodynamics called the Constructal Law.

The Constructal Law describes How Intelligent Systems emerge out of the Laws of Thermodynamics.

Heat is Random Motion

According to Thermodynamics the evolution of the universe has been characterized by an on-going transformation from a simple, restricted, highly condensed, homogeneous state (the Big Bang) to an increasingly complex, widely dispersed, dynamic, multipotent, granular diversity.

Energy always moves from High to Low Temperature. This is the main reason why most of the energy of Earth comes from our Sun.

With help of the Electro-Magnetic radiation of the Sun we call Light, plants, algae and bacteria transform simple molecules into complex organic structures (Biomass = Potential Chemical Energy) and store the energy of the Sun. Bacteria and Fungi brake up these structures, produce heat, reduce the complexity, and close the Cycle.

The Input of the Heat of the Sun has to be compensated by a comparable Output of Radiation otherwise Earth would be burned instantly. The difference between the Input and Output of Energy is transformed into Movements on Earth.

The Earth’s climate is a huge flow system. It circulates air and water from the tropics to the poles and back. These flows develop as air and water move from hot to cold regions, a result of variation in the heating of the Earth’s surface by the sun.

The Earth is not only heated by the Sun. The hot kernel of the Earth is surrounded by a small crust. At certain places on Earth and the Sea the rotating pool of the magma-ocean produces a lot of energy.

The Earth with its solar heat input, heat rejection, and wheels of atmospheric and oceanic circulation, is a heat engine without shaft. Its maximized mechanical power cannot be delivered, but is instead destined to dissipate through air and water friction and other forms of heat loss. It produces maximum power, which it then dissipates at a maximum rate.

The earth is wasting a lot of energy because the transmission of  most of its energy is not contained in a Tube-like-system.

The system maximizes the sum of the work done driving the planetary circulation, and the heat rejected back to space at the cold end of the heat engine.

When the speed of the planetary circulation is low, so are the turbulent losses. As speed increases, up to a certain point the sum of work done (circulation speed) and heat rejected also increasing until the turbulence starts to interfere with the circulation and  actually decrease the total of work done and heat rejected. That is the point, “the edge of turbulence”,  at which the system will naturally run.

Every flow system tries to find the perfect balance between movement and friction.

The interesting point is that friction is movement in opposition to itself.

Clouds on the Edge of Turbulence

The biomass on Earth is direct or later (Oil, Gaz, Coal) consumed by all kinds of organisms. In Food-Webs Organisms are consumed by other organisms.

We, the Humans, describe the interaction of the organisms in the Food-Webs as a Struggle for Live, a Survival of the Fittest, but the flow of energy in the food webs could also be seen as an adaptive chain of chemical reactions that are performed in self-moving, self-reproducing chemical factories.

Energy is Potential Movement. The potential energy stored in the biomass on Earth is transformed into circular movement (kinetic energy). During this transformation Heat is produced. Heat is the waste produced by a restrained non-effective movement.

The energy of the Sun produces all kinds of movements on Earth. Examples are river-delta’s, our respiratory system and the weather-system.

Flow Systems that move from one  to many always look like a leaf or a tree

Flow Systems that move from One to Many always look like a Leaf or a Tree. Left a River Delta is shown. Right a picture of the Lungs is displayed.

According to Constructal Theory the generation of design (configuration, patterns, geometry, shape, structure, rhythm) in nature is a physical phenomenon that unites all animate and inanimate systems. Design in Nature always shows itself as systems that flow and improve themselves.  Systems improve themselves because the movement from chaos to order takes many discrete steps.

Biological organisms are flow systems. River basins are flow systems. A Tree is a flow system. The Traffic System is a flow system. The City of Paris is a flow system. The Financial Market is a flow system. The Internet is a flow system. A Proces-Manager is a flow system.

Flow Systems

All these flow systems must be structured (architecturally designed) in such a way that the currents that flow (Life Energy, Water, Air, Electricity, Gaz, People, Cars, Trains, Airplanes, Content, Money) increasingly get to where they need to go. This gives shape and structure to everything that evolves over time.

Flow Systems always remove the Resistance to the Flow themselves.

Constructal Law is formulated as follows: “For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it.”.

Nature always provides Resistance to Change (Heat, Erosion, Friction) and this resistance is provided by the Systems of the Past. The current Order resists the potential Chaos of Free Moving Energy.

The Paths of Change of our Universe always move in the same way. The pattern looks like a Moebius Ring, the number 8, because this is the only way to cope with the polarity of the movements. In a Moebius Ring two Opposite Cyclic Movements are perfectly United.

A Moebius Ring is a 2D-structure that operates in 3D because it needs a Twist to operate.

According to Constructal Theory the Energy of the Sun creates two types of Systems called Engines and Brakes.

Engines are the most effective and efficient states in the evolution of our Universe. They have found the best way to cope with the resistance that is provided by the Brakes. Brakes are the old systems like trees, forests and natural or constructed dams that slow down the movement so the potential energy can be stored.

If the Energy of the Sun is not stored,  the Self-Reproducing Systems are unable to evolve to a higher level of organization.  If every Resistance would be gone the Flow-Systems would Flow in a Radial Pattern to every Direction possible.

If we remove the Natural Brakes (Forests, River-delta’s) the Expansive Flow-systems will take over control and the Smart Highly Improbable Next Generation Brake Systems, the Organisms would die.

We, the Humans, are here to Resist the Engines that give the Flow-Systems the opportunity to make Everything the Same. Every Thing has to become the Same because the Universe does not discriminate. Every configuration has the same chance to be alive.

Dependent on the type of flow there are standard solutions to the dualities that are provided by the eternal battle between engine (movement) and brake (not-movement).

Paris is a Flow System

The standard solution can be calculated using the Laws of Thermodynamics. Tree (or Leaf)-shaped fractal systems are for example the best solution for a One-to-Many (or Many-to-One) Flow. The Flow Systems always converge to these Shapes.

Our Earth is rotating, spinning and moving around our moving Sun. The influx of Energy on every area on Earth is changing all the time.

This means that every Flow-System on Earth has to adapt to the short term and long term changing Energy Input of the Sun.

The stability of a Flow-System, the Resistance to Change, is highly dependent on the Resilience of the System.

According to the theory of Panarchy the Resilience of a System is dependent on the Potential and the Connectedness of a System.

In  the Exploitation/Conservation phase of the Adaptive Panarchy-Cycle the Constructal Engines become more efficient (producing more Potential) and Connected. This means that the amount of possible paths in the River-delta of the System is reduced. This amount determinates the resilience of the system.

At the state of maximum production and maximum connection (conservation) the system is spending all its potential at self-reproduction and is highly vulnerable to external disturbances. At a certain moment when a disturbance destroys the majority of the paths the system collapses (release).

Revolt: A small Wildfire can destroy a complete Ecology

The bits and pieces, the components, of the scattered ecology are reassembled sometimes in a different way (reorganization) and the system starts again most of the time on a higher level of organization.

Kleiber's law

Kleiber’s Law is an example of the Constructal Law. It shows the many levels of organization that have occured since the chemical factories of the bacteria started to work together. Kleiber’s Law can be extented to the level of the Community and the City.

The Panarchy Cycle is Self-Similar. It repeats itself on many Scales. The higher the scale the bigger the impact and the slower change takes place.

The Scales are interconnected. A huge disturbance on a lower scale (Revolt) can have an huge impact on a higher scale when both cycles are moving into the Release State.

A Slow moving High Level Cycle can help to restructure a fast moving cycle by providing the template of the way the components were originally assembled (its CommunityDNA) (Remember).

Constructal Law is a Law of Thermodynamics. Thermodynamics has evolved into a Statistical Theory called Statistical Mechanics. To calculate a probability you have to define the Space of all Possibilities of a System, the Phase Space.

When you don’t know what a system will look like it is impossible to calculate the Phase Space. To calculate the possibilities of Constructal Law you have to imagine the possible constructions of Mother Nature.

It is possible to play with Probability Theory without knowing the exact Phase Space. In this case you have to look at the probabilities of the probabilities. These probabilities are described by the Triangle of Pascal. Behind the Triangle of Pascal a new Triangular Structure shows itself. This pattern goes on and on until the primary Triangle of Creation, the Holy Trinity, is reached.

The Trinity is a combination of Desire (Mother, Female, Expansion, FLOW), Control (Father, Male, Compression, BRAKE) and the combination of both Expansion AND Compression also called the Void or the Tao. In Physics the Void is called the Vacuum or the Zero-Point-Field.

The Triangular Pattern behind the Triangle of Pascal

When we have moved back in time and have reached the state of the Trinity we are very close to the Big Bang. The Big Bang is a result of the Union of the Opposites (Yin/Yang, Male/Female, Matter/Anti-Matter, Expansion/Compression) that both came out of the Void.

The Vacuum is the Eternal Flow System of Creation. It contains everything that is possible even the impossible, the Empty Set.

The Tao is like a well: 

used but never used up.

It is like the eternal void:

filled with infinite possibilities.

It is hidden but always present.

I don’t know who gave birth to it.

It is older than God.

(Tao Te Ching 4)

According to Constructal Law every finite-size system must evolve in such a way that it provides easier access to the imposed currents that flow through it.

The Tao is the Channel of the Energy of Life. If Constructal Law is Right we have to give easier access to this Flow. We have to become the River to move to our final destination, the Sea of the Nothing.


A video about Constructal Theory

About the Higgs Field and the Big Bang

Why the Objects in the Universe are of Different Size

About Innovation and Thermodynamics 

About the Big Bang

About Friction

About the Big Freeze

About the Constructal Theory of David Deutsch

The Website about Constructal Theory

An Introduction into Constructal Theory

About Morphology

About Panarchy

About Kleiber’s Law

About Genetic Similarity Theory

Why a City is an Organism

About the Triangle of Pascal and the Holy Trinity

About the Holy Trinity

About Constructal Law and the Trinity

Attraction + Obstacles = Excitement 

About the Law of three & Gurdjieff



About the Vedic Square

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.

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 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

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 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.

File:Michelino DanteAndHisPoem.jpg

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.

About Indian Vastu Science

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.

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.


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

About the Trinity

About the Vedic Square

About the Tetraktys and the Lo Shu

About the Lo Shu

About the Harmonics of the Universe

About the Hyperdiamond

About the Void

About Harmonics and Entrainment

About Good and Bad Vibrations

About the number 24

About the Jitterbug of Buckminster Fuller

A Simulation of the Jitterbug Pattern (CUBIC WONDER)

About the Game of Chess

About Plato and the Sri Yantra

About the Rodin Torus

About Vastu Science

About Gematria

About Vastu Science and the Borobudur