Local energy-companies are part of a movement aimed at Autarky. Autarky is the quality of being self-sufficient.
The word Autarky comes from Greek αὐτάρκεια (autárkeia), itself from αὐτός (autós) = self and ἀρκέω (arkéo) which in its passive form means be satisfied, content or “Out of my own Power“.
The need to become self-sufficient has certainly to do with the current economic-, ecological- and energy crisis. People are afraid that the supply of vital energy and food will suddenly stop or become priceless. This is not the only reason.
Some people think we are wasting our money, food and energy by producing nondurable throw-away goods in which the packaging is sometimes more important than the content. They want to create durable products and services. This is not the only reason.
Some people don’t trust the suppliers of their food and energy, their politicians and/or the caretakers of their savings, the banks, anymore. They believe that their food is intentionally manipulated, their savings are wasted, their country is sold out to the rich and their environment is polluted for just one simple reason, Greed, the opposite of “Being Satisfied”.
The people of the autarkic movement want to recreate the necessary functions (bank, energy, food, work, healthcare, government) themselves in a cooperative way in their own living context.
The local cooperatives are now joining on a higher (mostly country) level and are in need of a shared (real-time)infrastructure. The current infrastructure does not support flows that move between many participants (so called peer-to-peer-networks).
The current system is a Hierarchic Top Down System containing Huge Highways that are connected to Big Production-facilities. This makes it impossible to consolidate, share and distribute resources peer-to-peer on a local level.
The Top Down System has created Global Markets where the local production is sold to the highest bidder.
The Global Markets are the playing field for advanced speculators that make a lot of money. After the deal is made the local production is transported to a far away destination. That is the reason why the Top Down Systems are integrated in world-wide Logistic Systems.
If we want to coordinate all the local initiatives we need advanced systems that Up- and download, control, predict, connect, share, balance and distribute vital local resources on the short and long term.
All of these fysical Flow Systems, that map 1 Resource-> on Many Resources or Many Resources->on 1 Resource, are based on the third Law of Thermodynamics called Constructal Law.
In the philosophy of the Stoics and Cynics of Ancient Greece the perfect human was not only arete (“all-around excellent”) but also independent of every need. Arete was associated with a full life as a citizen of a city.
Autarky was the ideal of a complete man who needed no one and no organization. This ideal can also be found in Zen Buddhism, Taoist practise, Early Christians (Nazarites), the Catholic Order of the Franciscans and the Protestant Quakers.
In reality it is almost impossible to be completely self-sufficient as a lonely human being. We need others when we are young, sick and old and we need others because we are specialized (“Talented”). Autarky is a possible aim for a small community.
The religious, cultural and political movements at the left and the right aimed at living the “good life” (“Living Slow“) are converging in the movement of the Cultural Creatives. The Cultural Creatives are the new Elite that is moving to the AND-state (Left AND Right).
They don’t believe in the old Competitive Ideologies (Socialist, Capitalist, Religious, Liberal, Nationalist, Conservative, Globalist, etc..) but want to focus on the Shared Collaborative Values between all Humans. We all need Love, Food and Shelter.
Autarky has always been a major political aim of countries. Autarky is about the possibility to control your own future and to become independent of other powers that could use your dependency to tell you what to do to get what you desperately want.
At this moment the shortage of fossil fuels, especially oil and gaz, is used to influence the policy of other countries and is the cause of local conflicts in the Middle East.
China is buying all kinds of vital materials to make sure their economy will not become dependent on others. If they will succeed and nobody doubts about that, they will become the economic controlling power of the world.
Reaching a state of local autarky, being satisfied with what your environment provides you, has become the aim of the wordwide sustainability movement.
A strive for (regional) Autarky pushes the need to innovate to a higher level because not all regions contain the right materials, knowledge or experience to realize everything everybody needs. To become fully Autarkic the Production and Consumption Systems in the Region have to be based on simple concepts and parts that are available everywhere. Simple concepts are concepts that are easely copied.
The only way to combine Simple and Complex concepts is to use the Concept of Self-Reference (Fractals). The only way to make use of parts that are available everywhere is to make use of the Simple Building Blocks of Nature. A good example is the use of algea in food processing.
The concept of Autonomy not only plays an important role on the many levels of human (personal, organization, country) and animal society (Competitive vs Collaborative (Ants)). It is also a major subject in the research of (Second-Order) Cybernetics (Control Theory).
Second Order Cybernetics is concerned with the Control of the System of Systems (the Meta-System). Hierarchy is not the solution to this problem because it leads to the logical problem of Infinite Regress (“Turtles on turtles on … all the way down”).
The Meta-System is controlling, reproducing and balancing itself by a Fractal Feedback-Cycle that moves Backwards (Consumption) and Forwards (Production).
A System is an Autarky if all its Levels are capable of Self-Reproduction.
If a human wants to become self-sufficient he has to accept and develop his own talent and find complementary people so he is able to create a Self-Reproducing Unit.
The smallest Self-Reproducing Unit in Human Society is the Family living in one House. This House has to produce more than enough Energy and the Garden has to produce more than enough food to support the Family.
If we cluster all these Self-Reproducing Families/Houses & Gardens in a small City the chances are high that the combined talents of the families will be able to supply every profession needed if the people of the city want to Live a Simple Life.
It is also possible that the surplusses and shortages of the families can be compensated by creating a Market on the level of the small City or one level higher, the Region.
An Autarkic Movement becomes an Aggressive Competitive Nationalistic Movement when the Citizens of the Country feel that their Country is plundered by outsiders and their Leaders promise to get it all back by expanding the Space to Live (“Lebensraum“) to facilitate everything that is needed.
This happened when Germany was preparing for War after the enormous reparation payments of 132 billion gold marks to the Winners of World War I.
The Autarkic Movement could also become a Peaceful Cooperative Social Movement like the Movement of Gandhi in India when the people accept the Good Life they already live and want to share the surpluses with others who are in a worse state.
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.
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 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.
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.
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).
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).
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 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.
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.
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.
Alberti regarded mathematics as the common ground of Art and the Sciences. “To make clear my exposition in writing this brief commentary on painting,” Alberti began his treatise Della pittura, “I will take first from the mathematicians those things which my subject is concerned”.
Alberti stressed that “all steps of learning should be sought from nature“. The ultimate aim of an artist is to imitate nature.
Alberti did not mean that artists should imitate nature objectively, as it is, but the artist should be especially attentive to beauty, “for in painting beauty is as pleasing as it is necessary“.
The work of art is according to Alberti so constructed that it is impossible to take anything away from it or add anything to it, without impairing the beauty of the whole.
Beauty was for Alberti “the harmony of all parts in relation to one another,” and subsequently “this concord is realized in a particular number, proportion, and arrangement demanded by harmony“.
Alberti’s thoughts on harmony were not new – they could be traced back to Pythagoras – but he set them in a fresh context, which well fit in with the contemporary Aesthetic Discourse.
One of the earliest Artists to produce a book on how to draw in perspective was Albrecht Dürer. As well as discussing geometric methods, he also illustrated his book with a set of woodcuts showing practical tools for accurate perspective drawing. Other Artists like Leonardo Da Vinci developed comparable tools.
About Projective Geometry
Projective Geometry formalizes one of the Central Principles of Perspective Drawing and of Human Perception: Parallel Lines Meet at Infinity.
In Euclidean geometry, constructions are made with Ruler (Line) and Compass (Circle). Projective Geometry only require a Ruler.
In Projective Geometry one Never Measures Any Thing, instead, one Relates one Set of Points to another by a Line.
Alberti was the first to ask what two pictures have in common if two drawing screens are interposed between the viewer and the object, and the object is projected onto both resulting in two different pictures of the same scene.
The basic idea behind Linear Perspective is simple: in every painting an artist creates a “floor” or area of the painting where the figures and/or objects will be placed. The floor ends at a horizon line, and the horizon line has a Vanishing Point or Point of Convergence on it.
The artist then draws parallel lines radiating from the vanishing point outward. Images closest to the vanishing point should appear smaller and closer together, and images farthest from the vanishing point should appear larger and farther apart, giving the impression of depth and space in the painting.
Pappus of Alexandria ( 290 – 350), one of the last great Greek mathematicians of Antiquity, proved that that given one set of points A, B, C on a line, and another set of points a, b, c on another line, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are also on a Line. This line originates at the Vanishing Point.
Later Desargues (1591-1661) proved Pappus Theorem for Triangels. He proved that the three vertices of one triangle a, b, and c, and those of the other A, B, and C meet in a third point, and that these three points all lie on a common line called the Axis of Perspectivity.
Finally 400 years later in 1803 Lazare Carnot found the answer to the Question of Leone Battista Alberti.
In his book “Géométrie de Position” he proved that the so called “Cross-Ratio” is always preserved in a Projection with One Point of View.
The Cross-Ratio is the Ratio of the Ratio of the Four Points A,B,C,D lying on a Line that intersects the Four Lines defined by a,b,c,d that orginate at the Vanishing Point O. Carnot proved that (AB)(CD)/(AC)(BD) = (A′B′)(C′D′)/(A′C′)(B′D′).
The Cross Ratio is not only the Cross-Ratio of the Four Points A,B,C,D. It is also the Cross-Ratio of the Four Lines and the Three Angles that originate out of the Vanishing Point. In this case (AB)(CD)/(AC)(BD)=Sin(AB) x Sin(CD)/Sin (AC) x Sin (BD).
This principle is called Duality. In 2D-Projective Geometry Connected Points and Lines, called an Incidence, are interchangeable.
There is a corresponding Duality in three-dimensional Projective Geometry between Points and Planes. Here, the line is its own Duality, because it is determined by either two points or two planes.
Two examples: 2D: Two distinct Points determine a unique Line and two distinct Lines determine a unique Point. 3-D: Three distinct Points not all on the same Line determine a unique Plane and three distinct Planes not all containing the same Line determine (meet in) a unique Point.
The 24 Permutations of A,B,C and D produce 6 possible values of the Cross-Ratio, depending on the order in which the points are given. If k = 1, the other Cross-Ratio’s are 0 and infinity. This happens when point A = point D. In this case the Geometric Entity is an Equilateral Triangle.
If k = -1, the other Cross-Ratio’s are -1/2 and 2. This is called a Harmonic Cross-Ratio. This happens when the Internal Ratio of AC determinated by B is Equal to the External Ratio of AC determinated by D. When k=-1/2 and 2 the Geometric Entity is a Square.
If the Four Points of the Cross-Ratio are on a Line or a Circle, then the Cross Ratio is a Real Number, otherwise the Cross-Ratio is a Complex Number.
About the Four Layers of Geometry and the Four Points of the Cross-Ratio
In 1872 Felix Klein, published a new Mathematical Research Program called the Erlangen Program under the title Vergleichende Betrachtungen über neuere geometrische Forschungen.
In this program Projective Geometry was emphasized as the Unifying Frame for all other Geometries.
Although lines in the Projective Plane meet in one point of Infinity Klein argued that there could be two points of Infinity if the Projective Plane was a Surface Closed in Itself.
When we look at the Origin of Projective Geometry, the Artist painting A Sphere, Earth, on a Flat Surface, it is not difficult to realize that this Closed Surface is a Sphere.
Geometry is now divided into Four Layers. Each Layer adds a set of Assumptions that creates certain Invariants for that Layer.
An invariant is a property of a configuration of Geometric Entities that is not altered by any transformation belonging to the specific layer.
A transformation is an operation applied to a Geometric Entity. The most common transformations are Translation, Rotation and their Combinations (Reflection).
What we will see is that the Four Points of the Cross-Ratio are highly related to the Four Layers. In Every Higher Layer One Point of the Cross-Ratio is moved to the Internal Landscape, the Mind, of the Observer. This principle is called “As Within so Without“.
The First and most fundamental Layer is the Layer of Projective Geometry. This is the Layer of Human Perception and is invariant under the Cross-Ratio also called a Moebius Transformation.
The Second Layer is called Affine Geometry. In this layer Parallel Lines are preserved and the Assumption of a Plane at Infinity, the Horizon, is created. This Plane generates Parallel and Orthogonal Relationships between Geometric Entities by placing One of the Four Points of the Cross-Ratio in the Plane at Infinity.
Affine Geometry is Euclidean Geometry with congruence (something is the same when Shape and Size are the Same) and a metric (a definition of a Distance) left out. An affine transformation preserves straight lines and ratios of distances between points lying on a straight line.
This the layer of the Emotions, “the harmony of all parts in affinity to one another” of Leone Batista Alberti. In Affine Geometry the Frame of Reference of the Painter, the Floor, is created. On this Floor, the Horizon, the Plane at Infinity is drawn. From the Point of View of the Observer, it is not possible to Judge Distance or the-Sameness because a given Visual Object may be Large and Far away or Small and Close.
The Third Layer, Metric Geometry, adds the concept of Distance (Metric, Scale) to the Affine Layer. A Distance is a relation between Two Points of the Cross-Ratio. This is the Layer of the Painter, Creator, Observer, who creates his own Distance to the Distances he is Painting in the External Frame he is looking at.
Since Projective Space is the Space of Actual Perception, the true function of Metric Space is the Coordination of Various Perspectives. This is illustrated in Perspective Drawing, in which there are always Two Perspectives being Coordinated—the Vanishing Point and a Point just behind the Eyes of the Observer. Each of these is an instance of the One Point at Infinity.
The last, Fourth, Layer is the layer of Euclidian Geometry. In Euclidean Geometry Distance is Fixed (not Relative) and Scaled to a Unity (Meters). This leaves us with just One Point of the Cross-Ratio, the Unity, the One.
Between 1926 and his death in 1983 Alfred Tarski worked on the Axiomatization of Euclidean Geometry. In Tarski’s system there is only one type of Object: the One, The Point.
There are just two Geometrical notions between Points: the Ternary Relation of “in-Betweenness” and the Quaternary Relation of “Equidistance” or Congruence. Betweenness captures the Affine aspect of Euclidean Geometry; Equidistance/Congruence, its Metric aspect.
This is the Layer of the Human Expectation who has not Distanced Itself and is Looking Outside Through a Small Hole at the Future. The Expectation considers the Future as an Extension of the Past (Memory).
In Euclidian Geometry we have to Move Outside to the Objects to Measure their Distance and the Sameness with a Pre-Defined Unity.
If we don’t move to the Object we will measure an Illusion.
About Space measured by Time and Space measured by Space
The Ancient Sumerians knew that the Length of one side of a Hexagon is the same length as is the Radius of the Circle that circumscribes this Hexagon (See Pascals Theorem).
They also knew that the Stars of the Constellations of the Zodiac shift Counterclockwise, at the rate of 72 years/degree, because of the Precession of the Earth’s axis. One Wobble takes 25.920 years.
The length of one side of the Earth Hexagon is therefore the Distance traveled by the Constellations of the Zodiac along the horizon during 25.920/6=4,320 years.
They then subdivided this distance by 7,200 which produced the Royal Mile that was subdivided into 1,760 Royal Cubits.
Before 3.117 BC the complete Earth was covered with a uniform Geodesic System that was based on the Rotation and Precession of the Constellations of the Zodiac. This System was implemented in the Megaliths.
At that Time Space was measured by Time.
The Physical System of the Megaliths was destroyed by the Great Flood of 3.117 BC but the Metric System survived thousands of years until the French Revolution.
On the 8th of May, 1790, Charles Maurice de Talleyrand at the end of the French Revolution, proposed before the National Assembly in Paris a Change to a Decimal Measurement System.
The Academy of Science recommended that the new definition for a Meter be equal to 1/10,000,000 of the distance between the North Pole and the Equator, and this was accepted by the National Assembly in 1791.
From that Time on Space was measured by Space.
Why the Future contains Infinite Possibilities
A Painter or a Human always looks Forwards in Space. If there are Parallel Lines visible they all Converge to a Vanishing Point at The Horizon. The Horizon is a Parallel Line at Infinity.
We all know that the Parallel Line of the Horizon is caused by the fact that the Earth is a Moving Sphere. This Moving Sphere moves around other Spheres that move around other Spheres.
The Rotation of the Spheres in Space is used as a Clock. Every time when the Cycle of the Spheres repeats itself we add one unit to a standard Time-Measure. One Sun Cycle is named a Year. One Cycle of the Precession takes 25,920 Years.
We Measure our Time by the Cyclic Movement of Objects in Space. The Cyclic Movements caused by the Rotations of Objects around Objects (around Objects) is really a Rotating Rotating Rotation, a Spiraling Spiral.
We don’t live in the Sphere but we live and Move on the Surface of a Moving Sphere. That is why our Geometry is an Elliptic Geometry (or Projective Geometry).
Euclides and Pythagoras were aware of Projective Geometry (See also Pappus) but the Scientists of the Enlightenmentstarted to interpret their Theory with a Different Eye. They saw Numbers in a completely different way the Greek did.
The Scientist disconnected Number from Magnitude (Form) and created an Abstract Static Number Theory and an Abstract Static Theory of Physics. The Movement of our Universe, the Ether, came to a stilstand.
The Theory of Perspective Drawing, based on our Real Perception of Reality, is the same theory that generated Projective Geometry. We are now back to Normal again.
He reinvented Physics and Projective Geometry in his own way, Calculated all the well known Physical Constants and based his Theory on a simple well know Assumption that Space and Time are Reciprocal (A Ratio of Ratio’s , the Cross-Ratio) because Velocity= dS(pace)/dT(ime).
Moving Movement is the Essence of our Reality.
What Larson calls “Motion” is the Ether Wind (or the Higgs Field) the Velocity which was measured in the North-South (Z) direction at 208 km/s in 1933 by Dayton C. Miller.
Einstein believed the results of the Michelson-Morley Experiments in 1887 that “proved” that the Moving Ether was non-existent. Now we know that the experiments were wrong. Einstein also did not believe that Space was Expanding. At this moment the Expanding Universe is confirmed by many observations.
When Space Expands, Time Compresses and fills up the Space to Keep Balance. Time Scales Space.
Space moves to Infinity. Time Moves to the Inverse of Infinity, Zero. Space is measured by the Linear Visible Scalar Number-System. Time is Measured with the Rotational Invisible Imaginary Number System.
Expanding Space is Linear Motion. Time is the Spiraling Motion of the Vortex.
Space is Yang and Time is Yin. Yin is feminine, Curved, Rotational, smooth or cold. Yang is masculine, Straight, Linear, rough or hot.
Space and Time are Complementary Duals that Move Around the Void.
If we look at the Human Perspective we now are able to understand what Time and Space Really are.
When we Look Forward we Experience Space. Time is always Behind our Back.
Human Senses and Conventional, Scientific Equipment can only Look Forward and therefore Measure Space (Distance) and the Change of Space (Velocity, Acceleration).
This is a Limitation of our Physical Sensory System, which evolved to measure Space, Scaled by Time, to produce what we Perceive as Causality–a Linear Ordering of Events.
Without the Cause-and-Effect System, the Sensory World of Space would just become Chaos.
We have lost the relationship between Number and Form or Number and Magnitude as the Ancient Greeks called their Forms.
A few years ago a Revolution in Mathematics and Physics has started. This revolution is caused by Geometric Algebra.
In Geometric Algebra the Ancient Theories of Euclid and Pythagoras are reevaluated.
Numbers are Scalar (Quantum) Movements of Geometric Patterns and not Static Symbols of Abstractions that have nothing to do with our Reality.
Movements and not Forces are the Essence of Physics.
The basic rule Movement = Space/Time (v=s/t) shows that Time and Space are two Reciprocal 3D-Spaces. Our Senses Experience Space and not Time.
The Simple Rule N/N=1/1=1 balances the Duals of Space and Time. One Unit Step in Space is always Compensated by One Unit Step in Time.
Geometric Algebra has a strange relationship with Pascals Triangle. This Triangle, also called the Binomial Expansion, contains all the Possible Combinations of two Independent Variables. Our Universe is a Combination of Combinations exploring Every Possibility.
The last and perhaps most important Discovery in Mathematics called Bott Periodicity shows itself in Pascals Triangle.
Bott Periodicity proves that we live in a Cyclic Fractal Universe, the Wheel of Fortune, that is Rotating around the Void, the Empty Set. The Empty Set contains Every Thing that is Impossible in our Universe.
This blog is not a Scientific Article. I have tried to connect the Old Sciences and the New Sciences in my own Way.
It contains many links to Scientific Articles and even Courses in Geometric Algebra.
So if you want to Dig Deeper Nothing will Stop You.
About the One and the Dirac Delta Function
Every Thing was created out of No Thing, the Empty Set, ɸ, the Void, the Tao. The Empty Set contains 0 objects.
The Empty Set is not Empty. It contains Infinite (∞) Possibilities that are Impossible.
Every impossibility has a probability of 0 but the sum of all possibilities (1/∞=0) is always 1. In the beginning ∞/∞ =1 or ∞x0=1.
This relationship is represented by the Dirac Delta Function. It is used to simulate a Point Source of Energy (a Spike, an Explosion) in Physics.
The Delta is reprented by the Symbol Δ, a Triangle. The Delta is called Dalet in the Phoenican and Hebrew Alphabet. Daleth is the number 4 and means Door.
The original symbol of the Delta/Daleth contains two lines with a 90 Degree Angle. Two orthogonal lines create a Square or Plane.
The Dirac Delta Function is defined as a Square with an Area of 1, a Width of 1/n and a Height of n where n->∞.
The Dirac Delta Function is a Line with an Area of 1.
In the Beginning a Huge Explosion took place that created the Universe.
The Dirac Delta Function δ (x) has interesting properties: δ (x) = δ (-x), δ (x) = δ (1/x). It has two Symmetries related to the Negative Numbers and the Rational Numbers.
When we move from 2D to 1D, the Number Line, the Delta Function becomes the Set of the Numbers N/N =1.
The Monad (1) of the Tetraktys of Pythagoras, the Top of the Triangle, was created by Dividing the One (1) by Itself without Diminishing itself. The Monad (1/1=1) is part of the 1D Delta Function.
Creation is an Expansion of the 1/1 into the N/N, adding 1/1 all the time, until ∞/∞ is reached. At that moment every Impossibility has been realized.
To move Back to the Void and restore the Eternal Balance of the One, Dividing (Compression) has to be compensated by Multiplication (Expansion).
At the End of Time N/M and M/N have to find Balance in the N/N, move Back to 1/1, Unite in the 0 and become The Void (ɸ) again.
About the Strange Behavior of Numbers
The big problem of the Numbers is that they sometimes behave very differently from what we Expect them to do.
This Strange Behavior happens when we try to Reverse what we are doing.
It looks like the Expansion of the Universe of Numbers is Easy but the Contraction creates many Obstacles.
It all starts with the Natural Numbers (1,2,3,).
When we Reverse an Addition (Subtract) and move over the Line of the Void Negative Numbers appear. Together with the Natural Numbers they are called the Integers.
The same happens when we Reverse a Division and the Fractions (the Rational Numbers) (1/3, 7/9) suddenly pop up.
An Integer N is a Rational Number divided by 1 (N/1).
The Integers are the Multiples of 1, the Fractions are its Parts.
Numbers behave even stranger when we want to Reverse a Repeating Repeating Addition (Irrational Numbers) and want to calculate a Rational Power (2**1/2).
The Complex Numbers (or Imaginary Numbers), based on the Square Root of -1 called i, are a combination of the Negative Numbers and the Irrational Numbers.
Irrational Numbers ( the Pythagorean Theorem), Fractions (a Piece of the Cake) and Negative Numbers (a Debt) are part of our Reality but the Strange Number i represents something we cannot Imagine.
About the Duality and the Expansion of Space
In the beginning the only One who was in existence was the 1.
When the One divide itself again the number -1, the Complement of 1, came into existence.
1 and -1 are voided in the No Thing, the Empty Set, 0: -1 + 1 = 0.
The Two, the Duality, both started to Expand in Two Opposite Directions (<– and +->) both meeting in the ∞/∞. This expansion is what we call Space.
Space is a Combination of the Strings S(1,1,1,1,1,…) and -S = (-1,-,1,-,1,-1,…) where S+S=(0,0,0,0,0,0,…).
The Expansion pattern of Space is a Recursive Function S: S(N)=S(N-1)+1 in which + means concatenate (or add) the String “,1”.
An Addition X + Y is a concatenation of S(X) and S(Y). A Substraction X-Y is a concatenation of S(X) and -S(Y). In the last case all the corresponding combinations of 1 and -1 are voided. (1,1,1,1)-(1,1,1)=(0,0,0,1)=(1).
Multiplication XxY is Adding String S(Y) every time a “1” of S(X ) is encountered: 111 x 11 = 11 11 11. Dividing X/Y is Subtracting S(X) every time a “1” of S(Y) is encountered:.111 111 1/111=11 1/111. In the last example a Fraction 1/111 appears.
According to Euclid “A Ratio is a sort of relation in respect of size between two magnitudes of the same kind“.
A Magnitude is a Size: a property by which it can be compared as Larger or Smaller than other objects of the Same Kind. A Line has a Length, a Plane has an Area (Length x Width), a Solid a Volume (Length xWitdth x Height).
For the Greeks, the Numbers (Arithmoi) were the Positive Integers. The objects of Geometry: Points, Lines, Planes , were referred to as “Magnitudes” (Forms). They were not numbers, and had no numbers attached.
Ratio, was a Relationship between Forms and a Proportion was a relationship between the Part and the Whole (the Monad) of a Form.
Newton turned the Greek conception of Number completely on its head: “By Number we understand, not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same Kind, which we take for Unity”.
We now think of a Ratio as a Number obtained from other numbers by Division. A Proportion, for us, is a statement of equality between two “Ratio‐Numbers”.
This was not the thought pattern of the ancient Greeks. When Euclid states that the ratio of A to B is the same as the ratio of C to D, the letters A, B, C and D do not refer to numbers at all, but to segments or polygonal regions or some such magnitudes.
The Ratio of two geometric structures was determinated by fitting the Unit Parts of the first geometric Stucture into the Other.
An Example: The Tetraktys is a Triangle (A Monad) and contains 9 Triangles (a Monad). The 1x1x1-Triangle Δ, a Part of the Tetraktys, is Proportional to the Whole of the Tetraktys (T) and has a Ratio T/Δ = 3= Δ -> T = Δ (3) x Δ (3) = 9.
The Mathematics of Euclid is not a Mathematics of Numbers, but a Mathematics of Forms.
The symbols, relationships and manipulations have Physical or Geometric Objects as their referents.
You cannot work on this Mathematics without Knowing (and Seeing) the Objects that you are Working with.
About Hermann Grassman, David Hestenes and the Moving Line called Vector
In his time the most important mathematicians did not understand what he was talking about although many of them copied parts of his ideas and created their own restricted version. None of them saw the whole Grassmann was seeing.
When he was convinced nobody would believe him he became a linguist. He wrote books on German grammar, collected folk songs, and learned Sanskrit. His dictionary and his translation of the Rigveda were recognized among philologists.
Grassmann took over the heritage of Euclid and added, Motion, something Euclid was aware of but could not handle properly.
Grassmann became aware of the fact your hand is moving when you draw a 2D Geometric Structure. He called the Moving Lines, that connect the Points, Displacements (“Strecke”).
In our current terminology we would call the Displacements “Vectors”.
The Trick of Grassmann was that he could transform every transformation on any geometrical structure into a very simple Algebra. Multi-Dimensional Geometric Structures could be Added, Multiplied and Divided.
The Greek Theory of Ratio and Proportion is now incorporated in the properties of Scalar and Vector multiplication.
About a 100 years later David Hestenes improved the Theory of Grassmann by incorporating the Imaginary Numbers. In this way he united many until now highly disconnected fields of Mathematics that were created by the many mathematicians who copied parts of Grassmanns Heritage.
About Complex Numbers, Octions, Quaternions, Clifford Algebra and Rotations in Infinite Space
Grassmann did not pay much attention to the Complex Numbers until he heard of a young mathematician called William Kingdon Clifford (1845-1879).
Complex numbers are ,just like the Rationals (a/b), 2D-Numbers. A Complex number Z = a + ib where i**2=-1. Complex Numbers can be represented in Polar Coordinates: Z = R (cos(x) + i sin(x)) where R = SQRT(a**2 + b**2). R is the Radius, the Distance to the Center (0,0).
When you have defined a 2D-complex Number it is easy to define a 4-D-Complex Number called a Quaternion: Z = a + ib + jc + kd or a 8-D Complex Number called an Octonion.
William Rowan Hamilton, the inventor of the Quaternions, had big problems to find an interpretation of all the combinations i, j and k until he realized that i**2 =j**2 = k**2 = ijk=-1.
What Hamilton did not realize at that time was that he just like Grassmann had invented Vector Algebra and Geometric Algebra.
This all changed when William Kingdon Clifford united everything in his new Algebra. Clifford’s algebra is composed of elements which are Combinations of Grassman’s Multivectors.
The Clifford Algebra that represents 3D Euclidean Geometry has 8 = 2**3 components instead of 3: 1 number (Point), 3 vectors (Length), 3 bivectors (Area) and 1 trivector (Volume).
It turns out if you use combinations of these elements to describe your geometric objects you can do the same things you did before (you still have 3 vector components).
In addition, you can have additional data in those other components that let you find distances and intersections (and a lot of other useful information) using simple and (computationally) cheap numerical operations.
The most important Insight of William Kingdom Clifford was that the Complex Numbers are not Numbers all.
They are Rotations in higher Dimensional Spaces.
About Pascal’s Triangle and Mount Meru
The String 1,3,3,1 of Clifford’s 3D Geometry is related to the 4th Level of Pascal’s Triangle. Level N of Pascal’s Triangle represents N-1-Dimensional Geometries.
The Sum of every level N of the Triangle is 2**N. This Number expresses the Number of Directions of the Geometric Structure of a Space with Dimension N.
A Point has 0 Direction, while a Line has 2 Directions, relative to its Center point, a Plane has 4 Directions, relative to its Center Point, and a Cube has 8 directions, relative to its Center point.
Pascal’s Triangle is also called the Binomial Expansion. This Expansion shows all the Combinations of two letters A and B in the function (A+B)**N. Level 1 of the Triangle is (A+B)**0 = 1 and level 2 is A x A + 2 A x B + B x B -> 1,2,1.
The Diagonals of Pascal’s Triangle contain the Geometric Number Systems (Triangular Numbers, Pyramid Numbers, Pentatonal Numbers, ..) and the Golden Spiral of the Fibonacci Numbers.
Pascal’s Triangle is a Repository of all the Possible Magnitudes and their Components.
The Normal Distribution shows that the first level of the Triangle (the Tetraktys) is much more probable than the last levels.
The first four Levels of the Triangle of Pascal contain the Tetraktys of Pythagoras.
The Tetraktys is an Ancient Vedic Mathematical Structure called the Sri Yantra, Meru Prastara or Mount Meru.
About Numbers, Operations and the Klein Bottle
The Complex Numbers are not “Numbers” (Scalars) at all.
They are “Operations” (Movements) that can be applied to Magnitudes (Geometries) and Magnitudes are Combinations of the Simple Building Blocks of the Tetraktys, Points and Lines.
The Tao of Ancient China was not for nothing represented by a Flow of Water. According to the Ancient Chinese Mathematicians Every Thing Moves. In the Beginning there was only Movement.
In the Beginning only the One was Moved but when the Duality was created the Two moved around each other never getting into contact to Avoid the Void.
When we look at the Numbers we now can see that they are the result of the Movements of the first Diagonal of Pascals Triangle, the 1’s (Points) or better the Powers of the One: 1 **N (where N is a Dimension).
Even in the most simple Number System, the Unary Number System, Concatenation is an Operation, An Algorithm.
The Mathematician John Conway recently invented a new Number System called the Surreal Numbers that contains Every Number you can Imagine.
The Surreal Numbers are created out of the Void (ɸ) by a simple Algorithm (Conway calls an Algorithm a Game) that describes Movements (Choices of Direction: Up, Down, Left, Right, ..) that help you to Navigate in the N-Dimensional Number Space.
The Ancient Chinese Mathematicians played the same Game with the Numbers.
Algorithms were already known for a very long time by the Ancient Vedic Mathematicians. They called them Yantra’s.
Geometry is concerned with the Static Forms of Lines and Points but there are many other more “Curved” forms that are the result of Rotating Expansion and Compression. These forms are researched by the modern version of Geometry called Topology.
The most interesting 4D Topological Structure is the Klein Bottle. The Klein Bottle is a combination of two Moebius Rings. It represents a Structure that is Closed in Itself.
It can be constructed by gluing both pairs of opposite edges of a Rectangle together giving one pair a Half-Twist. The Klein Bottle is highly related to the Ancient Art of Alchemy.
The movement of the Duality around the Void can be represented by a Moebius Ring the Symbol of Infinity ∞.
Later in this Blog we will see why the Number 8 is a Rotation of ∞ and the symbol of Number 8 is a combination of the symbol of the number 3 and its mirror.
First we will have a look at the Reciprocal Relation between Space and Time.
Larson believed that the failure to recognize that Motion is the most basic physical constituent of the universe has handicapped the progress of the traditional study of physics, which focuses on Forces.
The definition of Motion stems from the Equation of Velocity, v = ds/dt.
Instead of depending upon the change of the location of an object to define an arbitrary “quantum” of space per “quantum” of time, such as miles per hour, or meters per second, the RST assumes that the observed universal passage, or progression, of time is one aspect of a universal motion that necessarily must be accompanied by a universal “passage,” or progression, of space.
The Units of Time fill up the Units of Space. Space and Time are Duals.
Space is not-Time and Time is not-Space. Time is Non-Local, Cyclic and represented by the Rotating Imaginary Numbers. Space is Local, Linear and Represented by the Scalar Numbers. Space is the Vacuum and the Nothing and Time is the non-vacuum, the Every Thing, the Solids represented by the Cube of Space.
The Cube of Space is the structure behind the Tetraktys but also behind the Book of Genesis.
Our Reality contains two Reciprocal 3D-structures related to Space and Time. Space and Time are related by the Simple Formula N/N=1/1=1, the Formula of Diracs Delta Function.
We are able to perceive the Real 3D-Structure of Space. The 3D-Structure of Time is Imaginary. It is situated in the Imaginary Number Space of i.
Larson, a Self Thought Genius like Grassmann, developed Geometric Algebra without knowing anything about Geometric Algebra but he also invented String Theory long before String Theory was invented. The Mathematics of Larson is also the Mathematics of the Tetraktys of Pythagoras without even knowing anything about it.
Larson was able to Calculate all the important Physical Numbers without any problem and was also able to Calculate Chemical Structures and Reactions.
About the Bott Periodicity
The fourth line of Pascals Triangle and the Tetraktys contains 8 Directions in the Four Geometric Dimensions: 0, 1, 2, and 3.
Mathematicians are intrigued with this number 8, because they find it popping up unexpectedly in advanced mathematics.
In fact, expanding the Binomial Expansion to 8 dimensions just creates an inverse copy of these first Four Dimensions, and then the pattern just repeats itself with a half-twist and back from there, ad infinitum.
The mathematician John Baez wrote an article in which he relates this 8-fold Periodicity to the Scalars (1), the Complex Numbers (2), the Quaternions (2×2), and the Octonions (2x2x2 = 2**3).
The Universe of Numbers and Magnitudes is Cyclic and Fractal.
Our own Reality, symbolized by the Tetraktys, repeats itself in Higher Dimensions until Infinity.
The Tetrad, represents Completion, because it contains all its Previous Numbers, the 1, 2, 3, and itself, 4, in One Number, 10 = (The One) + 9 (= 3 (Trinity)x 3 (Trinity) = Tetraktys).
As you can see in the Picture above the Fractal Pattern of 8 contains two kinds of Trinities/Triangles, an Upside and a Downside (Rotated by 180 Degrees) Triangle. When you Rotate by 180 Degrees the 1 becomes -1 and 1 + -1 =0 is the Void.
The Multi Dimensional Rotations of the Octonions always Come Back to Square 1/1=1, the One and keep Rotating around the Center, the Nothing, Until Infinity.
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).
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).
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 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.
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.
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.
About Mystical Number-Patterns
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 (Platonic) Musical 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).
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 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.
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.
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.
About the Triangular Numbers
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 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).
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.
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.
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.
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 Possiblebut every Expansion of the Triangle is less Likely to Occur.
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.
Early astronomical traditions identify the “Primeval Sun” as the planet Saturn.
Saturn was id]]><![CDATA[entified with Osiris in Egypt and Shiva in India. The Babylonians, the founders of Astronomy, called Saturn the "Light of Heaven, the Sun-God Shamash (or Šamaš).
Tacitus records the Jews as worshipping the planet Saturn, Shabbatai, as their god. In Plato’s Timaeus, the word for the planet Saturn is Helios, the “Sun” god. Popular Greek traditions identified Saturn as Kronos (Father Time). At that time Saturn ruled “over the Pole“.
The Pole was seen as the Entrance to the Other World. The Shaman using his Light Body had to Climb Jacobs Ladder to get there.
In Sankrit, Suryaputrah, “Son of the Sun” is the name for the planet Saturn. The Medieval Alchemists called Saturn the “Best Sun”.
The Kabbalah divides the Universe into Ten Spheres, or Sephiroth, which have Planetary and other Correspondences. The Sphere of Saturn is known as Binah (“Understanding“) and the terrestrial sphere, known as Malkuth, is said to have its foundation in Binah.
Long ago Saturn exercised the supreme power on Earth, his reign being remembered as the Golden Age, his time revered, a time which Man longed to return to. At this moment the Cycle is ruled by Kali. The Mother Goddess Kali represents the Moon in Saturn.
Semitic civilizations referred to the god Saturn as “El”. El was the source of the Great Flood. El was represented by a Black Cube. The Kaaba (“the Cube of Space”) is a cube-shaped building in Mecca, Saudi Arabia, and is the most sacred site in Islam. The building predates Islam, and, according to Islamic tradition, the first building at the site was built by Abraham.
In Ancient History the World was ruled by the Female and the Mother Goddess. The Vulva or the Yoni (Circle), of the Ancient Mother Goddess was always combined with a Black Stone (with a Point), the Symbol of the Phallus (Lingham) of the Male Creator. The God and the Godddess created the Universe in a short moment of Extreme Extacy.
The Pelgrims have to encircle the Black Stone of El Seven Times just like the Rings of Saturn encircle the Center.
About the Number Seven
The Blog “About the Sum of Things” is about the so called Seal of Saturn and the related 3×3 Magic Square of Saturn also called the Lo Shu in China.
Saturn is the Seventh Planet and Saturday (Sabbath) the Seventh Day of the Week.
In the Blog it is shown that the 3×3 Magic Square generates Two Cyclic Systems. One System, the Wheel of Karma is a Fractal Torus and is related to the Expansion and Compression of the Universe and the number 2. The Fractal Torus contains Seven Levels associated with the Chakra’s, the Seven Heavens, The Seven Tones and the Seven Colors. The Tower of Bable and the Egyptian Pyramids are a Simulator of the Torus.
The other is a Cyclic Vortex System, the Cycle of Death, related to the number 3 and the Triangel.
The Number Seven is the Repeating Factor of the Torus System. After Seven Steps a new Fractal Level of the Expansion/Compression-Cycle is reached. On the Seventh Day a new Stage in the Creation Process of our Universe is started.
The Torus looks like a Yoni and as you can see the Lingham of Shiva is Inside the Torus. This is the reason why Saturn is associated with Binah and Shakti, the Female Part of Shiva. It also explains why Saturn is related to Fertility.
The Sequence of the Trinity (A-U-M) controls the Creation/Destruction-Cycle of our Universe. This is the Cycle of Shiva, the Creator and Destructor.
Shiva and Shakti are the Circle (Torus, Yoni) and the Lingham (the Point, Singularity, Lingham). They are the 0 (or the 9) and the 1.
The only thing you need to Create our Universe is de Tao (0), the One, the Two and the Three and they will Generate Everything.
Why is Saturn the Son of the Sun?
There are two explanations. The first one is a Physical Explanation related to Plasma Physics. In the stage of the Golden Age our Solar System contained only Three Planets and Saturn was the Best Sun. Plasma Physics is able to explain many ancient Symbols.
The Second explanation goes back to the Numbers. The Magic Square of Saturn is the First Magic Square that can be constructed. 2×2 is not existent and 1×1 is a Point. The Magic Square of the Sun is the 6×6 Magic Square with Sum 666! and Constant 111. The 3×3 Square can be extracted (is a Son) out of the 6×6 Square.
About Kronos, Father Time
The Greek called Saturn Kronos, Father Time.
Most of the Souls are not aware of Cycling Cycle of the Fractal Torus of the Number Two of Duality, the Wheel of Karma. They stay in the same Place and Their reincarnations are a Merry Go Round.
Some Souls move around the Torus and experience many experiences in the Seven Universus.
All of them are Kept in de Rotating Matrix of the Demiurg.
If you really want to be Free you have move out of Duality into the Cycle of the Divine Trinity (Shiva).
You Twist when you Cross the Cycle of Life and Enter the Cycle of Death of the Trinity. When you join this Cycle you leave the Seven Heavens and move to the Void in the Middle of the Death Cycle.
The Void, the Kingdom, Cleans every thing and makes every thing New.
You Merge with Yourself when you integrate the Inside and the Outside, the Up and the Down, the Male and the Female.
“If we then become children, would we thus enter the kingdom?” Jesus said unto them, “When ye make the two one, and when you make the inside like unto the outside and the outside like unto the inside, and that which is above like unto that which is below, and when ye make the male and the female one and the same, so that the male no longer be male nor the female female; and when ye fashion eyes in place of an eye, and a hand in place of a hand, and a foot in place of a foot, and a likeness in place of a likeness; then will ye enter into the kingdom.” (Gospel of Thomas, Logion 22).