## Posts Tagged ‘point’

Thursday, February 2nd, 2012

Medieval Perspective

Introduction

When we Look with our Eyes and not with our Mind we can See that Space looks very different from what we Think it is. In Our Space Parallel Lines meet at Infinity.

Around 1400 during the Renaissance Painters started to look at Space with their own Eyes and discovered the Rules of Perspective Drawing.

Between 1600-1800 Perspective Theory changed from a Theory of Art to a Theory of Mathematics called Projective Geometry.

It took 400 Years before a few Mathematicians realized that Projective Geometry was the Foundation of Mathematics and it took another 100 years before Projective Geometry started to influence Physics.

In 1908 Hermann Minkowski discovered that Einstein’s Theory of Special Relativity could be analysed using Projective Geometry. Minkowski created a 4D Space-Time Metric Geometry in which he added one Time Dimension.

Many experiments now show that 4D-Space-Time  is not sufficient to incorporate what Time Really is.

The essence of Our Universe is Movement, Expanding Space,  and Movement = Space/Time (Space Divided By Time).

Both Time and Space are 3-Dimensional and represent a Different,  Reciprocal, Complementary (Dual), View on Movement.

We can move independently in Time OR Space and in Time AND Space (The Chronotope).

The Future is Expanding Space with Infinite Potential.

Time moves behind Space and Scales Space.

A Mathematical Theory of Perspective Drawing could only be developed when the Renaissance freed painters to depict Nature in a way closer to what they Observed.

In the Middle Ages Social Status was very important. Important People or Buildings were always emphasized.

In the Renaissance the Artists started to look with their own Eyes and Created Pictures where the Viewer looked through the Eyes of the Painter.

The Florentine architect Filippo Brunelleschi (1337-1446) studied Greek Geometry, developed a theory of perspective and undertook painting just to apply his geometry.

The first treatise, Della pittura (1435) by Leone Battista Alberti (1404-72) furnished most of the rules.

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.

Perspective Machine designed by Albrecht Durer

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.

The Perspectograph of Leonardo Da Vinci

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.

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.

Different Points of View

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.

Pappus' Theorem

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.

Desargues' theorem

Later Blaise Pascal (1623-1662) proved Pappus Theorem for Conics (Circles, Parabola, Hyperbola) in his  ”Essay on Conics” (1640) when he was 16 years of Age.

Pascals Theorem

About the Four Points of the Cross-Ratio

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

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.

The Cross-Ratio k is dependent on the Order of the Four Points. It converges to 0 and infinity when k=1 and to -1/2 and 2 when k= -1.

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.

Ruler (Space) and Compass (Time): The Tools of the Geometer

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.

In Eucludian Geometry A Ruler contains a Fixed Scale

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 Expectation looks through a Small Hole.

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.

In the blog about Geometric Algebra I wrote about a very Independent Thinker called Dewey B. Larson.

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.

The Spiraling Spiral of Time

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.

Time, History, is Always Behind Us and Space, The Future, The Adjacent Possible, with Infinite Possibilities, is Always in Front of Us.

Let’s Move.

Paul Dirac: About Physics and Projective Geometry

About Expanding Space and Not-Euclidian Geometry

Pictures of the Projective Plane

About the Reciprocal Theory of Dewey B. Larson

About Projective Geometry and our Senses

About Projective Geometry and Geometric Algebra

About the Four Points of View

Why the Future is Open Space

About Alchemy and the Klein Bottle

### The Relationship between Chaos Theory, Fractals, Lyapunov, Minkowski and the TAO

Thursday, November 1st, 2007

Chaos theory is the theory of Dynamic (Non-Linear) Systems. Dynamic systems converge to a state called an Attractor.

A fractal is a shape that can be subdivided in parts, each of which is a copy of the whole. This property is called Self-Reference.

Self-Referencial Dynamic Systems are represented by a Strange Attractor .

I want to start with a few citations of HUMAN DIMENSIONS OF CHAOS THEORY: Consciousness, Physiology, Perception, and Psychology by Iona Miller and Graywolf Swinney, ©1992.

Chaos underlies the ability of the brain to respond flexibly to the outside world and to generate novel activity patterns, including those that are experienced as fresh ideas (also fresh behavior, emotions, belief systems, mythologies, etc.).

Chaos results in meaning-laden perception, a gestalt, that is unique to each individual.

Chaos is implicated in human perception as a multi-sensory phenomenon. The controlled chaos of the brain is more than an accidental by-product, like “putting your brain in neutral.”

Indeed, it may be the chief property that makes the brain different from an artificial-intelligence machine. One profound advantage chaos may confer on the brain is that chaotic systems continually produce novel activity patterns“.

Research may reveal that assessments of personality or a personality trait over time generate data that leads to a fractal correlation dimension. Such evidence would confirm that personality or that a particular personality trait may be desribed as a strange attractor of associated behavior”.

In the article they apply Chaos Theory and Fractals to the Human. What they are showing is that Self-Reference (A Fractal) is the Result of a Chaotic Process.

Out of the Chaos of Creativity a Repeating Pattern Emerges.

Chaos is not total randomness, but implies an implicate, “hidden”, or occult order within the nature of reality“.

The Repeating Patterns looks Chaotic but when we find the Right Fractal behind the Order in Chaos we can know where the Path is Moving to (the Attractor).

To dig deeper into Chaos we have to dig deep into mathematics. We have to dig so deep that the language becomes almost incomprehensible but let us have a try.

A citation out of More than Synchrony: Chaoticity may be necessary for conscious brain functioning (Juergen Fell ea): “Indeed, estimation of nonlinear EEG characteristicslike dimensions and Lyapunov exponents revealed a pronounced decrease not only of complexity, but also of chaoticity during unconscious states like deep sleep”

” Sufficient dynamical complexity of brain activity, seems to be one more necessary condition.

Without chaoticity our brains might be unable to switch immediately between different oscillatory patterns and thereby different modes of operation”.

A citation out of Some Thoughts on Time, Quantum Mechanics, Geometry and Free Will (Linas VEPSTAS) : “The strange thing for us is, we live in this hyperbolic Minkowski space, but we don’t seem to notice much positive Lyapunov exponents in our lives.

Other than that nature all around us seems to be fractal and self-similar … other than that, we don’t notice.

I sometimes wonder if the quantum-entanglement problems of quantum-mechanics are really some manifestation of life in Minkowski space, some deep connection we haven’t noticed just yet“.

The Lyapunov exponents determine the predictability of a dynamical (Chaotic) system. When we don’t seem to notice much positive Lyapunov exponents in our life we experience life as a “random” process. The main reason that we don’t notice the self-reference is that we are stuck with our own attractor. We are repeating our own patterns all the time and are unable to resonate with the pattern of the Universe.

When the exponents decrease Life become simple again. This happens when we dream. So dreaming is a way “to make things simple“, to “understand what is happening“.

Minkowski space is a four-dimensional space. It was used by Einstein to describe space/time (The Chronotope).

When Space/Time contains positive Lyaponov exponents Space/Time is highly predictable.

This happens when Quantum Entanglement takes place.

Quantum Entanglement (QE) is the Union of seperate Parts into a Whole. QE makes it possible to be in Two or more States at the same moment.

It even makes it possible to Unite with Everything.

QE gives us the feeling that there is something that is connecting us to an Entity on a Higher Level.

When QE happens we are in the AND-state and experience a feeling of Harmony.

When we are not in the AND-state we are in the OR-state.

In this OR-state there are Four “Fractal Attractors” possible. They also give us a “feeling”.

We feel we are Stuck” (Point-Attractor) or

we feel we are “Going Round in Circles (Cycle-attractor)” or

Spirals (Torus-Attractor).

The most Attractive Fractal Attractor, the AND-attractor gives us the feeling that “We are in the Flow (Tao) or in The Now (Chaos)”.

Our Universe is “looking” for Quantum Entanglement. This state of Harmony cannot be perceived in our current 4D-Reality (The Minkowski-Space).

According to Tiller our Unconsciousness connects to another 4D-Reality that is highly related to what the Scientists call Dark Matter.

Out of this Reality The Force of Life (Chi, Prana) is moving into our Body.

If we look at the current state of our World it does not feel as a Strange Feeling that we have to connect to the Force of Life again.

Connecting to this Force is Difficult for us (and me).

The knowlegde about it is hidden in old “Mysteries” and “Magical practices“.

Many people don’t believe in this or even don’t want to believe in it because it is connected to the Dark Forces.

Strangely enough the Force is also connected to Dark Matter.

It seems we need to enter the Dark to find the Light.

We have to fight and embrace our Dark Shadow (Jung).

Friday, August 31st, 2007

Many people think I am a very intelligent person. I am able to solve very complex problems in a very short time. In reality I am a normal person. I use a trick. The trick is called Mapping.

I map a complex unknown domain to a well known simple domain. I solve the problem in the well know domain and map the solution back to the unknown domain. When I do that the unknown domain becomes simple again.

The well-known domain I use are the Seasons of The Year. People don’t believe the Seasons are a representation of the world we live in.

About a year ago I discovered the reason.

Everybody in the West believes the theory of reasoning of the Greek philosopher Aristotle. The theory of Aristotle is called causal reasoning.

I use causal reasoning when I am writing this article. I need causal reasoning to convince you. The only way to convince Western people is to use causal reasoning.

What I will show you is that causal reasoning creates many problems that are unsolvable. To solve these problems you have to move to another way of reasoning. What Western people don’t want to accept is that there are other and more effective ways to reason. The other way to reason was used before Aristotle was born.

This approach is more than 2000 years old. It originated in the Middle East (China & Persia). Western people don’t accept this. They are trained to believe that the people that used these “more advanced” approaches were stupid. They believe “The West is the Best” in everything.

They also believe that when we go back in history we see a down-ward spiral of knowledge. Western civilisation is always “Going UP”. We will become better and better and better until infinity.

What they don’t see is that Causal Reasoning creates a big problem.

When you believe everything has a cause you start to look for the cause of the cause and when you start to look for the cause of the cause you never stop looking for the next cause.

Because you don’t have infinite time to look for the infinite chain of causes you have to find a solution. The solution is named “Creation”.

Somewhere there is a beginning and in the beginning something or somebody is creating “Something out of Nothing”. When you are religious you name the creator God. When you are not religious you name the first cause The Big Bang.

After you have named the creator you stop wondering what “creation out of nothing” really is. You force yourself to believe. You force yourself to believe the Church or you force yourself to believe Science. When you believe the Church you have to believe in a Supreme Power.

When you believe in Science You have to believe Others. Most people believe the Others of Science are very intelligent. If you believe these others are very intelligent you have to believe there are also people that are not intelligent. We have named “not intelligent” “Stupid”.

The Belief that causal thinking is the only way to think does not only generate the idea that you are stupid. It also generated the idea that you are guilty of something. You are guilty of something when you are the cause of something that went wrong. When a human is doing something wrong it is named “Sin”.

When you believe in Sin and you believe in causal thinking you have to believe there is a Creator of Sin. The creator of Sin is named the Devil. Because God created everything you have to assume God also created the Devil. When you believe in a Supreme Power you don’t understand why a Supreme Power is not able to beat The Devil. Because you are “stupid” you stop wondering. You have to believe.

A “creation out of nothing” is a Miracle. When you believe in a Miracle you believe in Luck. You hope you will be lucky someday. When you are religious You don’t understand why God is helping the other and not You. To explain this You believe you are doing something wrong. You are stupid or sinful.

When you believe the intelligent people, you have to believe in the theory of probability. To improve your behaviour you start to collect data. You use statistical packages to find the pattern. This is not helping you at all because still some people are lucky and you are not. You are doing something wrong (you are stupid) or you do not believe in Him. You are sinful.

If you use causal reasoning you are able to find a solution. You have to assume that the beginning is the end. You have to close the causal chain in itself. When you close the chain the chain is transformed in a circle. If you do that you have moved to another way of thinking and the other way of thinking is “impossible”.

It creates a problem you cannot solve. In a Circle it is very difficult to find the beginning. Every point in a circle is a beginning. If you assume this point is the beginning it is also the ending. The causal solution is to assume that the circle is contracted into one point. Everything is one point. Everything is One.

This violates your Sensory perception. You observe Many Many Others. This is a contradiction and you stop this path of reasoning.

The “old cultures” have found a solution. They use the principle of unfolding and enfolding.

A Flower is an unfolding of One Seed. A Flower enfolds in Many Seeds. In Spring the Seeds grow into a Plant. In Summer the Plant produces Flowers and the Flowers produce Seeds. In Autumn the Seeds are covered by Earth and in Winter Earth protects the Seed.

What we observe with our Senses is a Cycle and the Cycle contains Stages. If we look around we see many Cycles and all of the Cycles contain the same stages.

If we look carefully we observe five stages. We observe the Four Seasons and a Fifth Season between Summer and Autumn. In other climate zones the fifth season is easy to experience. In the US it is called Indian Summer.

Before Aristotle the principle of unfolding based on a cycle divided in five stages was the most important theory to explain the world.

To explain the world every thing that happened was mapped to the Cycle of the Seasons.

The solar system, the universe, the body, the state, history and every else was governed by the same cycle. They are Wholes that are moving around Wholes that are moving around Wholes …..

They are The Same because every thing is moving through the Same Stages. They are also Different because they operate on a different Level.

Levels are a problem Western Civilization cannot handle. We believe in one level, the chain of cause and effect. We are constantly connecting different levels. Connecting different levels at the wrong connection point is really the main cause of the mess we are in.

The cycles on different levels are moving with a different periodicity (frequency). They act like waves. The Scientists of that time discovered that the frequencies were also related by a pattern of fives.

Thinking in waves is a way of thinking Western Civilization cannot handle. We think in particles that are colliding. Again this is a simple consequence of causal thinking.

Slow moving cycles are the unfolding of fast moving cycles. An unfolding cycle is a Spiral. Because everything is closed the Spiral had to be closed in itself. It is transformed into a Spiralling Spiral. The Spiralling Spiral is moving UP AND moving DOWN. This violates Western Thinking because we only believe in Spiralling UP. We don’t believe Western Civilization will disappear someday just like all the other civilizations before us.

We don’t believe the Spiralling Spiral works like a Clock. The cycles are moving in a highly predictable pattern. In the West we believe in statistics and statistics shows that every predictable pattern has its exceptions.

Every cycle is governed by a Clock. You are also governed by a Clock and if you don’t listen to your Clock you become ill. You get “out of rhythm”. Going “out of rhythm” is the main cause of all the Western diseases.

We really don’t believe Einstein. Everybody believes the Myth that he has failed. Many people don’t know that one of his pupils David Böhm found the solution. Einstein believed that “God is not throwing dice”. He also believed “God is a geometer”.

Although the new Psychics are showing that Einstein and Böhm were right and Niels Bohr (Quantum Mechanics) was wrong we are really afraid to believe this. It violates the foundation of Western Civilization.

We love the theory of Quantum Mechanics because it proves the world is a Miracle. People don’t understand Quantum Mechanics at all. Quantum Mechanics acts like a Religion and a Religion is based on the Mystery of “Creation out of Nothing”. Quantum Mechanics proves this is “really happening”. This gives us a good feeling. The foundation of Western Civilization is proved again.

We don’t want to believe that Civilizations rise and fall.

We also don’t want to believe is that long ago these civilizations were more advanced than our society.

What we really don’t want to believe is that they were destroyed because they fell into the same trap Western Civilization is falling. We don’t accept the lessons of history. We believe the old myths were fantasy stories. Just like the old civilizations we don’t see that we are now living in our own fantasy world. The fantasy of eternal youth and eternal life.

What we don’t want to believe is that the old civilizations even found a solution for this problem. The solution they found is not a material solution. They found out that the power of the human imagination is tremendous. When Western Science is proving this we don’t want to believe Western Science.

We are really very funny people! The solution for everything we want is round the corner and we don’t accept this solution.

Now I am able to explain my trick.

Step 1. Determine the level of observation. The level of observation is related to the periodicity of the cycle. An organization is often related to a yearly cycle.

Step 2. Determine the five stages in the cycle. Use the seasons. Observe carefully what every season mean to you.

If you want to reuse the knowledge of the scientists before Aristotle I advice you to study Chinese Philosophy. People with a mathematical interest have a look at Pythagoras or (better) Topology. People with a psychological interest read Jung, Will McWhinney or George Lakoff (!!). People with an interest in literature study Bahktin (!!). People with in interest in medicine study Acupuncture. People with an interest in Christian religion read the gospel of Thomas (The Essenes). People with an interest in Paintings have a look at Leonardo da Vinci. Physics look at Super String Theory and David Böhm (!!). Study Alchemy or the Zohar. Interesting philosophers are Erasmus and Schopenhauer. Lovers of music listen to Bach or John Cage. Cultures: Read Carlos Castaneda (!!). Study the Hopi’s, The Sufi’s, Zoroaster and The Mayas. Look at the Medicine Wheel. Study the old Cathedrals (Chartres!), Read Steiner, Blavatsky, Krisnamurti (a close friend of Bohm), Joseph Campbell (!) and Goethe. Read Poetry (!!). Last but not least Observe Nature.

Step 3. Connect the cycle to the cycle one level higher and the cycle one level lower. This shows the possible spiralling UP en the spiralling DOWN. The lower level of an Organization is called a Human. The higher level you could call a Value Chain (Cycle).

Step 4. Look for all the places where the Cycle is blocked or connected to the wrong level.

Step 5. Restore the Cycle and the Levels.

Step 6. Transform your observation in the language of the people that are paying you. This is the most important step. Carefully listen to the language. Everybody is talking in Seasons but we use many many words to say the same thing over and over again.

The reason? The solution is Too simple or/and Your clients believe in Luck or Destiny or/and They look for the Sinner and want to punish the Sinner or/and they Feel they are Guilty and Deny this or/and they want to use Statistics. If they do this they prove that the world is really very Complex.

Step 8. Try to explain them that they have fallen in the trap of causal reasoning. Tell them they are not Sinners or Stupid. Give them the feeling Life is meant to be a beautifull experience of the Whole.

Step 9. If they don’t follow your advide come back one cycle period later. The situation has surely increased in complexity. They need you again. Go to Step 1.

Do you want to read more about Cycles read the theory behind cycles (pdf). Sorry this document is written in the Dutch Language.

Do you want to know more about the Cycle in Litterature and Art?

Do want to know more about Reasoning and Logic?

Do want to know how Physics easily explains Acupuncture?