Posts Tagged ‘Harmony’

About Perspective

Thursday, February 2nd, 2012
File:Reconstruction of the temple of Jerusalem.jpg

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.

About Perspective Drawing

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.

IMAGE: Leonardo's perspectograph

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.

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.

Two projections of the same object.

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.

ratios.jpg

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.

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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.

alchemy

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.

golden mean spiral2

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.

LINKS

About Pythagoras and Heliopolis (Egypt)

About Pythagoras

Paul Dirac: About Physics and Projective Geometry

About the Middle Ages

About Expanding Space and Not-Euclidian Geometry

About the Renaissance

About Resistance and Mass

About the Mathematics of Perspective

About the Ether

About Clean Space

About Projective Geometry

Pictures of the Projective Plane

About Geometric Algebra

About As Within, So Without

About the Reciprocal Theory of Dewey B. Larson

A Video about Mobius Transformations

About Projective Geometry and our Senses

About Projective Geometry and Geometric Algebra

About the Cube of Space

About Time and Paranormal Experiences

About Geometry

A Textbook about Metric Geometry

About the Four Points of View

About the Void

Why the Future is Open Space

About Alchemy and the Klein Bottle

About the Human Sensory System

Stuart Kaufmann: About the Adjacent Possible

About Mystical Number Theory and Pascal’s Triangle

Friday, December 2nd, 2011

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

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

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

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 (PlatonicMusical Harmonies: the Fourth (4:3), the Ffth (3:2), the Octave (2:1) and the Double Octave (1:4).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A Hexagram is a Two Dimensional Cube

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

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

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

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

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

Star of David in The Israeli Art Genesis-2

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

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 Triangular Numbers is a Square

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

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

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

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

28 is a Hexagonal Number

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

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

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

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

Star Numbers are a Combination of Two Triangular Numbers

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

The Symbol of the Heart Chakra contains Two Triangles.

About Pascal’s Triangle

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

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

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

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

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

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

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

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

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

Triangle of Pascal

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

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

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

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

This pattern repeats itself all the time.

File:Yanghui triangle.gif

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

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

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

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

File:Pyramid of 35 spheres animation.gif

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

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

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

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

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

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

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

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

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

the Fibonacci Numbers as a Cube.

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

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

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

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

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

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

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

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

The Triangle of Pascal Modulo 3

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

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

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

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

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

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

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

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

Life is not only about Me and the Other.

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

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

LINKS

The Content of the Sepher Yesirah

About the Sepher Yesirah

About the Cube of Space

About the Tetractys

About the Cube of Space and Psychology

About the Sepher Yesirah and the I Tjing

A correspondence table of the Cube of Space

About Bereshit

About Genesis

About Patterns in the Bible

About Saturn

About the Trinity

About the Sri Yantra and Plato

About the Lo Shu and the I Tjing

All kind of strange relationships between Triangular Numbers

A website about Mystical Number Theory

About the Figurative Numbers

About Combining the Combinations

About the Golden Spiral and Plato

About the Logic of Creation

About Pascal’s Triangle and the Normal Distribution

A complete course in elementary Number Theory

About the Psychology of the Cube of Space

About the Tetraktys and the Zodiac

About the Process Theory of Paul Young

About the Theory of Dewey B. Larsson

Mysteries of the Equilateral Triangle

About Visual Patterns in Number Theory

About Pascal’s Triangle and Cell Division

About the Sum of Things

Thursday, September 22nd, 2011

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

lo-shu-color

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

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

The Chinese Pollen Path

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

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

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

SriYantra

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

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

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

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

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

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

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

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

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

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

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

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

The Seal of Saturn or the Dance of Yu

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

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

The Lo Shu and the Well

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

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

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

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

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

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

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

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

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

Lo Shu Torus

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

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

4

9

2

-5

-1

4

-3

3

5

7

-5

-2

0

2

8

1

6

-5

3

-4

1

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

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

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

About the Sum of Things

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

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

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

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

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

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

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

LINKS

About the Book “the Sum of Things” (Free Download)

About the I Tjing

The Calculations  behind the Number of Things

The Relationship between the Lo Shu and the I Tjing

About the Lo Shu Torus

About Saturn, the Son of the Sun

About the Trickster

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

The Sistine Chapel (Rome) and the Seal of Saturn

About the Relationship between the Lo Shu and Music

About the History of the Lo Shu

About the Wheel of Eternal Order and the Wheel of Time

About the Relationship between the Lo Shu and Chinese Alchemy

About the Great Flood

About the Orion Myth

About Plato, Pythagoras and the Lo Shu

About the Nothingness

About the Trinity

About the Heart Chakra

About the Sri Yantra

About the Lo Shu Magic Square and the Chinese Domino Game

About the Mirror Universe

About Acupuncture

About Tit for Tat

About the Prisoners Dilemma

About the Bronze Mean

About the Sheng Cycle

Everything You want to Know about Sacred Geometry

About the Influence of the Lo Shu on Recent Art

The Lo Shu Square with Imaginary Numbers

A City is an Organism

Saturday, February 12th, 2011

The System of Life is growing in size and in complexity, and so do the infrastructures sustaining it.

In 2008 the point was reached where more than 50% of the world population lives in cities. That percentage is growing. By 2030, over 60% of the world’s population – nearly five billion people – will be living in urban area.

The System of Life is generating Super Organisms and the City is one of them.

Urbanization is only a relatively recent phenomenon. As far as we now know urbanization started in Mesopotamia around 4000 – 3500 BC.

For the Ancient Mesopotamians, the cities were the centers of life. Their beginning of time was not a Garden of Eden, but rather an ancient site called Eridu, which they believed was the first city ever to be created.  The creator of Eridu was Enki, the God who came out of the Deep Waters of the Ocean  (the Abzu). The city of Eridu was destroyed by the Great Flood in 3117 BC.

Every time one of the Rulers of a City took control over many Cities and a State was created with a new Center, the Capital. In the Center of the Capital the Palace of the Ruler was situated. What we see is an Expanding Pattern in which the same Structure repeats itself all the Time.

Initially the two life-essential resources for cities were Water and Food.

As Cities grew and as technology evolved, the supply of these resources became dependent on energy for processing, for pumps and for transportation. In the process, cities and their resources evolved into a system of life energized by the individuals that populate cities, and fuel its dynamics.

mexico-city

Mexico-City

Kleiber's law

Kleiber's law

About the Law of Kleiber

The Law of Kleiber shows that Bottum-Up Structures are more efficient than Top-Down Structures.

It also shows that Big Structures are more efficient than Small Structures. The Law is applicable from the level of the Bacteria to the Level of the City and higher.

These two statements are a contradiction until we realize that there are two ways of looking at the same structure. One view is the Top-Down View. The other is the Bottom Up View. When we look Down we see a huge Hierarchy and start to believe that Big is Beautiful.

When we look Up we see that all the work is done by Small Cooperative Components starting at the Bottom with the Bacteria.

The Law of Kleiber is explained by the fact that all the organisms (including the Cities) contain the Same Fractal Structure.

Bacteria exchange their DNA

Bacteria exchange their DNA

About DNA

In the Beginning the Chemical Soup generated Self-Reproducing Chemical Factories. These Factories combined into bigger Factories by exchanging and combining their Factory-Designs (DNA-parts).

Every time when the Factories fused they became more Efficient, Bigger, Stronger, Older and therefore more Competitive. When the Factories became more Competitive they started to Protect themselves against the other Factories.

To Protect themselves the Multi-Cellular Systems began to Sense in Many Directions and became better and better in Predicting the Future.

One of our Ancient Ancestors is the Sponge which is a static Cluster of Bacteria. Scientist have detected that the DNA of the Sponge has a lot in common with the Human DNA. The Sponge contains the Basic Fractal Structure, the Tube.

In a later stage our Ancestors developed something we call a Brain. The Brain is used to Sense the Now and Predict the Future.

The Human is a Moving Sponge who is able to Imagine. With his Imagination the Human is capable to Change the Future (and the Past).

Sponge

The Sponge, the Tube, is the Basic Structure

One of the Bilateria: The Octopus

The Octopus is a Sponge with a Brain.

About Innovation

The Law of Kleiber can be explained by the principle of Self-Reference (Similarity,  Fractals).

The Small Structures at the Bottum combine into comparable structures on a higher level.

What we don’t see is that the structures on every level can be decomposed in the Same (DNA-)Components with a different Scale and a different Name.

Innovation is nothing but a  Combination of  the Components in many different ways until all the combinations are made.

At that moment One of the Combinations becomes the new DNA and a new Level is explored.

Life explores every possible path.

About Cities

A City is a Combination of Houses.

A City protects a Cluster of Houses. A House protects a Cluster of Humans and a Human protects a Cluster of Organs.

All the way down to the Bacteria,  Cooperative Clusters are formed that give the participants a more Secure Way to Live.

A House is a Body. It supports the Body but it also contains all the Bodily Functions.

If we provide the House with these Bodily Functions the House becomes more Sustainable. It is possible to regenerate the Waste that the Humans produce into Energy just as the Liver is doing in our Body.

If we Cluster the Bodily Houses into a City and we provide the City with the same functions of the Body we are able to create structures that are independent of the input of external Top-Down Sources of Energy and Food.

If we are able to detect the DNA of the House and the DNA of the City we are able to Compose new Types of Houses and Cities that are Adaptable, Scalable,  highly Sustainable and are also in Balance with Nature.

It will also be  much more easy to Destroy or Move buildings and cities to another Place without the production of Waste.

looking-to-toledo1

Toledo (Spain)

About Synchronization

In the Miedeval City the Major Cycle was the Cycle of the Seasons. In the Modern City we have become almost independent of the Cycles. We import Food and Energy out of every place in the World.

It takes a Lot of Energy to transport the Food and Energy and Many Places on Earth are dependent on a small amount of other Places that contain the Sources of our always increasing Need to Survive the Effects of our Environment.

When a lot of People (The Slaves) are dependent on a small amount of people (The Masters), the Masters start to dominate the others and increase their Power until Eternity. Power is the most important Addiction of the Humans.

The problem becomes bigger when we realize that many people are producing Food and Energy but are themselves not able to use the Food and the Energy they are producing.

The independence of the Cycles has created a huge unbalance. This unbalance is not only visible in the City but in every place on Earth that is providing Energy and Food to sustain the Modern City.

The only solution to this problem is to create self-sustaining (autarkic) Cities that are able to supply their own food, water  and energy.

Paris

The kernels of Paris

When we want to restore Balance we have to Synchronize the Cycles in the City.  This can be done with the help of Smart Technology (Sensors, Actuators, Analytics) but we can also make use of very old fashioned techniques that were used in the Middle Ages.

When Humans communicate from the Heart and focus on the same subject, the Humans start to Synchronize. They become a  Group.

The most important structure in the Miedeval City to Synchronize its inhabitants was the Building of the Heart, the Church.   The Church was always situated in the Center and was visible from a long distance. The Church was also the only place where people were always Protected.

The Human Heart is doing the same thing as the Church is doing in the City. It connects the Mind with the Organs of  the Body and the Pericardium provides one of the major Rythms of the Human Body.

If we want to Synchronize the City we have to create a Center where People are able to Commun(e)nicate about the Shared Goals of the City. The most important way to share can be found in the Human Culture. The Centres have to contain all the ways Human Culture is performed.

chartres4

The Spiraling Spiral in the Memory Palace of Chartres, the Symbol of the Fractal Universe

About the Memory Palace

Art, Music and Singing were United in the Medieval Cathedrals. The Medieval Cathedrals also contained all the Knowledge (The Memory Palace)  that was needed to understand the Universe.

If we want to Synchronize the Modern City we have to build Modern Cathedrals in the Center of the Center and use them as Places to Commune, to Educate, to Make Music, to Dance and to Sing.

We have to design them as Modern Memory Palaces (Simulators) to teach the Masses about the Fractal Universe and give them Feedback about the State of their own Environment.

sagrada-familia-barcelona-projection

The Sagrada Familia in Barcelona (Antoni Gaudi)

LINKS

This is a  presentation about The City as an Organism

Why the Tube is such an Important Structure

About The Law of Kleiber

Why Innovation is a Combination of Combinations

An Interview with Geoffrey West of the Santa Fe Institute about the Law of Kleiber

About Thermo-Economics

About the Autarkic City

About Cybernetics

About Synchronization

How Gaudi used a Machine to design the Sagrada Familia

About Machines that generate Architectures

How the Heart Synchronizes the Body

About Super Organisms

About Smart Technology

About Addiction

About the Master/Slave-relationship

About the Heart

Why the Cathedral contained all the Knowledge that was Needed

About the start of Civilization

About the Good, the Bad and the Nothing

Monday, July 5th, 2010

Sometimes You want do the Good Thing and the result is a Bad Thing.

Sometimes You start an Activity with a Good Intention but the Activity runs completely Out of Hand because You were unable to Predict all the effects.

Sometimes an Activity starts with a Good Intention and Ends with a Good Result but much later You discover that you have started a Chain of Activities and the Result of the Chain is Bad.

Good and Bad are Context Dependent.

Look at this little Story:

A Chinese farmer awoke one morning to find a horse grazing in one of his fields. Since nobody came to claim the horse, the farmer used it to plow his uncultivated land. “What a good this horse is for you,” said his neighbor. “We’ll see,” said the farmer. Soon after, the farmer’s 18-year-old son was riding the horse when it shied at a snake. The youth fell off and broke his leg. “What an evil this is for you,” said his neighbor. “We’ll see,” said the farmer. Soon after, the warlord’s agents combed the village, conscripting young men for his army. Because of his broken leg, the farmer’s son was not taken. “What a good this is for you,” said his neighbor. “We’ll see,” said the farmer.

The Devil in the TAROT

The Tarot Card the Devil: The Devil, the God Pan, is the Force of Desire. On the Card we can see that Humans are able to Lift the Chains of Desire.

About the Golden Rule of Ethics

The big problem is that we attach a Value to the Intentions or the Consequences of an Activity and this Value has just two Names named: “Good” or “Bad“.

When the E-Valuation is “Good” we Feel Good. When the E-Valuation is “Bad” we feel Anger, Sadness, Guilty, Depressed or Worse. The Names of the Evaluation trigger the Emotions Deeply.

Most of the time we feel Guilty without any need to feel Guilty because we are not in Control of Everything that Happens.

The Ego, our Identity, is not able to stand the Lack of Control it has on the Outside World.

Humans need the “Illusion of Control” to sustain their Identity. Deep within they don’t want to believe that Other Human Beings or the Forces of Nature (“the Acts of God“) are also Active Agents.

The Science that is trying to define the Good and the Bad is called Ethics.

Ethics is the Philosophical Study of  “the Moral Value of Human Conduct and of the Rules and Principles that ought to govern it” or “the principles concerning the distinction between Right and Wrong or Good and Bad Behavior’

The most important Rule of Ethics is the so called Golden Rule : ” We should do to others what we would want others to do to us” or “do not do to others what you would not like to be done to you“.

The Golden Rule of Empathy and Forgiving is the most important Ethical Rule in many Cultures all around the World.

The Golden Rule uses the Principle of Self-Reference. Something is Applied on Itself.

When we apply the Principle of Self-Reference to the Bad, we move into the Domain of Lady Justice.

Lady Justice

Lady Justice: An Eye for an Eye (Revenge)

About Justice and Revenge

When Bad People don’t want to use the Good Golden Rule some cultures use the Rule “An Eye for an Eye” (Revenge) to do Justice to the Bad Ones. When you do Bad to a Person, the Bad Behavior is Applied to Yourself. “Eye for an eye” is used to make Bad People Aware of the Good Golden Rule.

The interesting point is that the combination of the Golden Rule and the “Eye for an Eye“-Rule  (called Tit-for-Tat) is the best way to solve the Iterated Prisoners Dilemma, a Chain of Conflicts.

Strangely enough The Good (now called Cooperation)  is Accomplished when you just Imitate the Behavior of the Others.

There is One (fatal) Exception to the Strategy of Tit-for-Tat. If you want to operate On Your Own and You think You don’t need the Others to Survive the best Way to Operate is to Betray Every Body. Bad Behavior provides you with a Huge Economic Win but also with a  Huge Social Loss.

When you Betray the others, You have To Protect Yourself for the Revenge of the Others.

To Compensate your Social Loss You have to Buy the Others (Lawyers, Mercanaries, ) or (worse) you have to Force the Others with Violence to Cooperate with You.  In the first Case you lose a lot of Money and in the last case the Amount of Revenge increases exponentially.

To Survive you always have to Balance the Social (“the Other“) and the Economic (“the Ego“). The “easy way’ is to do the Good and the “difficult way“ is to do the Bad.

Cooperation is a highly Effective and Efficient Method to accomplish the Good for Your Self and the Others because Good Behavior attracts Good Behavior and Bad Behavior attracts  Bad Behavior.

Seven Virtues

The Seven Virtues

About the Seven Virtues.

To do Good or to do Justice you have to Apply the Same to the Same because The Same Attracts the Same.

The Same not only Attracts the Same. The Same Repels the Not-the-Same.

The Same Law:  ”Look-A-Likes Like Each Other and Repel the Organisms that are not looking alike” (called Genetic Similarity Theory) also explains Evolution. This rule was detected by Charles Darwin when he wanted to explain the huge amount of Cooperation in Nature.

When you Apply the Same to the Same (Self-Reference) it generates a Fractal Structure in which the Same Structure Repeats Itself on Every Level (“As Above, So Below“).

It Looks Like there is a Universal Moral Law that explains  Behavior on Every Level of the Universe.

Some Scientists believe the Moral Laws are Eternal Truths. This believe has a long History that goes back to Ancient Rome (Cicero), Ancient Greece (Plato, Socrate) and Ancient Egypt (Ma’at).

During the Renaissance (“Rebirth“) the Moral Principles of Rome, Greece, and Egypt were reborn and transformed into Moral Systems based on the Four (Prudence, Justice, Temperance and Fortitude) or the Seven Virtues (The Four Virtues & Faith, Hope and Love).

When you understand and apply the Universal Moral Laws you use the Virtue of Prudence (Wisdom, Knowledge).  To Apply the Golden Lesson you need to have Courage and a Strong Will, the Virtue of Fortitude. When you keep the three Virtues (Wisdom, Justice and Will) in the right Balance you are practising the virtue of Temperance.

They last three virtues are derived out of the First Letter to the Corinthians written by Paul:

“If I speak in the tonguesof men and of angels, but have not love, I am only a resounding gong or a clanging cymbal. If I have the gift of prophecy and can fathom all mysteries and all knowledge, and if I have a faith that can move mountains, but have not love, I am nothing. If I give all I possess to the poor and surrender my body to the flames,but have not love, I gain nothing.

Love is patient, love is kind. It does not envy, it does not boast, it is not proud. It is not rude, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, always perseveres.

Love never fails. But where there are prophecies, they will cease; where there are tongues, they will be stilled; where there is knowledge, it will pass away. For we know in part and we prophesy in part, but when perfection comes, the imperfect disappears. When I was a child, I talked like a child, I thought like a child, I reasoned like a child. When I became a man, I put childish ways behind me. Now we see but a poor reflection as in a mirror; then we shall see face to face. Now I know in part; then I shall know fully, even as I am fully known. And now these three remain: faith, hope and love. But the greatest of these is love”.

Balance of Ma'at

The Golden Rule of Ma'at (Forgiving and Empathy)

About the Temperance of Ma’at

Ma’at (later  named Sophia ) was one of the (3×3) Nine Egyptian Neteru . She represented the Virtues of Truth, Balance, Justice, and Order. Just like Lady Justice she uses a Scale.

The dead person’s Heart was placed on the scale, balanced by the Feather of Ma’at. If the Heart weighed the same as the Feather of Ma’at, the deceased was allowed to go on to the Eternal Afterlife and gained Immortality (Heaven).

If the deceased had been found not to have followed the concept of Ma’at during his life (if he had lied or cheated or killed or revenged) his heart was devoured by a demon and he died the final death and was Reincarnated (Hell).

The Human,  the Talking Baboon,  had to start all over again to Learn and Apply the most important Golden Lesson of the Universe.

The way of Ma’at was to Act out of the Center of Conscioussness, the Heart. Acting out of the Heart is called the Virtue of Love.

Ma’at was also the God of Balance, Rythm and the Cycles. The Good and the not-Good moved the Cycles of the Universe.

The Good and the not-Good came out of the Same Source, the Void, the Sekhem, the Source of Infinite Potential and Love.

The Tree Faces of Shiva

The Tree Faces of Shiva (Creator, Destructor and Brahman (Void, Love, Infinite Potential))

About the Trinity of the Mind

In the Eastern (Gnostic, Egyptian, Indian, African, Mystic) Believe System the Good and the Bad are both created by the Demiurg (or the Logos, “In the beginning was the Word and Word was God“) one of the Creations of the Void. The Void exists outside our Level of Existence. The Void is a Mystery.

The Good and the not-Good, the Bad, are Complementary Opponents. Their Union is the Whole. They Share the Void, the Empty Set. The Empty Set contains Everything that is Impossible.

The Good and the Bad Move the Cycles of the Universe. They represent the E-Motions (Expansion (Desire, Creation) and Contraction (Control, Destruction)) of the Void.

The Good and the Bad Cooperate by Competing to bring the Lonely Souls back to the Void.

The Good, the Creator, and the Bad, the Destructor, are Two Faces of the Same Entity, the Spiritual Trinity, who came out of the Nothing.

The connection with the Void can be found in the Middle (the Golden Mean) of the Body, the Heart, the Link between the Triangel of the Material World (the Body) and the Triangel of the Spiritual World (The Mind).

The Heart, the connection with the Void,  is the Seat of Consciousness. It is also the Seat of the Emotions (Soul).

When the Heart Weights like a Feather, the Upper (Mind) and the Lower Trinity (the Body) connected by the Emotions are in Balance.

The Bad and the Good are Attractors. When the Attraction becomes too High you get Addicted to the Bad or the Good. A huge Expansion is compensated by a huge Contraction to Balance the Cycle. When you want to avoid the Pain of Contraction you have to Balance your Life.

The Two Triangels with the Heart in the Middle are the Same As the Seven Chakra’s (2×3+1) and the Seven Virtues of the Renaissance.

The Seven Chakra’s together with the Spiritual Trinity are Similar to the Tree of Life, The Egyptian Pesedjet/Neteru and the Jewish Sephirot (3×3+1).

The Seven Chakra's

The Seven Chakra's

About the Trinity of the Body

When Body and Mind are in Balance, the Ideas of the Spiritual Trinity are received by the Trinity of the Mind and are Implemented by the Trinity of the Material Body in the Material World (Earth).

When the Mind is not connected to the Body and the Spirit, the Ideas become Abstractions (only Words) or the Impulses of the Desire of the Body Takes over Control.

When the Mind is not connected to the Spirit Innovation stops and Old Ideas (the Seeds) Repeat Themselves All the Time. The Human becomes a Machine Caught in the Wheel of the Good and the Bad.

The One, the Spiritual Trinity, who is The Void, the Good (the Creator) and the Bad (the Destructor), uses the Strategy of Tit-for-Tat, the Trickster.

The Trinity Imitates the Behavior of the Humans or We the Humans Copy their Behavior (“We are made in the Image of God“)!

The One wants to make the Humans Aware of the Fact that the only way to Survive the Infinite Chain of Conflicts (Dilemma’s, Contradictions), which is Life, is to Cooperate (Live out of the Heart) with the Infinite Potential, the God of Love.

Implosion

The Big Implosion

About Destruction

The Golden Rule of Ethics tries to avoid the Negative Emotions, the Emotions of Loss. It wants to Create a situation were Everybody is Happy.

The Golden Rule of Ethics helps a lot to Evaluate our Intentions and the Consequences of our Intentions but the effect of our Collective Actions on the Bigger Structures is almost Unpredictable.

It would help a lot if we could Expand our View from the Emotional Human Other to “all the Other Life-Forms” that are in Existence.

In this case we have to Understand the Effects of our Behavior on the Negative Emotions of the Bacteria, the Plants, the Animals, Earth and even the Universe.

It is not clear if our Emphatic System is able to Imagine the Inner World of the other Life Forms.

If the Act of Destruction is felt the same by all these Organisms it will be simply impossible to Live.

We have to Destroy Others to stay Alive.

The Universe Destroys to Stay Alive.

The Destruction of the Old is needed to Create the New.

At this moment we are confronted with a World-Wide Crisis on many Levels of Existence. The Collective Behavior of the Human Race is Destroying our Environment.

The Collective Desire to Create and Expand has reached the Limits of the System we Live in.

It is almost impossible to Control this Expansion. The Control Systems have also reached their Limits.

The Two Complementary Forces, The Good and the Bad, Expansion and Contraction, Desire and Control,  have both reached their Outer Limits.

If they both reach the Limits the Expanding Balloon of our Outer Universe will Explode in Itself (Implode) and move through the Void to another State. There is Simply no Space Left to Expand.

If this happens the Expansion of the Outer World will be Transformed into an Expansion of the Inner World.

The Big Bang, the Big Explosion, the Sublime Act of Creation, will Repeat itself and the Big Implosion will Create a New Universe.

The Flow of Time

The Flow of Time

About Not-Acting

Wu Wei (Not Acting, Acting without Effort) is an important concept of Taoism.

When you want to apply this concept you have to Know Everything There is to Know about the Cycles and the Forces of Nature. Wu Wei is only possible when you have aquired the Knowledge and the Wisdom of Ma’at/Sophia.

Most of the Time the Power of the Cycle is so Strong that it is impossible for Us to influence the Flow of Time.

Sometimes on a Rare Moment on a Special Place the Cycles can be Influenced.

Only at that Special Magic Moment the Human is able to do the Good or the Bad.

At all the other Moments History Takes its Course. The Good and Bad Forces, the Acts of God, Act without our Consent on the Higher Levels of Our Universe. It is a Waste of Time and Energy to Oppose these Forces.

Most of the Time we are Pushed by the Forces of Nature. We Act without Effort.

In Ancients Times the Forces were called Gods, Angels or Devils. At this moment we Name these Forces with Different Names (Gravity, the Strong and Weak Force and the Electro-Magnetic Force) but the Effect of the Forces on our Field of Existence stays the Same.

At a Magic Moment Everything, even the Impossible, is Possible. At such a moment we are ”in Tune with“  the Empty Set, the Void, the Sekhem, the Tao, the Field, the Love, the Mystery, the Unspoken.

The Emotions of Loss are created when we Lose Something.

If you Own Nothing You have Nothing to Lose.

If you Expect Nothing, Every Thing will Happen.

When people see things as beautiful,
ugliness is created.
When people see things as good,
evil is created.

Being and non-being produce each other.
Difficult and easy complement each other.
Long and short define each other.
High and low oppose each other.
Fore and aft follow each other
.

Therefore the Master
can act without doing anything
and teach without saying a word.
Things come her way and she does not stop them;
things leave and she lets them go.
She has without possessing,
and acts without any expectations.
When her work is done, she takes no credit.
That is why it will last forever
.

(Tao Te Ching,  Chapter II)

LINKS

About Game Theory and Ethics

About Cybernetics and Ethics

About Social Balance Theory

About Addiction

How to Resolve a Conflict

About the Seven Virtues

About the Illusion of Control and other Human Biases

About Morality and Ethics

About Ethical Dilemma’s

About Tit-for-Tat

About Tit-for-Tat and World Politics

About Genetic Similarity Theory

About the Renaissance

About Ma’at and the Measure of the Heart

About the Nine Egyptian Concepts (the Neteru)

About Plato and the New Harmony

About Socrate and the Golden Rule

About the Demiurg

About the Mystic Experience

About the Heart

About the Void

About the Sekhem

About the Trinity

About the Trickster

About Understanding Mathematics

Thursday, July 1st, 2010

When I was a student I made some money by helping young people to understand Mathematics. I Listened to the way they tried to Solve the Puzzles their Teachers gave them and tried to find out their Personal Way of Reasoning. 

When I found their reasoning problem I gave them a new Way to Think.

This approach was very effective. Sometimes I just needed one hour and the highly frustrated pupil transformed into a Lover of Mathematics. They saw that Mathematics is the Art of Beauty and Harmony.

The main problem most of these pupils had was the Why-Question. They wanted to know what was “behind” mathematics. In Mathematics there is no Why. Mathematics is “Just the Way it is“. It is a System of Rules you have to Apply.

When you are a Highly Creative Person Mathematics is a Highly Frustrating Science.

Highly Creative Persons are Highly Associative.  They See Links between Every-Thing. They Build Bridges between Islands almost nobody knows they exist.

Mathematics is about creating Abstractions.

Highly Creative Persons Think with their Imagination. They Jump from Images to Image.

The Images of the Mathematicians are Abstract Symbols and most of the time the Mathematical Symbols are chosen without any relationship to the Situation they want to represent.

If Creative, Associative, Persons are not able to See what Mathematics is trying to represent they are Lost in Space.

When we go back in History the Imagination (and not Thinking/Reasoning) was the most important “Thinking-Tool“. The meaning of the word “Think” originally meant “to See“. 

The Art of Reasoning was based on detecting Correspondences.

The Ancient Scientists observed Nature and found out that certain situations “looked alike“. They used and linked Visual Metaphors to explain what they observed. 

Abstract Concepts were always represented by something others could See.

The interesting point is that New Scientists are founding out what the Old Scientists already knew. Our Basic Reasoning System is based on Images.

When we move to the Abstract Domain we use Images (called Prototypes) to Reason. Abstract terms like Furniture are always represented by a prototype we have Seen.

This explains why most of the People are not able to See Mathematics and why most of the People have a big problem Understanding Ancient Knowledge.   

They try to Map the Ancient Visual Landscape of Knowledge to the Modern Space of  Abstract Terms they don’t understand.

They are looking for something that is already there.

They are asking the Why-Question while the  Ancient Knowledge is Visible All Around Us.

LINKS

About the Foundation of Mathematics

About Old Fashioned Thinking

About Topology, the Art of Seeing

About the Art of Creation

About Magic and the Memory Palace

Tuesday, June 1st, 2010

In the 5th century AD, Martianus Capella defined the Seven Liberal Arts: Grammar, Dialectic, Rhetoric, Geometry, Arithmetic, Astronomy, and (Musical) Harmony. The System of Capella became the model of the Medieval University.

The term liberal means “appropriate for free men“.  The Liberal Arts represented the kinds of skills and general knowledge needed by the Elite. The Servile Arts represented Specialized Tradesman Skills and knowledge needed by persons who were employed by the Elite.
liberal arts

The Seven Liberal Arts

The Number Seven was not taken by Accident. This number represented the Seven Days of the Week, the Seven Spheres, the  Seven Planets, the Seven Notes and the Seven Colors.

The Seven Planets of the old cosmology included the Sun (Sol) and the Moon (Luna), which we now don’t regard as planets at all.   The other Five were Mercury, Venus, Mars, Jupiter, and Saturn, the 7th planet and the Son of the Sun.

seven planets of Ptotelemy

The Seven Planets in the System of Ptolemy

In Ancient Babylonia,  the Seven Spheres were represented by a Building , A Ziggurat, that looked like the Tower of Bable. This Babylonian Step Tower was a Visual Model of the StairWay to Heaven.  The Seven Spheres were divided in three Heavens and three Hells with the Earth in the Middle. The Pyramids in Egypt are based on the same model.

The Seven Heavens

Dante and The Tower of Bable

The Three Heavens contained Three Hierarchies of Angels, Nine in total. At the Top, the Eight Step,  the Creator was situated. Again the Creator consisted of three Entities, The Father, The Son and The Holy Gost (The Trinity).

As You can See the Number Three plays a very important role in the Ancient Models. The main reason could be that this number is the most efficient number when you want to create a Number System.

If you use this number as the Base System the Universe could be explained as Triangels within Triangels. With the help of the Trinity-Fractal Ancient models like the Tree of Life, and the Egyptian Pesedjet are easy to understand. The Trinity-Fractal comes back in modern Physics as the Triangel of Pascal.  In Ancient China the Triangel of Pascal was called the Wei Chi. The Wei Chi was used to play the Game of GO.

The The Three Levels and the Nine Hierarchies of Angels

Earth (the Grave), The Three Heavens and the Nine Hierarchies of Angels

In 1275 Raymond Lull designed a tool to Classify and Generate all the Knowledge of the Universe based on a Set of  Nine (3×3) Concepts.  His invention was described in his book Ars Magna (“The Ultimate General Art”).

Lull’s logic machine consisted of a stack of concentric disks mounted on an axis where they could rotate independently. The disks, made of card stock, wood, or metal, were progressively larger from top to bottom. As many as 16 words or symbols were visible on each disk. By rotating the disks, random statements were generated from the alignment of words. Lull’s most ambitious device held 14 disks.

The disk contained the Nine most important (Christian) Concepts of the Universe:  Bonitas (Goodness), Magnitudo (Greatness), Eternitas (Eternity), Potestas (Power), Sapientia (Wisdom), Voluntas (Will), Virtus (Virtue), Veritas (Truth), and Gloria (Glory).

Lull’s fame as a Mystic spread throughout Europe during the late Middle Ages and Renaissance. Centuries later, the German mathematician Leibniz acknowledged the importance of Lull’s Machine.

The idea for the machine came to Lull in a Mystical Vision that appeared to him after a period of fasting and contemplation. In reality, Lull may owe his Divine Inspiration to the Magi. They used a Device called the Zairja, an Idea Generator that used the 28 letters of the Arabic alphabet to represent the 28 Categories of Thought in Arab Philosophy.

Ramon_Llull_-_Ars_Magna

In 1516 Thomas Moore published his book Utopia, a description of an Ideal State. Much later Tommaso Campanella (The City of the Sun), Francis Bacon (The New Atlantis) and Johannes Valentinus Andreae (Description of the Republic of Christianopolis) wrote comparable books. The books were based on Plato’s Republic. All of the books contained Architectual Designs based on Ancient Geometries.

Utopia Moore

Utopia Thomas Moore

Around 1300 Wealthy and Powerfull Merchants (de Medici) in Italy regained interest in the Ancient Knowledge of the Egyptians, the Arab, the Greek and the Romans. This knowledge was almost completely destroyed by the Church of Rome.

Scientists, Poets and Artists (Petraca, Dante, Boccaccio) visited Byzantium and discovered ancient manuscripts written in Greek and Latin.  Greek scholars were encouraged to come from Byzantium to Italy. Byzantium and Europe was ransacked for copies of the long unused Latin classics and copyists multiplied them.

Libraries were founded, and schools for the study of both Greek and Latin in their classic forms were opened at Rome, Mautua, Verona, and many other towns. After the fall of Constantinople in 1453 the proces gained a further impetus because of a number of Greek humanists who moved from Byzantium to Italy.

In 1462 the Platonic Academy was opened in Florence under the patronage of Florentine Duke Cosimo de’ Medici. Its leader became Marcilio Ficino.

Around 1463 Cosimo de’ Medici commissioned Marsilio Ficino to translate a book called the Pimander with high priority  into Latin from a Greek manuscript brought from Macedonia. This book contained the Corpus Hermeticum that was attributed to Hermes Trismegistus.

Hermes Trismegistus “Thrice-Wise“was named Thoth or Djehuti in Ancient Egypt. The Djed of Djehuti represents the Power of the Snake (the Kundalini) in the Spinal Column. Thoth was the Scribe of the Gods, the inventor of writing, and was usually depicted in hieroglyphics with the head of an Ibis Bird. This head of the IBIS carried the Moon, the Planet of the Mother God, The Night and of Magic.

Thoth and the Human represented by the Ape. The Ape uses the Eye of Horus (the third Eye) to look at Thoth

Thoth, the Ibis with the Moon on his Head and the Human represented by the Ape. The Ape uses the Eye of Horus (the third Eye) to look at Thoth. The Eye of Horus is the Pineal Gland, the Seat of the Imagination in the Brain.

At that time Scientists believed that the Older the Truth the Bettter. They believed there was once a Golden Age in which the Eternal Truth was known. From that time on History was in a Downfall until the time of the Resurrection.

Marsilio Fucino dated Hermes Trismegistus “long before the sages and philosophers of Greece, but yet after Abraham, Isaac, Jacob, and Joseph“. His dating of the Corpus “proved” that the writings of Hermes contained the Primal Source of Wisdom.

In 1945 Coptic fragments of the Corpus Hermeticum together with many other Gnostic Writings were found at Nag Hammadi, Egypt.

At this moment we know that the Corpus Hermeticus just like the Emerald Tablet ,another text dedicated to Hermes, was written in Alexandria around 200 BC.

Alexandria, created by Alexander the Great, was the place where the Ancient Knowledge of Heliopolis ,the City of the Sun, was stored in its immense libraries after its destruction. A part of this library was later moved to Byzantium, the Center of the East when the library of Alexandria was destructed.

The most important message of Hermes is that the Physical World is the Manifestation of the Spiritual World. This is summarized in the Hermetic adage “As above, so below.” At this moment we would name the concept of “As Above, So Below” a Self-Reference or a Fractal.  The old Scientists knew that one Concept returned on every Level of Our Inside and Outside Reality.

The Corpus Hermeticus provided a seminal force in the development of Renaissance thought and culture. After the Dark Middle Ages, controlled by the Rule System of the Church of Rome, the Scientists started to do their Own Research and  Thinking.

The Cycle of Culture moved from the Stage of Order to the Stage of Imagination.

Hermes Trismegistus in the Cathedral of Siena

Hermes Trismegistus in the Cathedral of Siena

One of these Scientists was Giordano Bruno. Bruno was born in 1548. At the age of 17, he entered the Dominican Order, was trained in the Liberal Arts of that time  and became an ordained priest in 1572 at age 24. During his time in Naples he became known for his skill with the Art of Memory and on one occasion traveled to Rome to demonstrate his System before Pope Pius V and Cardinal Rebiba.

The Art of Memory belonged to the rhetoric tradition as a technique which enabled the orator to deliver long speeches from memory with unfailing accuracy.

giordano_bruno

Giordano Bruno

At that time people had to Learn a lot of information by Heart. To do this they used a Mnemonic System called a Memory Palace. The memory images were linked with visualized locations that could be found in architecture, books, sculpture and painting, which were seen by practitioners of the Art of Memory as Externalizations of Internal Memory.

Artists used the Images that were part of the Memory Palace as a reference. The Images were linked with Universal Concepts. In the time of Middle Ages and the Renaissance people knew the Meaning of every Image that was used. Geometric Structures, Paintings, Poems and Buildings were carriers of many Levels of Meaning. Some of these levels were only known by the Inner Circle of Ancient Secret Societies.

In the old days the Inside World was seen as the most important place to Explore. The Outside was nothing but a Projection of the Inside.

Examples of Ancient Memory Palaces are Ancient Egypt, the City of Meru (Sri Yantra), the City of Heliopolis, the Babylonian Ziggurat, the Egyptian Pyramid, The Temple of Solomon,  and the most important Memory Palace,  the Zodiac. In the Middle Ages Cathedrals (Chartres), Palaces (Alhambra) and Temples (Taj Mahal) were designed and used as Memory Palaces.

new jerusalem coming down out of heaven
The New Jerusalem

One of the most important Christian Memory Palaces is the New Jerusalem. It is a Vision about the New Earth that will arise after the Seven Seals are Broken and the Seven Trumpets have Sounded.

The Seven Trumpets are a symbol of the Seven Chakra’s. The Apocalyps, the opening of the Seventh Seal, is a symbol of the Jump of the Kundalini to the Kingdom, the New Jerusalem.

A large portion of the final two chapters of the Bible deals with John of Patmos’ Vision of the New Jerusalem. He describes the New Jerusalem as “‘the bride, the wife of the Lamb’“.  After John witnesses the new heaven and a new earth “that no longer has any sea“, an angel takes him “in the Spirit” to a vantage point on “a great and high mountain” to see the New Jerusalem’s descent.

According to John, the New Jerusalem is “pure gold, like clear glass” and its “brilliance [is] like a very costly stone, as a stone of crystal-clear jasper.” The New Jerusalem is a Symbol of  the Renewal of Religion that will take place and which secures the Prosperity of the Human World.

Alhambra
Alhambra

“Some place is chosen of the largest possible extent and characterised by the utmost possible variety, such as a space house divided into a number of rooms. Everything of note therein is carefully committed to the memory…The next step is to distinguish something [to be remembered] by some particular symbol which will serve to jog the memory…. These symbols are then arranged as follows. The first thought is placed, as it were, in the forecourt; the second, let us say, in the living-room; the remainder are placed in due order all round… and entrusted not merely to bedrooms and parlours, but even to the care of statues and the like. This done, as soon as the memory of the facts requires to be revived, all these places are visited in turn and the various deposits are demanded from their custodians, as the sight of each recalls the respective details” (Quintilian Institutio Oratoria, Book XI).

The Temple of Solomon

The Temple of Solomon

The Art of Memory was a very important Art because Manuscripts were very rare and the use of Manuscripts and Symbols was seen as the major cause of the detoriation of society. Books produced “fake” scientists who are not wise but only appear to be wise.

For this invention will produce forgetfulness in the minds of those who learn to use it, because they will not practise their memory. Their trust in writing, produced by external characters which are not part of themselves, will discourage the use of their own memory within them. You have invented an elixir not of memory but of reminding; and you offer your pupils the appearance of wisdom, not true wisdom, for they will read many things without instruction and will therefore seem to know many things, when they are for the most part ignorant and hard to get along with, since they are not wise, but only appear wise” (Plato, Phaedrus).

Chartres

Chartres

Giulio Camillo was one of the most famous men of the 16th century. With a commission of 1200 ducats from the King of France, Francois I, he created a theatre that contained divine powers: whoever entered the theatre would emerge with a complete memory of all the knowledge that had ever existed. The design of his theatre is described in his book L’idea del teatro written in 1554, the last year of his life. The theatre consisted of Seven Levels and contained the complete Knowledge of the Corpus Hermeticum.

Borobudur (Meru Temple) contains the Sri Yantra

Borobudur (Meru Temple)

Following the order of the creation of the world, we shall place on the first levels the more natural things…those we can imagine to have been created before all other things by divine decree. Then we shall arrange from level to level those that followed after, in such a way that in the seventh, that is, the last and highest level shall sit all the arts… not by reason of unworthiness, but by reason of chronology, since these were the last to have been found by men” (Guilio Camillo).

meru

The Design of the City of Meru is based on the Sri Yantra-generator. This Generator generates Quasi-Crystals and Penrose Tilings.

At the time of Gordiano Bruno Copernicus detected that the Sun was the Center of our Planetary System. His discovery was in conflict with the Theories that were educated at the Universities that were controlled by the Church of Rome.

Gordiano Bruno saw that the new theories were in line with the teachings of the Corpus Hermeticum in which the Sun (of Heliopolis) was in the Center. Bruno started to Fuse the Knowlegde of the Corpus Hermeticum with his already highly advanced Memory System.

At that time the power of the Church of Rome was going down rapidly. Europe was divided by religious wars between Catholics and Protestants. The merchants, bankers and artisans of Europe’s largest cities and towns resented the fact that local bishops of the Church controlled all of their commercial and economic activities.

Priests married and then took mistresses, holy offices were bought and sold for the highest price, incompetence among the clergy became the rule, the congregation of more and more people in towns and cities perhaps exposed the amorality and immorality of the clergy.

Taj Mahal (Shiva Temple)

Taj Mahal (Shiva Temple)

Gordiano Bruno believed he had to create a new Religion based on the Ancient foundation of Religion he found in Egypt.

Gordiano was a free-thinker and he never stopped expressing his own believes. When he learned that an indictment was being prepared against him in Naples he fled the convent and started to travel Europe.  He visited the French King Henry III, the English Queen Elizabeth, Wittenburg , the Center of the Protestant Movement, the city of Prague and Emperor Rudolf II.

At every place he came he started to educate the others in his Expanding (forbidden) Vision of the Infinite Universe. At every place his free-thinking brought him into a conflict and he had to move on.

In 1592 he was lured to Venice, arrested by the Inquisition and transported to Rome. In 1600 he was convicted and publicly burned at the Campo de’ Fiori in Rome. In 1889 a Statue was erected at the place of his execution.

The Statue of Giordano Bruno at the Campo Fiero

The Statue of Giordano Bruno at the Campo de' Fiori

At the time of Gordiano Bruno many of the important people where highly interested in Magic, Alchemy and Astrology. One of the people Gordiano visited,  the Emperor Rudolf II,  practised Alchemy in his own private laboratory and spared no expense in bringing Europe’s best alchemists to court, such as John Dee, the personal alchemist of Queen Elizabeth.

There is no evidence that Giordiano Bruno met John Dee when he was in England but he was in contact with the most important people that were part of the “Hermetic Circle” of John Dee. There is a lot of evidence that Bruno was the creator of the Rosicrusians.

Queen Elizabeth and John Dee

Queen Elizabeth and John Dee

Bruno’s books on the Art of Memory were in reality Courses in the Solar Magic taught by Hermes Trismegistus.

Bruno knew that the most important tools of the Magician are the Power of his Imagination and the Connection with the Major Powers in the Hierarchy in the Heavens, the ArchAngels and the Demiurg, the Creator of the Matrix.

The System of Bruno was created to make is possible to generate powerfull images and Magical Rituals to make the Right Connection (Bonding) with the Major Creators of the Universe we live in.

There are three gates through which the hunter of souls (the Magician) ventures to bind: vision,  hearing and mind or imagination. If it happens that someone passes through all three of these gates, he binds most powerfully and ties down most tightly.” “He who enters through the gate of hearing is armed with his voice and with speech, the son of voice. He who enters through the gate of vision is armed with suitable forms, gestures, motions and figures. He who enters through the gate of the imagination, mind and reason is armed with customs and the arts.” (Giordano Bruno, Theses de Magia, Vol. LVI).

An Image to make a connection with the Power of the Sun

An Astrologial Image of Bruno. It contains the Lyre of David (an Ancient Pole Star).

Robert Fludd was born in England in 1574,  thirty years after Giordano Bruno was born. Between 1598 and 1604, Fludd studied the Liberal Arts on the European mainland. Robert Fludd was one of the last Hermetic Scientists of the Renaissance. In his life he tried to describe everything that was known until that day. Robert Fludd was a Rosicrucian.

One of his products is a Musical Memory Palace and a Memory Theatre, the Globe, that was used by Lord Chamberlain’s company to which Shakespeare belonged. In the view of Fludd a theatrical play was a Projection of a part of a Memory Palace.  The Theatre was a tool to educate the Elite (and the Masses) into the Hermetic Whole.

Robert Fludd, The Temple of Music

The work of Bruno and Fludd was taken over by many other Scientists but the relationship between the Art of Memory and Magic slowly faded away.

At the end of the Renaissance Reason, Logic, became much more important than the Seven Liberal Arts.  The Scientists of the Age of Enlightment wanted to get rid of the Delusions of the Imagination. The only thing that counted were Facts observed by the Senses.

Leibniz (1646-1716) was one of the exceptions. He was an Old Fashioned Renaissance Magician and believed in the Harmony of the Spheres. He invented many theories that proved his Point of View and are now proven to be Right.

He believed that scholarship or “perfect knowledge of the principle of all sciences and the art of applying them ” could be divided into three equally important parts : the Art of Reasoning, the Art of Inventing (Combinatorics) and the Art of Memory and the Imagination (Magic). He even wrote an unpublished manuscript about the Art of Memory.

Leibniz got into a big conflict with Newton about the Role of the Imagination and Scientific Discovery in Society. Afther his death Newton and his Protectors, the Medici of Florence, started an effective campaign to prove that Newton was the inventor of the Theories of Leibniz. This is the reason why Newton became the major Scientist of his time.

Rosicrucian_Rose

The Rosicrucian Rose with the Trinity of the Chained Lion of the Sun, the Unicorn of the Moon and the Crossed Triangels of the Center, the Heart Chakra.

After the Renaissance, the Stage of the Imagination, the Cycle of Culture moved to the Stage of the Emotions, the Time of the French Revolution. The Old Social Structures of Europe started their big Transformation.

Around 1740 the Industrial Revolution started in England and the Theories of Newton, Descartes and Kant about the Cold Mechanical Reasoner without Feeling and Imagination took the Lead. The ultimate products of the Industrial Revolution, the Computer and the related Factory-Model are now the Leading Paradigm.

The link between the Machine that Controls and Generates our Universe (the Yantra), the Ideas/Concepts that govern our Universe and the related mostly Astronomical Symbols is now completely lost.

Almost nobody is able to understand and value the Ancient Knowledge of Heliopolis that Scientists like Gordiano Bruno and Robert Fludd were trying to reconstruct during the Renaissance.

The Medici and their Successors, the Big Bankers and Merchants, finally succeeded in the Destruction of the Imagination and the Emotions.

Starting 1950 the Cycle is moving into a new Renaissance.

After the Invention of Quantum Mechanics (inspired by Robert Fludd) it becomes clearer and clearer that The Imagination is the Major Power that Creates and Sustains our Universe.

The Ideas of Bruno, Fludd, Plato, the Corpus Hermeticum and the Ancient Knowledge of Heliopolis are coming Back.

One of the most interesting scientists of the Art of Memory is Alan Turing. He constructed a very simple machine now called a Turing Machine that contained a Memory and a Writing Device that could move backwards and forwards. With the help of this simple Device Turing was able to solve many Mathematical Problems. At the end of his Life just before he committed suicide he started to research Morphology, the Science of the Forms.

What to Do?

The Current University System is highly fragmented. It needs a general principle to Unite all the now diverging Subjects of Investigation. The complexity of our Science is enormous. The link between Science, Art and Religion is gone.  We are in the same state as the World was at the end of the Middle Ages.

Perhaps we could introduce the Memory Palace again. It would certainly help to improve the Memory of the Students who are now Copying and Pasting the Internet.

Behind The Memory Palace lies the theory of the Fractal Universe (“As Above, So Below“) and the Trinity-Pattern. The Same Pattern Repeats itself on every Level of Existence.

The Fractal Universe simplifies Science enormously!

Perhaps the Modern Scientists could start to look for Repeating Structures on many Levels and Exchange these Structures. They could start to listen to the Scientists of Physics and Mathematics who have already found the fractal patterns.

LINKS

An English translation of the Pimander

About the Power of the Imagination

About the Nine Concepts of Egyptian Science

About Alan Turing and the Art of Memory

About the Egyptian Djed

About the 7th Planet, Saturn

Why the Trinity is the Most Efficient Number Base

Why Leibniz was a Magician

About the Triangle of Pascal

About the Power of Reason

About Plato and Harmony

How Robert Fludd influenced Quantum Mechanics

About the War between Newton and Leibniz

How Socrate changed the Cycle of Western Culture

Why the World is one big Memory Palace

About the Renaissance

Why the Industrial Revolution was really a Social Revolution

About Epigenetics and the Power of the Imagination

About the Placebo Effect

Arthur Zajonc: Catching the Light

About Plato and Harmony

Monday, January 5th, 2009
SriYantra

Sri Yantra

The teachings of Ancient Civilizations are often Self-Referential.

The same knowledge is showed on many levels.

What we see with our Senses is an Illusion. Behind this Illusion lies a deeper structure.

The last and deepest level of the ancient teachings is always related to Numbers, Geometry and the Trinity.

This blog is about one of the most important geometric structures of the Trinity called the Sri Yantra.

In the ancient teachings a problem is defined and the teacher gives a clue how to solve the problem.

If the pupil has solved a problem he is able to move to a deeper level.

The whole idea is that real knowledge, wisdom, is only discovered, when the pupil has solved the puzzle of life him- or herself.

Let’s have a look at a Deeper Level.

A deep level is related to a number called Phi. Phi is called the Golden Ratio or the Divine Proportion. It is the real solution of the quadratic equation x**2-x-1.

It is also a solution of the proportion a:b=b:a+b, the sequence of Fibonacci x(n+2)= x(n+1) + x(n), the geometric structure of the Pentagram, the Fifth Element (the Quintessence, the Ether) and the Logarithmic (Golden)Spiral.

Phi is the pattern behind the Egyptian Pyramids, the Stock Market, Harmony in Music and Architecture and many other fields of science including Physics.

Let us first have a look at the way the old teachers have hidden the knowledge of the Divine Proportion in their teachings.

plato

Plato

A beautiful example is Plato.

Plato was an initiate of the Mathematikoi, the Secret Society of Pythagoras. Pythagoras was initiated in the Secrets Societies of Egypt.

What do you think of this problem-statement:

What are the most perfect bodies that can be constructed, four in number, unlike one another, but such that some can be generated out of one another by resolution? … If we can hit upon the answer to this, we have the truth concerning the generation of earth and fire and of the bodies that stand as proportionals between them (Timaeus 53e)”

and

Two things cannot be rightly put together without a Third; there must be some bond of union between them. …and the fairest bond is that which makes the most complete fusion of itself and the things which it combines, and proportion (analogia) is best adapted to effect such a union”.

and

“For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it, and again, when the mean is to the first term as the last term is to the mean – then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one [Timaeus 31b-32a]“.

In the last citation Plato is formulating a mathematical problem related to the four bodys A,B,C,D with the three proportions A:B = C:D = (A+B) : (C+D) = (C+D) : (A+B+C+D). This problem is unsolvable if you don’t have a clue where to start.

This problem is solved when you realize that the Divine Proportion has many strange relationships that are very useful to solve the puzzle.

These relationships can be found if you know everything there is to know about Triangles and Triangles are again related to the Trinity (“Two things cannot be rightly put together without a Third“).

The Trinity comes back in the structure of the Dialogues of Plato. They are divided into Three Parts (and an introduction).

The structure of the dialogues relates to itself.

movmetatronThe knowledge of the self-reference of the Dialogues is also a clue to solve more complicated puzzles about the Dialogues themselves.

Just like the famous book of Hofstadter (Goedel, Escher, Bach) the dialogues show a new layer everytime they are read with a new aquired Insight.

The dialogues of Plato are organized according to the model he wants to teach. There are seven layers (-1,-2,-3,0,1,2,3) related to the Seven Mirror-Universes (or Hells and Heavens) of Our Universe (Eight, the Whole) in our Multi-Universe.

The Seven is a combination of Two Trinities with the Zero (The Void) in the middle. The Eight (2**3) State is the Dialog, The Whole, itself.

The Seven layers are divided into Three Sections (the Trinity) so the total amount of clues is 7×3= 21 + 1 (the Whole) = 22.

22 divided by 7 is an approximation of the number π (Pi).

Pi relates the Square to the Circle.

The Square represents the Playing Board of the Universe. On this Board we, the Humans, play our Game of Free Will.

Our Free Will is an Illusion. We are controlled by the Matrix.

The Circle represents the Wheel of Fortune, the Matrix, that Governs the Seven Universes in our Multi-Universe and the Game of Life.

What is Plato Trying to Explain?

golden ratio

The Golden Ratio

The Divine proportion is the basic concept behind Harmony.

The Divine Proportion is Not Symmetrical so Harmony is not related to Balance.

If everything would be balanced the Universe was never created.

Harmony is Balanced Unbalance.

An Architecture is beautiful when there is a slight unbalance in the Design.

This Unbalance shows the Sign of the Creator.

The Universe is created out of an Unbalance between Two Forces, the Positive and the Negative, the Good and the Bad.

The Two forces (-1,0,1) are divided into Four Forces (-2,-1,0,1,2) with the One in the Middle (Five) and are expanded into Seven Levels (-3,-2,-1,0,1,2,3). The Four Forces are “the most perfect bodies that can be constructed, four in number“.

The Seven Levels are related to the Circle. The Four Forces are related to the Square. The Universe oscillates between The Square and The Circle.

When the Square, the Game We Play with the Four Forces and the Circle (The Five Fold Cycle, Our Destiny) are in Balance the Human is in the Tao and Magic happens.

Every division is a split of the Trinity into Trinities until a state of Balance is reached. At that time the process reverses.

mt meru

Mount Meru

The unbalance of the Good (+1) and the Bad (-1) is an Illusion. They Cooperate to Create Harmony (0).

Behind the perceived Unbalance is a balancing principle, the Divine Proportion. This principle brings everything Back into the Balance of the One who is the Void (0).

What we don’t know is to be known if we understand the progression of the Divine Spiral.

The Future, the Third Step, is a Combination of Two Steps in the Past (The Fibonacci Sequence), nothing more.

Is there a Deeper Structure Behind the Divine Proportion?

golden mean spiral2

Golden Mean Spiral

The Four Forces (Control, Desire, Emotion(Compassion) and the Whole of the Trinity, Imagination) and the related sacred geometry were a guiding principle for the Imagination and the E-Motivation of many Western Scientists.

They tried to Control the Chaos of the Desires of the Senses by enforcing the Rules of Scientific Falsification.

The Proces of Falsification destroyed the Human Intuition.

The Western Scientists forgot to look at the Source of Intuition, the Center, the Quintessence (The Fifth, Consciousness).

The principle behind the Quintessence (Ether, Chi, Prana) is related to higher order symmetries and is a solution of a generalization of the generating function of the Divine proportion X**2 -X – 1

This generalization is X**2-pX-q or X(n+2) = pX(n+1) + qX(n). The solutions of this formula are called the Metallic Means.

When p=1 and q =1 the Divine proportion comes back again.

When the p=3 and q=1 a new sequence X**2-3X-1 or X(n+2) = 3*X(n+1) + X(n), the Bronze Mean appears.

The Bronze Mean generates the pattern: 1,1,4,13,43, the pattern of the Sri Yantra.

continued fraction golden meanIt shows a very beautiful pattern of “3″s when the Bronze Mean is evaluated in a Continued Fraction. The Golden Mean is a continued fraction of “1″s.

There are many more “Metallic Means” (other p’s and q’s). They are related to all kinds of symmetries and fractal patterns.

The Bronze Mean shows that behind the Trinity of the Golden Mean lies another Trinity (and another Trinity and …).

quasycrystal

Penrose Tiling

The Bronze Mean is the generator of so called QuasiCrystals.

Quasy Crystals play a very important role in the Electro Magnetic Structures of our Body, the Collagens.  Collagens are the most abundant protein in mammals,making up about 25% to 35% of the whole-body protein content. The Collagens in our body explain the Ancient Chinese Science of Acupuncture.

Quasi Crystals are “normal” Crystals with a very complex symmetry.

They are ordered AND not-ordered.

One of the most beautiful examples of the patterns behind Quasi Crystals are the Penrose Tilings.

They were developed by the famous physicist Roger Penrose. He used the tilings to show his insights about consciousness.

Penrose believes that Our Universe is just like our Body a Quasi Crystal, a Hall of Mirrors. We the Souls travel all the Paths of this Magnificent Fluent Crystal.

Was the Bronze Mean Known by the Ancient Architects?

lalita

Tripura

The most important implementation of the Bronze Mean can be seen in the Sri Yantra (“Sacred Device”).

The Sri Yantra is related to the Red Triple Goddess of Creation, Tripura also named Lalita (“She Who Plays“).

The Sri Yantra is generated out of the FiveFold Pattern or Creation and the Four Forces of Destruction.

It contains  9 interlocking isoceles triangles. 4 of them point upwards and represent the female energy Shakti, while the other 5 point downwards, representing the male energy Shiva.

The standard form of the Sri Yantra  constitutes a total of 43 triangles. The centre of the Yantra has a Bindu which represents the Void.

The FiveFold Pattern of Creation moves with the Clock. A pattern that moves with the clock is a  generating pattern.

It moves away from the void and generates space. The FiveFold Pattern is the pattern of the Universe. It creates Universes, Galaxies and Planets. The Pattern moves around the Cellestial Center of Creation,  the Black Hole.

The FourFold Pattern moves Against the Clock and is a destructing pattern. It dissolves space and moves back to the void.  The FourFold pattern is the pattern of the Human Being and Earth.

The combination of both patterns is a Moebius Ring (the symbol of infinity) with the celestial Centre in the Middle. The FiveFold/Four Fold pattern resembles the Ninefold Egypian Pesedjet and the Ninefold Chinese Lo Shu Magic Square.

From the fivefold Shakti comes creation and from the fourfold Fire dissolution. The sexual union of five Shaktis and four Fires causes the chakra to evolve” (Yogini Hridaya (Heart of the Yogini Tantra)).

In Pakistan the Mother Goddess (Sharika) is represented by a diagram that contains “one central basic point that represents the core of the whole cosmos; 3 circles around it and 4 gates to enter, with 43 triangles shaping the corners“.

The Penrose Tilings and many other quasi crystals can also be found in Ancient Roman, Islamic and Christian Architecture (Pompei, Alhambra, Taj Mahal, Chartres). The tilings are an expression of the Game of Life and were used to build Educational Buildings (Pyramids, Cathedrals,..) to teach and show the old teachings.

kepler_spheres-l

Keplers Model of the Solar System

Kepler (1570-1640), a German Mathematician and Astronomer (The Cosmographic Mystery) and Albrecht Durer (1471-1528), a German Painter, knew about the Penrose Tilings but until the discovery of the Penrose Tilings nobody knew that they knew.

The new scientists (re)discovered old patterns that were known by the old scientist

What is the Meaning of the Bronze Mean?

The Bronze Mean shows the effect of a continuous division of the Universe in Trinities.

It shows that the Universe (and other levels) is suddenly moving from an ordered state to a chaotic state.

This chaotic, not predictable, state is not chaotic at all when you understand the patterns behind chaos.

In our Universe chaos is always ordered. Chaos is an effect of something that is happening in a higher (not Sensible) Dimension or A Higher Consciousness.

m-brane

The Two Brains of Paul Steinhardt

The writer of an important article about Penrose Tilings and Islamic Art, Paul Steinhardt is like Roger Penrose a well known physicist.

He has created a new theory about the Universe based on Four Forces AND the Quintessence.

In this theory the Universe is Cyclic. It is expanding and contracting.

The expansion of the Universe ends when the Two Major Structures (-1,0, 1) in the Universe, called Membranes or Branes, are in Balance with the Center (0, the Void).

The membranes are higher dimensional Squares that are in parallel.

The Braines at both sides split into many similar cell-like structures. We live in one of the Cells of the Universe.

Adam_Kadmon

Adam Kadmon

The Others, our Twins, live on the other membranes and are not aware of our existence until the Brains are getting into Balance.

At that moment the Twin Universes are Connected.

Scientists don’t know when this will happen but the Old Scientists who could travel the Multi-Universe with their United Brains knew.

It would happen at a very special Alignment of the Five Fold Center of Creation of the Milky Way with the FourFold Cross of the Destruction of Earth.

The Bronze mean is the Master-Pattern of our Multi-Universe.

The pattern 1,1,4,13,43 is in its 42nd enfolding and soon we will experience the 43th step, a Merge of the Left and the Right Brain of the Super Conscioussness, Adam Kadmon.

LINKS

About the Divine Trinity Pattern

How to Raise the Djed

About the Nine-Fold Pattern of the Egyptian Pesedjed

Everything you want to know about the Divine Proportion

About The Indefinite Dyad and the Golden Section: Uncovering Plato’s Second Principle

About the Self-Referential Structure of the Dialogues of Plato

About the Law of Three of Gurdjieff

About Sharika, the Mother Goddess

About the Metallic Means

Paul Steinhardt, About Penrose Tilings and Ancient Islamic Art

About Penrose Tilings and the Alhambra

About the Geometric Patterns in Ancient Structures

An interview with Roger Penrose about the relationship between Conscioussness and Tilings

A lot of information (including simulations) about the Cyclic Universe of Paul Steinhardt

A simple model for the formation of a complex organism

How life emerged out of one Quasy Crystal

About Quasy Crystals and Sacred Geometry