## Archive for the ‘Physics’ Category

Thursday, April 28th, 2016

Res Bina (REBIS), Double Matter

This blog is about Logic, Alchemy and the relationship between Logic and Alchemy.

Logic is based on the Square of Opposition of Aristotle. This square was discussed for more than 2000 years. Recently scientists detected that the Square is really a Cube. It misses an extra dimension.

The Square describes  the six oppositions (“dual combinations”) of possibility (“some”, Particular Affirmative), necessity (“all”, Universal Affirmative), not-possibility (“not some”, Particular Negative) and not-necessity (“not all”, Universal Negative).

Geometric Logic is a part of Modal Logic which is a part of Logic.

In Geometric Logic the Linguistic Representations of Logic are mapped to the Graphical Representations of Geometry.

Modal Logic works with the notion that propositions can be mapped to sets of possible worlds.

The idea of possible worlds is attributed to the famous mathematician Gottfried Leibniz (1646-1716), who spoke of possible worlds as ideas in the mind of God and argued that our actually created world must be the best of all possible worlds. Possible worlds play an important role in the explanation of Quantum Mechanics.

Alchemy is based on the Square of the Four Elements. The Four Elements are oppositions. The oppositions of Alchemy are related to the oppositions of the Square of Opposition of Aristotle in an orthogonal model. They form an arc of 90 degrees, which makes Alchemy the complex part (imaginary) of Logic.

In his work De Arte Combinatoria Leibniz developed a general theory of science that was based on a fusion between Alchemy and Logic.

The Square of Leibniz

In this blog I will explain Alchemy and Logic and show that Alchemy and Geometric Logic share the same geometry, the Hexagon (the Seal of Solomon), which is a 2D-mapping of the Cube of Space of the Sefer Yetsirah.

Drawing of Leibniz’s calculating machine, featured as a folding plate in Miscellanea Berolensia ad incrementum scientiarum (1710), the volume in which he first describes his invention.

Then is all the secret found.
George Ripley (d. 1490).

One of the most complicated ancient architectures is the architecture behind Alchemy. The architecture of Alchemy is full of strange concepts and allegories.

Alchemy comes from the Arabic al-khemia meaningthe art of transformation.

Alchemy is an ancient science that was practiced all over the world. The most famous teacher was called Hermes Trismegistus in Greece and Thoth in Egypt. The old science of Hermes came back in the Renaissance.The essence of Alchemy is called REBIS (Res Bina), Double Matter. REBIS is the end product of the alchemical magnum opus or Great Work. It is the Fusion of Mind and Matter (the Body).

Double matter is sometimes described as the divine hermaphrodite, a reconciliation of spirit and matter, a being of both male and female qualities as indicated by the two heads within a single body.

The Great Work is the chemical and personal quest to create Double Matter, the Philosopher’s Stone. The stone is the agent of chemical transmutation, and the key ingredient in the creation of the elixir of life, said to heal all diseases, induce longevity and even immortality.

In current Physics  transmutation is proved at room temperature in a proces called Cold Fusion.

In Physics one explanation for the fusion between Quantum Mechanics and Gravity is called the Wheeler Feynman Absorber Theory. In this theory the Future causes the Past and vice-versa.

The Philosopher Stone

The aim of the Great Work is a fusion with the One (the Blazing Star, Saturn, Point) that is divided in the Two (Sun & Moon, Male & Female, Line) and the Four Planets (Square), the 12 (4×4-4) Constellations or more structures based on a power of a power (of a power….)  of 2. The Great Work moves from 2**N, …., Sixteen to Four to Two to One (or from one to two to four) in a Spiraling Spiral Motion.

Saturn is the 7th Star, connected to the 7th day of the week, Shabbath, the point of rest in the middle.

The Spiraling Spiral in Chartres Cathedral

The One, to which the elements must be reduced, is a little circle in the center of the squared figure (the cross). The cross is the mediator, making peace between the opposition of the planets.

The Spiral of the Great Work

The winged dragon represents First Matter (Quintessence, The Fifth Element) and suggests ascension, a merging of matter (body) and spirit (Creativity). Creativity (Enthusiasm, Spirit, Inspiration, Imagination) is the engine behind the Spiral Wheel.

The First Matter is the primordial chaos comparable to what we now call the vacuum, the state of lowest energy. This state contains all possibilities.

It can be looked on as an unorganized state of energy  that is the same for all substances and exists in an invisible state between energy and matter.

The First Matter is the One, the Point in the Middle of the Circle. In current physics a point is called a singularity or a black hole.

Alchemy is about re-combination summarized in the Latin expression Solve (break down in separate elementset Coagula (coming back together (coagulating) in a new, higher form).

Solvite Corpora et Coagulata Spiritus (“dissolve the body and coagulate the spirit”).

Earth (Body) becomes Water (Spirit) by dissolving it in some solvent. Water becomes Air (a moist vapor) by boiling, which further heating turns into Fire (a dry vapor).

Finally Fire becomes Earth by allowing the vapors to condense on a solid material (a stone).

The circulation may repeat if it is done in a reflux condenser, such as the Pelican or kerotakis.

Destillation

The Pelican feeds its children with blood from the Heart. It is a symbol of the Upper Cycle of the Mind.

According to the early alchemists, the four elements—fire, water, air, and earth—come into existence via the combination of specific qualities, recognized as hot, cold, wet, and dry, being impressed on to the Prime Matter. For example, when hot and dry are impressed on the Prime Matter, we have fire.

(Hot and Cold) and (Wet and Dry) are oppositions. Their fusion is the empty set.

(Hot or Cold) and (Wet or Dry) can be combined -> (Hot, Wet), (Hot, Dry), etc.

The Cold allows substances to get together. The Hot is the power of separation.

Wet things tend to be Flexible (Fluent, Movement, Self, Not-Agency) whereas Dry things are fixed and structured (Agency, Resistance, not-Movement).

When the qualities are changed, the elements themselves are changed. Adding Water to Fire substituting wet for dry; hot and dry becomes hot and wet: steam, or Air.As you can see in de picture below Hot & Cold and Wet and Dry are Binary Opposites (“lines”) that are voided by the central cross, the point of the Empty Set, The Void (A and not-A -> Empty).

Hot & Dry = Fire, Hot & Wet = Air, Cold & Wet = Water and Cold & Dry = Earth.

There are four types of Trinities the Passion-trinity (Dry, Wet, Warm), the Structure-trinity (Dry, Wet, Cold), the Resistance-trinity  (Warm, Cold, Dry) and the Movement-trinity (Warm, Cold, Wet).

The Square rotates. The wheel of the Square is driven by the four qualities. Wet on the rising side, Hot on the top, Dry on the descending side, and Cold on the bottom.

The start is powerless (Cold). By becoming more flexible (Wet) success and power comes (Heat).  At the top Rigidity (Dry) undermines the system and it falls back to a powerless state.

The Four Elements arise out of the Quintessence, The Fifth Element, The Symbol of The Heart Chakra, the Seal of Solomon

The four Elements all share the Triangle of the Holy Trinity. Every Trinity is seen from another perspective.

The two triangels, The Up-triangel and the Down-Triangel, are Opposites that are created out of two triangels that contain two Opposites that are fused. Humans are opposites in opposition.

Humans are part of the Bilateria,  animals with bilateral symmetry.

Humans can be described as a fusion of two mirrored bilateral triangles, the Up-triangel related to the Top of the Body (Mind) and the Bottom-triangel of the lower Body both connected by the Heart. The two triangels and the separate triangels have to be balanced.

The principle of Balancing the two Triangles of Top & Bottom of the Body is represented by the Egyptian Goddess Ma’at who weights the Heart (the connector of Up and Down) of the Dead with her Feather (Nothing).

The Balance of Ma'at

As you can see the Square of the Cycle of the Four Elements is visible in the picture below. The total picture shows a Hexagram with a Center (The Cross).

The points that are missing in the Square are the top- and the bottom of the two triangels which are the Whole (Up, Heaven) and the not-Whole, Emptiness, the Bottom (Hell).

The Seal of Solomon

A hexagon is a 2D-projection of a Cube.

The hexagon of Solomon can be transformed into the Cube of Space of the Sefer Yetsirah.

Cube of Space

The Cube of Space can be transformed into the Tree of Life.

The Tree of Life also called the tree of knowledge or world tree  connects heaven and underworld and all forms of creation and is portrayed in various religions and philosophies all over the world.

The Tree of Life shows the levels/scales (three) and possible paths of the Opus Magnum, the alchemical journey.

The Cube of Space can be transformed into the Tree of Life.

The Spiral of Alchemy is about the Chemical Transformation of the Body by the Conceptual Transformation of the Mind.

About Truths for a Fact and Truths for a Reason

People have always believed in the fundamental character of binary oppositions like Hot & Cold and Wet & Dry are used in Alchemy.

In this part we move to the Field of Logic. In this case Hot/Cold and Wet/Dry transform into Necessary /Not-Necessary and Possible/not-Possible.

Chemical Wedding (Marriage of Opposites)

Oppositions can be divided in:

1. Digital oppositions, contradictories, contain mutually exclusive terms (Gender (Male/Female)).
2. Analogue oppositions, antonyms  or contraries: contain terms that are ordered on the same dimension (Temperature (Minimum->Cold/maximum->Hot)).

Square of Opposition

The doctrine of the square of opposition originated with Aristotle in the fourth century BC and has occurred in logic texts ever since.

The logical square is the result of two questions: Can two things be false together? Can two things be true together?

This gives 4 possibilities: no-no: contradiction (A/O, E/I); no-yes: subcontrariety (I/O); yes-no: contrariety (A/E); yes-yes: subalternation (A/I, E/O) better known as implication (A->B).

The four corners of the square represent the four basic forms of propositions recognized in classical logic:

1. A propositions, or universal affirmatives take the form: All S are P.
2. E propositions, or universal negations take the form: No S are P.
3. I propositions, or particular affirmatives take the form: Some S are P.
4. O propositions, or particular negations take the form: Some S are not P.

The square of opposition was debated for many reasons for more than two thousand years.

One of the discussions is about the difference between possible and necessary. In the so called Master Argument Diodorus proved that the future is as certain and defined as the past. The essence of logic (necessaty) implies the non-existence of freedom (possibility). X is possible if and only if X is necessary.

The term ‘possible’, in Aristotle’s view, is ambiguous. It has two senses known as one-sided possibility and two-sided possibility (or contingency). Being two-sided possible means being neither impossible nor necessary, and being one-sided possible simply means being not impossible.

Leibniz distinguished between necessary Truths for a Reason, which are true for a reason—i.e, their opposite is a contradiction—and contingent truths, (Truth for a Fact) such as the fact that the president of France is François Hollande. A contingent truth cannot be proved logically or mathematically; it is accidental, or historical (based on facts (events)).

About Geometric Logic and n-dimensional “Squares” of Opposition

Modal Logic is about the fusion of necessity and possibility, contingency.

Logical Hexagon

In Modal Logic the propositions are modelled in a logical hexagon where:

1. A is interpreted as necessity: the two propositions must be either simultaneously true or simultaneously false. A is a Proof or Law.
2. E is interpreted as impossibility. E is an Observation that contradicts the Proof of A.
3. I is interpreted as possibility:  the truth of the propositions depends on the system of logic being considered . I is an Idea implied by the Proof that Contradicts E.
4. is interpreted as ‘not necessarily’. O contradicts A.
5. U is interpreted as non-contingency: neither logically necessary nor logically impossible, its truth or falsity can be established only by sensory observation.
6. is interpreted as contingency: propositions that are neither true under every possible valuation (i.e. tautologies) nor false under every possible valuation (i.e. contradictions). Their truth depends on the truth of the facts that are part of the proposition. Y is a possible proven theory.

The Logical Hexagon has the same geometry as the Seal of Solomon. It contains the Square of Opposition (A,E,I,O).

The Logical Hexagon can be transformed just like the Seal of Solomon to the Cube of Space now called The Cube of Opposition.

The Logical Hexagon is used in many fields of Sciences like Musical Theory or Scientific Discovery.

The Hexagon of Opposition is used in Musical Theory

Every opposite can be defined by a string of zero’s and one’s like 1010. An opposite is 0101. This means that there are 2**4/2 = 4×4/2= 16 / 2 = 8 possible opposites. This is called the Octagon of Opposites.

The Octagon contains six reachable and two unreachable points in the Center (0000, 1111), the Empty Set (the Void, the Hole, Contradiction) or its Opposite (The Whole, Tautology).

Logical Geometry

Octagon of Opposition

It is clear that it must be possible to create higher dimensional models based on n-based-logics.

When n becomes very big the geometry will tend to a n-Dim circle in which every point is in opposition to the point at the other side.

The big problem with n-based logics is language. We don’t have the names to articulate the many grades of opposition that are possible.

The other problem is that a negation is not a symmetric operation (A = not-not-A) but a rotation with an angle 360/n.

The n-opposite-geometries are highly related to Simple Non-Abelian Groups (SNAGs). They play an important role in biology.

The last step is to transform the static n-opposite geometries by making n very large (infinite) to a dynamic “opposition field”.

Examples of Comparable Models

In current Psychology Hot & Cold are called Communion and Moist & Dry, Agency. Together they create the so called Interpersonal Circumplex.

The four stages of Learning by Jean Piaget.

The mathematical model behind the theory of Piaget is called the Klein Four Group or Identity/Negation/Reciprocity/Correlation-model.

INRC-model of Piaget

In the Science of Ecology (Panarchy) Agency & Communion are called Connectedness and Potential. The Panarchy model looks like a Mobius Ring.

panarchy model

A model that describes the Four Perspectives on Security:

The Semiotic Square of Greimas:

Semiotic Square

The Chinese Sheng-Cycle: The Chinese Five Element Model is comparable to the Western Four Element model. It contains as a Fifth Element, the Whole-Part-relationship (Earth, Observer State). Emptiness (Nothing, the Tao) is represented by the recursive Pentangle inside.

The Sheng Cycle

Tetralemma (Ancient Indian Logic)

The lesson

When thou hast made the quadrangle round, then is all the secret found“.

The secret: It is impossible to move though the center because the paths that go through the center are a contradiction or a tautology (I am what I am, JHWH).

We have to move with the Cycle (making the rectangle round (a circle)), around the Singularity (the Hole of the Whole), going Up and going Down all the time.

The only way to solve this problem is to join the opposites by accepting only one thruth-value, by:

1. Accepting everything (Wu Wei),
2. Denying everything (Stoics) or
3. Diminishing the amount of dimensions (“becoming Simple”, “like Children”) by moving up in abstraction (Hot & Cold -> Temperature -> Everything is Energy) to the level of zero dimensions, a point, becoming One with the First Matter (Tao/Vacuum).

“If we then become children, would we thus enter the kingdom?” Jesus said unto them, “When ye make the two one, and when you make the inside like unto the outside and the outside like unto the inside, and that which is above like unto that which is below, and when ye make the male and the female one and the same, so that the male no longer be male nor the female female; then will ye enter into the kingdom.” (Gospel of Thomas, Logion 22).

Semiotics for Beginners

How to use the Square of Opposition as a Research Tool

About the Cube of Space of  Sefer Yetzirah

About Truth of Reason and Truth of Fact

Why Humans are Toolmakers

About the Ars Generalis Ultima of Ramon Lull (1305)

About the Six Domains of  the Polynomic System of Value

Why Innovation is Re-Combination

How to Balance the Seal of Solomon

Anti-Fragility and the Square of Opposites

An Introduction to Hexagonal Geometry

### About the Next Steps of the Future Center Smart Systems and the Leiden Center of Data Science

Sunday, May 4th, 2014

At 28-4-2014 the Future Center Smart Systems (FCSS) was launched At the same moment the Leiden Center of Data Science (LCDS) was announced. This is not a coincidence.

FCSS and LCDS are two parts of the same coin. LCDS is focused on the Science of Data and FCSS is focused on using the results of the Science of Data to build Smart Systems or to give the Science of Data interesting problems to solve. FCSS is Practice and LCDS is theory.

When we look at Wikipedia we see that the Data Sciences are defined as “The study of the generalizable extraction of knowledge from data”. Knowledge shows itself in this case as Models, Patterns and Laws (Rules).

The Data Sciences detect patterns based on facts and are an abstraction of the Scientific Process itself.  The Data Sciences are therefore also called E-Science.

E-Science is a combination of:  Signal processing, Mathematics, Probability models, Machine learning, Statistical learning, Computer programming, Data engineering, Pattern recognition and learning, Visualization, Uncertainty modeling, Data warehousing, and High performance computing.

Because E-Science is an abstraction of Science it can be applied to every science you can think of. In the case of the LCDS the current sciences that are involved are Physics, Astronomy, Bio-Science (Leiden Bioscience Park), Life-Sciences (Dutch Techcenter for Life Sciences, DTL), Medical Sciences (LUMC), Law (Leiden Law School), Aviation (NLR), Mathematics and Informatics (LIACS).

The FCSS is not only about E-Science it is also about using the patterns that are discovered to control processes and influence humans. In this case we are interested in the developments in Value, Case & Process-managers,  Domain Models and Sensor-Technology. Theory is static and practice is dynamic. Theory becomes practice by doing.

In the long view we expect that the combination of Senses & Actions & Thinking Machines will lead to Autonomous Systems that will help and support Humans. These systems are sometimes called Robots because we believe Autonomous Systems look like us but an autonomous system like our Universe does not have to move itself.

The next step of the Future Center is to connect with practice, start with technology transfer and find out what the need is of the market.

The Future Center is already involved in some projects and we wants to start some more. If you are interested to participate in one of the projects or want to start a new projects send an email to hans.konstapel@gmail.com.

• Smart System Architectures
Smart Systems need Smart System Architectures. They are a combination of existing architectures based on Process-Models & Sensor-Systems (the Sensory-Motor System) and new architecture that are related to Visualization (the Imagination), Analytics (Data, Process, Text, Software-mining (Thinking)) and Social Networks. (Emotions). The end state is called Global Brain (“The Singularity”).
• Disclosure of Open Data (Repository, Data Warehouses/Data Cards, Visualization)
A lot of Open Data is entering the market. It is not clear where this data is situated and what the data means. In this case we need a Repository and tools to define what part of the data we need at this moment (a Data Warehouse, Data Cards). We need also tools to hide the complexity of the current open databases and makes it possible to show what is available (Visualization).
• Smart Innovation (Business-model generation out of Big- & Open Data)

There are already many on-line tools available that support the innovation process. The data that is used in these tools (for instance market-data) has to be gathered to validate the models. We want to use the Data-sciences to gather appropriate market-data and detect interesting business-opportunities, To do that we want to use Big- and especially Open Data.

• Smart Value Chain Integration & Reversal (Product Configurators, Data/Proces/Software-Mining, Intelligent User-Interfaces)
The value-chains are integrating and the customer is moving into control. Currently a value chain consists of many companies that use many not integrated and not connected legacy-systems. These systems contain data and process-models. We want to use the Data-Sciences to detect these models (Process-mining. Data-Mining, Software-Mining) and map them to available domain-models (ACCORD, ARTS,…) or develop shared domain-models. The data that is used in the value-chain can be mapped to a product-configurator. Last but not least the user-interaction with the customers has te be designed.
• Smart Education (Just-in-Time Education)
Technology is changing fast. This has a huge impact on education. The current system of education is not able  to react to the fast moving technology-waves because it is created  to support the Industrial Revolution that ended around 1950. We are now beyond the next wave that is taking over manual work and moving into the next step in which the Human Brian is copied. The solution to this all, Life Long Education,  is already invented a long time ago but we are not implementing this solution because the Current Education System is aimed at preventive education, trying to train people in the first stage of their life. The solution is Just-in-Time-Education, training people at the moment they need knowledge (DeepQA) or experience (Simulators).
• Smart Buildings & Building Process (with ABN AMRO Dialogs House)
The Building Industry is already using 3D-Models (BIM). A BIM-model is a low-level product-configurator but the model can be moved to a higher level in which it is possible to share (and sell) complete Models (for instance a Hospital). The models can be used to simulate every attribute of a building and let the customers play with possible designs. The models can be used to calculate risk and most important optimize existing buildings (Facility Management). When we combine this with sensor-technology we are able to create adaptable buildings.
• Smart Food Chain & Dietetics
The Food Value Chain will revert and the customer will be in control. At this moment the customers are educated with theories about food that are changing fast. “Good” Food is time, context and body-dependent. We want to make tools that use personal sensors that show the customer what they have to eat to stay healthy and tell them where and how they can buy this food.
• Food & Health (with the International Alliance of Future Centers)
It is not clear what the impact is of Food on our Health. In this project we will analyze the complete food-chain.
• Smart Urban Space (The Self Actualizing City)
This is an integration of the concepts of the Smart City, Integrated Value Chains, the Creative City and Smart Social Networks.
• Next Generation Media

Media are at this moment based on the Sender/Receiver-concept. We want to use the Data Sciences to detect patterns, transform these patterns into text and images and implement a feed-back process with the customers.

• Next Generation Theatre
We want to use the concept of  5D Film to create a new type of sensory and emotional experience.
• Next Generation Energy Systems
Implementing Autarkic Energy Systems.
• Smart Mobility
Using new concepts like the Self-Driving Car & Sensor-Networks  to optimize mobility in Big Cities.
• Social Enterprises (Impact Finance, Social Bonds, Circular Systems, Cooperatives, Complementary Currencies)
The next generation enterprises has to be aware and reduce its impact on society and environment. In this project we also look the finance of Social Enterprises (Impact Finance, Social Bonds) but also at the circular economy, new organizational structures (cooperations, ) and new financial systems based on complementary currencies.
• Conflict-Resolution/E-Mediation
Preventing and resolving conflicts by implementing Smart Mediation.
• Future Jobs (Unemployement & Sense-making, with the International Alliance of Future Centers)
What are the consequences of Smart Systems on employment?
• Smart Care Systems

We want to use Smart Systems to help people that need Economic, Physical, Social and/or Mental Support.

• Virtual Future Centers (with the International Alliance of Future Centers)
Future Centers are currently location-based. Future Centers look like Smart Systems because they Sense their environment, react (Process Manager) on the Events that are happening, try to predict what will happen (Analytics), involve Networks (Communitiies of Practice, Interest and/or Affinity) and Scientific Knowledge Centers (like the LCDS), create a shared Vision and  put this vision into Action (Entrepeneurs, Innovators, Incubators). There are already many people busy with developing parts of the Virtual Future Center. We are currently designing an Architecture and will create an Alliance to join the efforts of all participants.
• Next Generation Secure Data Centers
Smart Systems need Smart Data Centers.

The Presentation of Prof. dr. H. J. van den Herik about Big Data and the Leiden Center of Data Science.

Tuesday, April 17th, 2012

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

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

Heat is Random Motion

According to Thermodynamics our Universe moves from a highly improbable extreme hot state called the Big Bang to a highly probable extreme cold state called the Big Freeze.

The Big Bang was caused by a conflict between expanding and compressing (inflating) space in the Ocean of the Nothing (the Zero Point-Field) that contains infinite possibilities. This conflict generated a high level of turbulence.

Our Universe is one of the many bubbles that drift in the Sea. At a certain moment the Bubble will dissolve and space will become evenly distributed.

To reach the Big Freeze in finite time every new state of our Universe has to be more probable than the old state. The Probability of a System is called Entropy in Thermodynamics.

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

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

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

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

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

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

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

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

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

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

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

Clouds on the Edge of Turbulence

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

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

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

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

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

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

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

Flow Systems

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

Flow Systems always remove the Resistance to the Flow themselves.

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

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

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

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

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

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

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

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

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

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

Paris is a Flow System

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

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

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

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

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

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

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

Revolt: A small Wildfire can destroy a complete Ecology

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

Kleiber's law

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

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

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

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

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

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

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

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

The Triangular Pattern behind the Triangle of Pascal

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

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

The Tao is like a well:

used but never used up.

It is like the eternal void:

filled with infinite possibilities.

It is hidden but always present.

I don’t know who gave birth to it.

It is older than God.

(Tao Te Ching 4)

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

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

About the Higgs Field and the Big Bang

Why the Objects in the Universe are of Different Size

About the Constructal Theory of David Deutsch

An Introduction into Constructal Theory

Why a City is an Organism

About the Triangle of Pascal and the Holy Trinity

About Constructal Law and the Trinity

Attraction + Obstacles = Excitement

About the Law of three & Gurdjieff

### Over Veerkracht (Resilience) (in Dutch)

Monday, April 2nd, 2012

De leer van de thermodynamica is in de 19e eeuw ontstaan uit natuurkundig onderzoek dat werd uitgevoerd om de efficiëntie van de stoommachine te verbeteren. Inmiddels heeft deze wetenschap belangrijke wetten over het Universum ontdekt.

Volgens de thermodynamica beweegt de energie in het Universum op de lange termijn zich van een onmogelijke extreem hete situatie, de Big Bang, naar een toestand van extreme kou en een hoge waarschijnlijkheid, de Big Freeze, waarin alles netjes uniform verdeeld is. Het universum beweegt van extreme chaos naar extreme ordening.

Energie stroomt daarom altijd van hoge (chaos) naar lage temperatuur (orde). Vrijwel alle energie op aarde komt van de zon in de vorm van electro-magnetische straling. Met behulp van deze zonnestraling transformeren planten, algen en bacteriën simpele chemische structuren in complexe biomassa (chemische energie). Bacteriën en schimmels breken deze structuren weer af waardoor de cyclus wordt gesloten.

Biomassa wordt door de rest van de organismen op aarde direct of soms veel later (olie/gas/steenkool) geconsumeerd. Het consumeren van de primaire zonne-energie vindt plaats in voedselketens waarin de organismen ook zelf weer worden geconsumeerd.

Vanuit ons menselijk perspectief interpreteren wij het consumeren van de participanten in een voedselketen als een competitie (“survival of the fittest”) maar we kunnen het ook zien als een uiterst flexibele keten van chemische reacties die worden uitgevoerd in bewegende zichzelf reproducerende chemische fabrieken.

Energie is de mogelijkheid om iets in beweging te krijgen of te houden. De opgeslagen (potentiële) energie in de aarde wordt soms in meerdere stappen omgezet in beweging (kinetische energie). Tijdens deze omzetting gaat er altijd warmte verloren (dissipatie, wrijving, weerstand, erosie).

De energiestroom van de zon wekt door de rotatie van de aarde stroom-systemen (flow-systems) op. Stroom-systemen hebben een boom-structuur als de stroom van een punt naar een oppervlak of een volume gaat. Een voorbeeld zijn het weersysteem en een meanderende rivier-delta. De longen en de bloedvaten in ons lichaam hebben precies dezelfde opbouw omdat hetzelfde proces ook in ons lichaam optreedt.

Volgens de Constructal Theory van Adrian Bejan verbeteren de stroomsystemen zich steeds opnieuw en dat doen ze door gebruik te maken van de permanente slijtage (dissipatie, wrijving, weerstand, erosie) die de stroombeweging veroorzaakt.  Uiteindelijk vindt er altijd weer ergens een doorbraak plaats in de meander.

Adrian Bejan heeft de Constructal Theory volledig analytisch afgeleid uit de wetten van de Thermodynamica. Hij  beschouwt het principe van de verbeterende stroomsystemen (“construction”)  als een Hoofdwet van de Thermodynamica. Volgens hem is ontwerpen niet een intelligente actie maar een volstrekt logisch gevolg van de werking van ons Universum.

De stroom van onmogelijkheid naar uniformiteit, van hoge onwaarschijnlijkheid naar hoge waarschijnlijkheid gaat gepaard met allerlei tussenstadia waarin de bestaande structuren de nieuwe structuren de weg wijzen.

De doorbraken in de meanderende stroompatronen maken beter gebruik van de huidige mogelijkheden maar de oude paden behouden lange tijd hun functie. Hierdoor is een stroomsysteem niet afhankelijk van één pad en is daardoor ook bestand tegen calamiteiten (bijv. een sterk verhoogde regenval in het stroomgebied van de rivier).

Het systeem waar de stroomsystemen onderdeel van uitmaken bestaat uit Machines (Constructal Engines) en Barrières (Brakes).

De machines en de barrieres zorgen ervoor dat de aarde niet te warm of te koud wordt.

De machines zoeken de weg van de minste weerstand, optimaliseren de stroom, bouwen structuren op, slopen de barrieres, produceren kinetische energie en minimaliseren de slijtage. De machines slechten het pad naar de Big Freeze waarin alles gelijk zal zijn en alles stil zal staan.

De barrières bevatten potentiële energie, leveren de weerstand om de minste weerstand te vinden, slopen de machines, optimaliseren daardoor de dissipatie en produceren een maximale hoeveelheid straling met een hogere golflengte als de binnenkomende zonnestraling die de aarde weer verlaat.

Produceren (Machines) en Consumeren (Vernietigen, Slijten) houden het wankele evenwicht van Moeder Aarde in stand.

De evolutie van ecologieën, steden, organismen (wet van Kleiber), talen (wet van Zipf), organen (de longen, bloedvaten), sporten en technologieclusters voldoen aan de Constructal Theory.

In 2002 hebben Holling en Gunderson het boek Panarchy: Understanding Transformations in Human and Natural Systems gepubliceerd.

Panarchy bevat een integrale visie op de evolutie en de adaptieve cyclus van  de constructal engines die men ecosystemen noemt.

Panarchy is ontstaan door het combineren van ogenschijnlijk conflicterende theorieën. Die theorieën leken conflicterend omdat ze betrekking hadden op de verschillende fasen (Zie Plaatje Phase Space hieronder) van de adaptieve cyclus die een ecologie doorloopt.

De combinatie van de vier mogelijke fasenruimten levert een op-en neergaande cyclus op waarin op een zeker moment een sprong (bifurcatie) wordt gemaakt naar een compleet nieuwe ruimte. Nadat deze sprong heeft plaatsvonden stopt de dynamiek van de oude ecologie.

De meest herkenbare fase is de Exploitatie-fase (B).  Dit is de opgaande lijn waarin de ecologie een hoge mate van orde bereikt en overschotten (Potential) produceert.

Aan het eind van deze fase neemt de potentie af en wordt er van alles gedaan om de levensduur van de ecologie zo lang mogelijk te rekken (C -Conservation). In deze fase zijn de structuren verouderd geraakt en zijn er op allerlei plaatsen reeds alternatieven (bypass) ontstaan die beter passen op de huidige vraag naar beweging.

De ecologie is in de conservatie-fase vrijwel tot stilstand gekomen en besteed al zijn energie aan het onderhouden  van zijn eigen structuur.

Tijdens de langzaam opgaande lijn van de Exploitatie-fase neemt de verbondenheid en vooral de afhankelijkheid (Connectedness) tussen de participanten onderling enorm toe waardoor de ecologie steeds kwetsbaarder wordt voor verstoringen.

De constructal engines in de ecologie gaan steeds efficienter samenwerken wat ten koste gaat van de diversiteit en de variatie in de ecologie.

De ecologie kan verstoringen opvangen totdat de  verschillende paden nog niet zijn dichtgeslibd maar een verstoring die alle paden tegelijkertijd raakt is dodelijk.

Op het laatst wordt de Veerkracht (Resilience) enorm op de proef gesteld en kan het gebeuren dat de Ecologie volledig ten onder gaat (Release) of na een grondige Reorganisatie weer aan een nieuwe levensloop kan beginnen.

De resilience van een ecologie wordt gemeten door de snelheid te meten waarmee een ecologie zich weer herstelt van een verstoring.

Volgens Panarchy beweegt een Ecologie zich op verschillende schalen en plaatsen en kan op die plaatsen in verschillende fasen verkeren waardoor er allerlei soorten van interactie mogelijk zijn. Hoe hoger de schaal hoe langzamer de cyclus verloopt. Hoge schalen dempen de fluctuaties van de lagere schalen.

Een snelle lokale reorganisatie kan zich plotseling heel snel verspreiden (een lopend vuur (Revolt)) terwijl een langzame geleidelijke ontwikkeling (een organisatie) een snelle ontwikkeling weer kan dempen of tot stilstand kan brengen (de organisatie van de Brandweer (Remember)).

Panarchy is een Zelf-Refererend Cyclus-Model dat bestaat uit Vier Fasen en (minstens) drie Schaalniveaus.  De panarchische golfbeweging bestaat uit twee stappen n.l. de  positieve weg Omhoog naar de Top (exploitation, conservation)  en de negatieve weg Omlaag naar het Dal (depressie, reorganization).

Op een bepaald moment kan de golfbeweging zich splitsen waardoor een nieuw (ander) niveau van organisatie optreedt. Deze sprongen doen zich volgens een vaste regelmaat voor.

Als we kijken naar terminologie en inhoud valt het op dat er een groot verband bestaat tussen Ecologische-, Economische- en Persoonlijkheidstheorien. De natuur, de economie en de mens lijken onderhevig te zijn aan dezelfde wetten.

De theorie over de Menselijke Natuur heeft erg veel weg van de Theorie die wij over de Natuur zelf hebben ontwikkeld. De mens is een zelfstandig bewegende ecologie, een stromend stroomsysteem waaraan een groot aantal participanten deelnemen.

Bacteria wisselen DNA uit

Die participanten hebben zich in de loop der tijd verenigd omdat een samenwerkingsverband veel efficienter bleek te zijn dan een solitaire ontwikkeling.

In het begin bestond het samenwerkingsverband uitsluitend uit simpele chemische fabriekjes, bacteriën, die genetische ontwerpinformatie (DNA) van hun fabriekjes uitwisselenden. De eerste fundamentele stap in de samenwerking van de bacterien was de statische spons. De mens heeft erg veel ontwerp-informatie gemeen met de spons.

Spons

Later gingen de organismen bewegen en waarnemen. Hierdoor gingen ze als individuen samenwerken met andere organismen die genetisch op hen leken (zie Genetic Similarity Theory).  Het uitwisselen en combineren van ontwerp-informatie ging gewoon door (sex).

In volgende fasen ontstonden er hogere ordeningen, zoals lichamen die lichamen konden bevatten (huizen, steden, vliegtuigen, onderzeeboten en auto’s).

Op dit moment is de (grote) stad het hoogste niveau van koppeling maar niets houdt de “Constructal Law” (of de Entropie) tegen om een ordening te gaan scheppen op  het niveau land of aarde.

Dat betekent dat er nieuwe slagaderen moeten komen om de ontwerp-informatie en de energie die tussen de grote steden gaat stromen te faciliteren. Overduidelijk is dat het Internet de basis vormt van het zenuwstelsel van de aardse informatievoorziening.

De thermodynamica is een statistische theorie. De kansverdeling van een Systeem wordt de Entropie van het Systeem genoemd.

Om een kans te kunnen berekenen moet je de kansruimte (de Phase Space) bepalen. Die bestaat uit alle mogelijkheden.

Het aardige is dat je niet hoeft te beschikken over kennis van alle combinaties van alle componenten van een specifiek systeem om erg veel te weten te komen over een kansruimte.

Je kunt gaan spelen met willekeurige componenten (a,b,c,d,e,….), de combinaties in een matrix stoppen, gaan tellen en patronen zoeken in de getallen die er dan worden gevonden. Je bent dan bezig met de statistiek van de statistiek of beter de waarschijnlijkheid van de waarschijnlijkheid. De mogelijke patronen worden beschreven in de Driehoek van Pascal.

Wat dan blijkt is dat er achter het patroon van Pascal weer een patroon zit (zie plaatje).  Dat patroon bevat wederom driehoeken en het is zeer waarschjinlijk dat er achter de driehoeken weer driehoeken zitten. Driehoeken in driehoeken in driehoeken in…. noemt men een fractal, een zelf-herhaling.

De  (on)waarschijnlijkheid is eigenlijk heel mooi en vooral ook simpel geordend en komt voort uit een zichzelf herhalend patroon, de Drie-Eenheid, dat in iedere godsdienst op aarde voorkomt.

Wetenschap en Godsdienst gaan uiteindelijk over hetzelfde.

Over de Big Bang

Over Thermodynamica

Over Constructal Theory

Waarom een stad een organisme is

Waarom de genen bepalen wie we aardig vinden

Over Pan en Anarchie

Over de Wet van Kleiber

Over de orde achter de wanorde (Over de Driehoek van Pascal)

Over de Uitgebreide Driehoek van Pascal

Over de drie-eenheid.

Thursday, February 2nd, 2012

Medieval Perspective

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

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

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

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

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

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

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

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

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

The Future is Expanding Space with Infinite Potential.

Time moves behind Space and Scales Space.

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

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

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

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

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

Alberti regarded mathematics as the common ground of Art and the Sciences. “To make clear my exposition in writing this brief commentary on painting,” Alberti began his treatise Della pittura, “I will take first from the mathematicians those things which my subject is concerned”.

Alberti stressed that “all steps of learning should be sought from nature“. The ultimate aim of an artist is to imitate nature.

Perspective Machine designed by Albrecht Durer

Alberti did not mean that artists should imitate nature objectively, as it is, but the artist should be especially attentive to beauty, “for in painting beauty is as pleasing as it is necessary“.

The work of art is according to Alberti so constructed that it is impossible to take anything away from it or add anything to it, without impairing the beauty of the whole.

Beauty was for Alberti “the harmony of all parts in relation to one another,” and subsequently “this concord is realized in a particular number, proportion, and arrangement demanded by harmony“.

Alberti’s thoughts on harmony were not new – they could be traced back to Pythagoras – but he set them in a fresh context, which well fit in with the contemporary Aesthetic Discourse.

The Perspectograph of Leonardo Da Vinci

One of the earliest Artists to produce a book on how to draw in perspective was Albrecht Dürer. As well as discussing geometric methods, he also illustrated his book with a set of woodcuts showing practical tools for accurate perspective drawing. Other Artists like Leonardo Da Vinci developed comparable tools.

Projective Geometry formalizes one of the Central Principles of Perspective Drawing and of Human Perception: Parallel Lines Meet at Infinity.

In Euclidean geometry, constructions are made with Ruler (Line) and Compass (Circle). Projective Geometry only require a Ruler.

In Projective Geometry one Never Measures Any Thing, instead, one Relates one Set of Points to another by a Line.

Different Points of View

Alberti was the first to ask what two pictures have in common if two drawing screens are interposed between the viewer and the object, and the object is projected onto both resulting in two different pictures of the same scene.

The basic idea behind Linear Perspective is simple: in every painting an artist creates a “floor” or area of the painting where the figures and/or objects will be placed. The floor ends at a horizon line, and the horizon line has a Vanishing Point or Point of Convergence on it.

The artist then draws parallel lines radiating from the vanishing point outward. Images closest to the vanishing point should appear smaller and closer together, and images farthest from the vanishing point should appear larger and farther apart, giving the impression of depth and space in the painting.

Pappus of Alexandria ( 290 – 350), one of the last great Greek mathematicians of Antiquity, proved that that given one set of points A, B, C on a line, and another set of points a, b, c on another line, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are also on a Line. This line originates at the Vanishing Point.

Pappus' Theorem

Later Desargues (1591-1661) proved Pappus Theorem for Triangels. He proved that the three vertices of one triangle a, b, and c, and those of the other A, B, and C meet in a third point, and that these three points all lie on a common line called the Axis of Perspectivity.

Desargues' theorem

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

Pascals Theorem

About the Four Points of the Cross-Ratio

Finally 400 years later in 1803 Lazare Carnot found the answer to the Question of Leone Battista Alberti.

In his book “Géométrie de Position” he proved that the so called “Cross-Ratio” is always preserved in a Projection with One Point of View.

The Cross-Ratio

The Cross-Ratio is the Ratio of the Ratio of the Four Points A,B,C,D lying on a Line that intersects the Four Lines defined by a,b,c,d that orginate at the Vanishing Point O.  Carnot  proved that (AB)(CD)/(AC)(BD) = (A′B′)(C′D′)/(A′C′)(B′D′).

The Cross Ratio is not only the Cross-Ratio of the Four Points A,B,C,D. It is also the Cross-Ratio of the Four Lines  and the Three Angles that originate out of the Vanishing Point. In this case (AB)(CD)/(AC)(BD)=Sin(AB) x Sin(CD)/Sin (AC) x Sin (BD).

This principle is called Duality. In 2D-Projective Geometry Connected Points and Lines, called an Incidence, are interchangeable.

There is a corresponding Duality in three-dimensional Projective Geometry between Points and Planes. Here, the line is its own Duality, because it is determined by either two points or two planes.

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

Two examples: 2D: Two distinct Points determine a unique Line and two distinct Lines determine a unique Point. 3-D:  Three distinct Points not all on the same Line determine a unique Plane and three distinct Planes not all containing the same Line determine (meet in) a unique Point.

The 24 Permutations of A,B,C and D produce 6 possible values of the Cross-Ratio, depending on the order in which the points are given. If k = 1, the other Cross-Ratio’s are 0 and infinity. This happens when point A = point D. In this case the Geometric Entity is an Equilateral Triangle.

If k = -1, the other Cross-Ratio’s are -1/2 and 2. This is called a Harmonic Cross-Ratio. This happens when the Internal Ratio of AC determinated by B is Equal to the External Ratio of AC determinated by D. When k=-1/2 and 2 the Geometric Entity is a Square.

If the Four Points of the Cross-Ratio are on a Line or a Circle, then the Cross Ratio is a Real Number, otherwise the Cross-Ratio is a Complex Number.

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

About the Four Layers of Geometry and the Four Points of the Cross-Ratio

In 1872 Felix Klein, published a new Mathematical Research Program called the Erlangen Program under the title Vergleichende Betrachtungen über neuere geometrische Forschungen.

In this program Projective Geometry was emphasized as the Unifying Frame for all other Geometries.

Although lines in the Projective Plane meet in one point of Infinity Klein argued that there could be two points of Infinity if the Projective Plane was a Surface Closed in Itself.

When we look at the Origin of Projective Geometry, the Artist painting A Sphere, Earth, on a Flat Surface, it is not difficult to realize that this Closed Surface is a Sphere.

.

Geometry is now divided into Four  Layers. Each Layer adds a set of Assumptions that creates certain Invariants for that Layer.

An invariant is a property of a configuration of Geometric Entities that is not altered by any transformation belonging to the specific layer.

A transformation is an operation applied to a Geometric Entity. The most common transformations are Translation, Rotation and their Combinations (Reflection).

What we will see is that the Four Points of the Cross-Ratio are highly related to the Four Layers. In Every Higher Layer One Point of the Cross-Ratio is moved to the Internal Landscape, the Mind, of the Observer. This principle is called “As Within so Without“.

The First and most fundamental Layer is the Layer of Projective Geometry. This is the Layer of Human Perception and is invariant under the Cross-Ratio also called a Moebius Transformation.

The Second Layer is called Affine Geometry. In this layer Parallel Lines are preserved and the Assumption of a Plane at Infinity, the Horizon,  is created.  This Plane generates Parallel and Orthogonal Relationships between Geometric Entities by placing One of the Four Points of the Cross-Ratio in the Plane at Infinity.

Affine Geometry is Euclidean Geometry with congruence (something is the same when Shape and Size are the Same) and a metric (a definition of a Distance) left out. An affine transformation preserves straight lines and ratios of distances between points lying on a straight line.

This the layer of  the Emotions, “the harmony of all parts in affinity to one another” of  Leone Batista Alberti.  In Affine Geometry the Frame of Reference of the Painter, the Floor,  is created. On this Floor,  the Horizon, the Plane at Infinity is drawn.  From the Point of View of the Observer, it is not possible  to Judge Distance or the-Sameness because a given Visual Object may be Large and Far away or Small and Close.

In Eucludian Geometry A Ruler contains a Fixed Scale

The Third Layer, Metric Geometry, adds the concept of Distance (Metric, Scale) to the Affine Layer. A Distance is a relation between Two Points of the Cross-Ratio. This is the Layer of the Painter, Creator, Observer, who creates his own Distance to the Distances he is Painting in the External Frame he is looking at.

Since Projective Space is the Space of Actual Perception, the true function of Metric Space is the Coordination of Various Perspectives. This is illustrated in Perspective Drawing, in which there are always Two Perspectives being Coordinated—the Vanishing Point and a Point just behind the Eyes of the Observer.  Each of these is an instance of the One Point at Infinity.

The Expectation looks through a Small Hole.

The last, Fourth,  Layer is the layer of Euclidian Geometry. In Euclidean Geometry Distance is Fixed (not Relative) and Scaled to a Unity (Meters). This leaves us with just One Point of the Cross-Ratio, the Unity, the One.

Between  1926 and  his death in 1983 Alfred Tarski worked on the Axiomatization of Euclidean Geometry. In Tarski’s system there is only one type of Object: the One, The Point.

There are just two Geometrical notions between Points: the Ternary Relation of “in-Betweenness” and the Quaternary Relation of “Equidistance” or Congruence.  Betweenness captures the Affine aspect of Euclidean Geometry; Equidistance/Congruence, its Metric aspect.

This is the Layer of the Human Expectation who has not Distanced Itself and is Looking Outside Through a Small Hole at the Future. The Expectation considers the Future as an Extension of the Past (Memory).

In Euclidian Geometry we have to Move Outside to the Objects to Measure their Distance and the Sameness with a Pre-Defined Unity.

If we don’t move to the Object we will measure an Illusion.

About Space measured by Time and Space measured by Space

The Ancient Sumerians  knew that the Length of one side of a Hexagon is the same length as is the Radius of the Circle that circumscribes this Hexagon (See Pascals Theorem).

They also knew that the Stars of the Constellations of the Zodiac shift Counterclockwise, at the rate of 72 years/degree, because of the Precession of the Earth’s axis. One Wobble takes 25.920 years.

The length of one side of the Earth Hexagon is therefore the Distance traveled by the Constellations of the Zodiac along the horizon during 25.920/6=4,320 years.

They then subdivided this distance by 7,200  which produced the Royal Mile that was subdivided into 1,760 Royal Cubits.

Before 3.117 BC the complete Earth was covered with a uniform Geodesic System that was based on the Rotation and Precession of the Constellations of the Zodiac. This System was implemented in the Megaliths.

At that Time Space was measured by Time.

The Physical System of the Megaliths was destroyed by the Great Flood of 3.117 BC but the Metric System survived thousands of years until the French Revolution.

On the 8th of May, 1790, Charles Maurice de Talleyrand at the end of the French Revolution, proposed before the National Assembly in Paris a Change to a Decimal Measurement System.

The Academy of Science recommended that the new definition for a Meter be equal to 1/10,000,000 of the distance between the North Pole and the Equator, and this was accepted by the National Assembly in 1791.

From that Time on Space was measured by Space.

Why the Future contains Infinite Possibilities

A Painter or a Human always looks Forwards in Space. If there are Parallel Lines visible they all Converge to a Vanishing Point at The  Horizon. The Horizon is a Parallel Line at Infinity.

We all know that the Parallel Line of the Horizon is caused by the fact that the Earth is a Moving Sphere. This Moving Sphere moves around other Spheres that move around other Spheres.

The Rotation of the Spheres in Space is used as a Clock. Every time when the Cycle of the Spheres repeats itself we add one unit to a standard Time-Measure. One Sun Cycle is named a Year. One Cycle of the Precession takes 25,920 Years.

We Measure our Time by the Cyclic Movement of Objects in Space. The Cyclic Movements caused by the Rotations of Objects around Objects (around Objects) is really a Rotating Rotating Rotation, a Spiraling Spiral.

We don’t live in the Sphere but we live and Move on the Surface of a  Moving Sphere. That is why our Geometry is an Elliptic Geometry (or Projective Geometry).

Euclides and Pythagoras were aware of Projective Geometry (See also Pappus) but the Scientists of the Enlightenmentstarted to interpret their Theory with a Different Eye. They saw Numbers in a completely different way the Greek did.

The Scientist disconnected Number from Magnitude (Form) and created an Abstract Static Number Theory and an Abstract  Static Theory of Physics. The Movement of our Universe, the Ether, came to a stilstand.

The Theory of Perspective Drawing, based on our Real Perception of Reality, is the same theory that generated Projective Geometry. We are now back to Normal again.

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

He reinvented Physics and Projective Geometry in his own way, Calculated all the well known Physical Constants and based his Theory on a simple well know Assumption that Space and Time are Reciprocal (A Ratio of Ratio’s , the Cross-Ratio) because Velocity= dS(pace)/dT(ime).

Moving Movement is the Essence of our Reality.

What Larson calls “Motion” is the Ether Wind (or the Higgs Field) the Velocity which was measured in the North-South (Z) direction at 208 km/s in 1933 by Dayton C. Miller.

Einstein believed the results of the Michelson-Morley Experiments in 1887 that “proved” that the Moving Ether was non-existent. Now we know that the experiments were wrong.  Einstein also did not believe that Space was Expanding. At this moment the Expanding Universe is confirmed by many observations.

When Space Expands,  Time Compresses and fills up the Space to Keep Balance. Time Scales Space.

Space moves to Infinity. Time Moves to the Inverse of Infinity, Zero. Space is measured by the Linear Visible Scalar Number-System. Time is Measured with the Rotational Invisible Imaginary Number System.

Expanding Space is Linear Motion. Time is the Spiraling Motion of the Vortex.

The Spiraling Spiral of Time

Space is Yang and Time is Yin. Yin is feminine, Curved, Rotational, smooth or cold. Yang is masculine, Straight, Linear, rough or hot.

Space and Time are Complementary Duals that Move Around the Void.

If we look at the Human Perspective we now are able to understand what Time and Space Really are.

When we Look Forward we Experience Space. Time is always Behind our Back.

Human Senses and Conventional, Scientific Equipment can only Look Forward and therefore Measure Space (Distance) and the Change of Space (Velocity, Acceleration).

This is a Limitation of our Physical Sensory System, which evolved to measure Space, Scaled by Time, to produce what we Perceive as Causality–a Linear Ordering of Events.

Without the Cause-and-Effect System, the Sensory World of Space would just become Chaos.

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

Let’s Move.

Paul Dirac: About Physics and Projective Geometry

About Expanding Space and Not-Euclidian Geometry

Pictures of the Projective Plane

About the Reciprocal Theory of Dewey B. Larson

About Projective Geometry and our Senses

About Projective Geometry and Geometric Algebra

About the Four Points of View

Why the Future is Open Space

About Alchemy and the Klein Bottle

Monday, January 9th, 2012

We have lost the relationship between Number and Form or Number and Magnitude as the Ancient Greeks called their Forms.

A few years ago a Revolution in Mathematics and Physics has started. This revolution is caused by Geometric Algebra.

In Geometric Algebra the Ancient Theories of Euclid and Pythagoras are reevaluated.

Numbers are Scalar (Quantum) Movements of Geometric Patterns and not Static Symbols of Abstractions that have nothing to do with our Reality.

Movements and not Forces are the Essence of Physics.

The basic rule Movement = Space/Time (v=s/t) shows that  Time and Space are two Reciprocal 3D-Spaces. Our Senses Experience Space and not Time.

The Simple Rule N/N=1/1=1 balances the Duals of Space and Time. One Unit Step in Space is always Compensated by One Unit Step in Time.

Geometric Algebra has a strange relationship with Pascals Triangle. This Triangle, also called the Binomial Expansion, contains all the Possible Combinations of two Independent Variables. Our Universe is a Combination of Combinations exploring Every Possibility.

The last and perhaps most important Discovery in Mathematics called Bott Periodicity shows itself in Pascals Triangle.

Bott Periodicity proves that we live in a Cyclic Fractal Universe, the Wheel of Fortune, that is Rotating around the Void, the Empty Set. The Empty Set contains Every Thing that is Impossible in our Universe.

This blog is not a Scientific Article. I have tried to connect the Old Sciences and the New Sciences in my own Way.

It contains many links to Scientific Articles and even Courses in Geometric Algebra.

So if you want to Dig Deeper Nothing will Stop You.

About the One and the Dirac Delta Function

Every Thing was created out of  No Thing, the Empty Set, ɸ, the Void, the Tao. The Empty Set contains 0 objects.

The Empty Set is not Empty. It contains Infinite (∞) Possibilities that are Impossible.

Every impossibility has a probability of 0 but the sum of all possibilities (1/∞=0) is always 1. In the beginning ∞/∞ =1  or ∞x0=1.

This relationship is represented by the Dirac Delta Function. It is used to simulate a Point Source of Energy (a Spike, an Explosion) in Physics.

The Delta is reprented by the Symbol Δ, a Triangle. The Delta is called Dalet in the Phoenican and Hebrew Alphabet. Daleth is the number 4 and means Door.

The original symbol of the Delta/Daleth contains two lines with a 90 Degree Angle. Two orthogonal lines create a Square or Plane.

The Dirac Delta Function is defined as a Square  with an Area of 1,  a Width of 1/n and a Height of n where n->∞.

The Dirac Delta Function is a Line with an Area of 1.

In the Beginning a Huge Explosion took place that created the Universe.

The Dirac Delta Function δ (x) has interesting properties: δ (x) = δ (-x), δ (x) = δ (1/x). It has two Symmetries related to the Negative Numbers and the Rational Numbers.

When we move from 2D to 1D, the Number Line, the Delta Function becomes the Set of the Numbers N/N =1.

The Tetraktys of Pythagoras

The Monad (1) of the Tetraktys of Pythagoras, the Top of the Triangle, was created by Dividing the One (1) by Itself without Diminishing itself. The Monad (1/1=1)  is part of  the 1D Delta Function.

Creation is an Expansion of the 1/1 into the N/N, adding 1/1 all the time,  until ∞/∞ is reached. At that moment every Impossibility has been realized.

The Dirac Delta Pulse

To move Back to the Void and restore the Eternal Balance of  the One,  Dividing (Compression) has to be compensated by Multiplication (Expansion).

At the End of Time N/M and M/N have to find Balance in the N/N,  move Back to  1/1, Unite in the 0 and become The Void (ɸ) again.

About the Strange Behavior of Numbers

The big problem of the Numbers is that they sometimes behave very differently from what we Expect them to do.

This Strange Behavior happens when we try to Reverse what we are doing.

It looks like the Expansion of the Universe of Numbers is Easy but the Contraction creates many Obstacles.

It all starts with the Natural Numbers (1,2,3,).

When we Reverse an Addition (Subtract) and move over the Line of the Void Negative Numbers appear. Together with the Natural Numbers they are called the Integers.

The same happens when we Reverse a Division and the Fractions (the Rational Numbers) (1/3, 7/9) suddenly pop up.

An Integer N is a Rational Number divided by 1 (N/1).

The Integers are the Multiples of 1, the Fractions are its Parts.

Numbers behave even stranger when we want to Reverse a Repeating Repeating Addition (Irrational Numbers) and want to calculate a Rational Power (2**1/2).

The Complex Numbers (or Imaginary Numbers), based on the Square Root of -1 called i, are a combination of the Negative Numbers and the Irrational Numbers.

Irrational Numbers ( the Pythagorean Theorem), Fractions (a Piece of the Cake) and Negative Numbers (a Debt) are part of our Reality but the Strange Number i represents something we cannot Imagine.

About the Duality and the Expansion of Space

In the beginning the only One who was in existence was the 1.

When the One divide itself again the number -1, the Complement of 1, came into existence.

1 and -1 are voided in the No Thing, the Empty Set, 0:  -1 + 1 = 0.

The Two, the Duality, both started to Expand in Two Opposite Directions (<– and +->) both meeting in the  ∞/∞. This expansion is what we call Space.

Space is a Combination of the Strings S(1,1,1,1,1,…) and -S = (-1,-,1,-,1,-1,…) where S+S=(0,0,0,0,0,0,…).

The Expansion pattern of Space is a Recursive Function S: S(N)=S(N-1)+1 in which + means concatenate (or add) the String “,1″.

An Addition X + Y is a concatenation of S(X) and S(Y). A Substraction X-Y is a concatenation of S(X) and -S(Y). In the last case all the corresponding combinations of 1 and -1 are voided. (1,1,1,1)-(1,1,1)=(0,0,0,1)=(1).

Multiplication XxY is Adding String S(Y) every time a “1″ of S(X ) is encountered: 111 x 11 = 11  11  11. Dividing X/Y is Subtracting S(X) every time a “1″ of S(Y) is encountered:.111  111  1/111=11 1/111. In the last example a Fraction 1/111 appears.

This Number System is called the Unary Number System.

About the Trinity and the Compression of Space called Time

The Strange Behavior of Numbers is caused by the Limitations of our Memory System. We are unable to remember long strings that contain the same Number.

To make things easy for us we Divide Space into small Parts so we were able to Re-Member (Re-Combine the Parts).

When we want to Re-member, Move Back in Time, we have to Compress Expanding Space.

Compressed Space is Time.

Time and Space have a Reciprocal Relationship called Movement (Velocity = Space/Time).

There are  many ways ( (1,1,1), (1,1,1),..) or ((1,1),(1,1))) to Compress a String in Repeating Sub-Patterns.

In the blog About the Trinity I showed that the most Efficient Way to group the One’s is to make use of a Fractal Pattern (a Self Reference) and Groups of Three Ones.

The Trinity applied to the Trinity ( A Fractal) is a Rotating Binary Tree. Binary Trees represent the Choices we make in Life.

The rotating Expanding Binary Trees generate the Platonic Solids (see linked video!) when the (number)-parts of the Binary Tree Connect.

The Ternairy Number System is represented by the Binary Tree

When we connect Three Ones (1,1,1) by Three Lines (1-1,1-1,1-1) a 2 Dimensional Triangle Δ is Created.

If we take the Δ as a new Unity we are able to rewrite the patterns of 1′s and -1′s into a much Shorter Pattern of Δ’s and 1′s: (1,1,1),(1,1,1),(1,1,1), 1,1 becomes Δ,Δ,Δ,1,1.

We can repeat this approach when there is still a Trinity left: Δ,Δ,Δ,1,1 becomes ΔxΔ,1,1.

This Number System is called the Ternary Number System.

According to EuclidA Ratio is a sort of relation in respect of size between two magnitudes of the same kind“.

A Magnitude is a Size: a property by which it can be compared as Larger or Smaller than other objects of the Same Kind. A Line has a Length, a Plane has an Area (Length x Width), a Solid a Volume (Length xWitdth x Height).

For the Greeks, the Numbers (Arithmoi) were the Positive Integers. The objects of Geometry: Points, Lines, Planes , were referred to as “Magnitudes” (Forms). They were not numbers, and had no numbers attached.

Ratio, was a Relationship between Forms and a Proportion was a relationship between the Part and the Whole (the Monad) of a Form.

Newton turned the Greek conception of Number completely on its head: “By Number we understand, not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same Kind, which we take for Unity”.

We now think of a Ratio as a Number obtained from other numbers by Division. A Proportion, for us, is a statement of equality between two “Ratio‐Numbers”.

This was not the thought pattern of the ancient Greeks. When Euclid states that the ratio of A to B is the same as the ratio of C to D, the letters A, B, C and D do not refer to numbers at all, but to segments or polygonal regions or some such magnitudes.

The Ratio of two geometric structures  was determinated  by fitting the Unit Parts of the first geometric Stucture into the Other.

The Perfect Triangle of the Tetraktys contains 9 = 3x3 Triangels. A Triangle contains 3 Lines and 3 Points.

An Example:  The Tetraktys is a Triangle (A Monad) and contains 9 Triangles (a Monad). The 1x1x1-Triangle Δ, a Part of the Tetraktys,  is Proportional to the Whole of the Tetraktys (T) and has a Ratio T/Δ = 3= Δ -> T = Δ (3)  x Δ (3) = 9.

The Mathematics of Euclid is not a Mathematics of Numbers, but a Mathematics of Forms.

The symbols, relationships and manipulations have Physical or Geometric Objects as their referents.

You cannot work on this Mathematics without Knowing (and Seeing) the Objects that you are Working with.

About Hermann Grassman, David Hestenes and the Moving Line called Vector

Hermann Grasmann lived between 1809 and and 1877 in Stettin (Germany). Grassmann was a genius and invented Geometric Algebra a 100 years before it was invented.

In his time the most important mathematicians did not understand what he was talking about although many of them copied parts of his ideas and created their own restricted version. None of them saw the whole Grassmann was seeing.

When he was convinced nobody would believe him he became a linguist. He wrote books on German grammar, collected folk songs, and learned Sanskrit. His dictionary and his translation of the Rigveda were recognized among philologists.

Grassmann took over the heritage of Euclid and added, Motion, something Euclid was aware of but could not handle properly.

A Displacement or Bivector

Grassmann became aware of the fact your hand is moving when you draw a 2D Geometric Structure. He called the Moving Lines, that connect the Points, Displacements (“Strecke”).

A Displacement and a Rotation of a Vector

In our current terminology we would call the Displacements “Vectors”.

Vector algebra is simpler, but specific to Euclidean 3-space, while Geometric Algebra works in all dimensions. In this case Vectors become Bi/Tri or Multi-Vectors (Blades).

The Trick of Grassmann was that he could transform every transformation on any geometrical structure into a very simple Algebra. Multi-Dimensional Geometric Structures could be Added, Multiplied and Divided.

The Greek Theory of Ratio and Proportion is now incorporated in the properties of Scalar and Vector multiplication.

Combining (Adding) Bivectors creates a Trivector

About a 100 years later David Hestenes improved the Theory of Grassmann by incorporating the Imaginary Numbers. In this way he united many until now highly disconnected fields of Mathematics that were created by the many mathematicians who copied parts of Grassmanns Heritage.

About Complex Numbers, Octions, Quaternions, Clifford Algebra and Rotations in Infinite Space

Grassmann did not pay much attention to the Complex Numbers until he heard of a young mathematician called William Kingdon Clifford (1845-1879).

Complex numbers are ,just like the Rationals (a/b), 2D-Numbers. A Complex number Z = a  + ib where  i**2=-1. Complex Numbers can be represented in Polar Coordinates: Z = R (cos(x) + i sin(x)) where R = SQRT(a**2 + b**2).  R is the Radius, the Distance to the Center (0,0).

When you have defined a 2D-complex Number it is easy to define a 4-D-Complex Number called a Quaternion:  Z = a + ib + jc + kd or a 8-D Complex Number called an Octonion.

William Rowan Hamilton, the inventor of the Quaternions, had big problems to find an interpretation of all the combinations i, j and k until he realized that i**2 =j**2 = k**2 = ijk=-1.

What Hamilton did not realize at that time was that he just like Grassmann had invented Vector Algebra and Geometric Algebra.

Quaternions are rotations in 4D-space

This all changed when William Kingdon Clifford united everything in his new Algebra.  Clifford’s algebra is composed of elements which are Combinations of Grassman’s Multivectors.

The Clifford Algebra that represents 3D Euclidean Geometry has 8 = 2**3 components instead of 3: 1 number (Point), 3 vectors (Length), 3 bivectors (Area) and 1 trivector (Volume).

It turns out if you use combinations of these elements to describe your geometric objects you can do the same things you did before (you still have 3 vector components).

In addition, you can have additional data in those other components that let you find distances and intersections (and a lot of other useful information) using simple and (computationally) cheap numerical operations.

The most important Insight of William Kingdom Clifford was that the Complex Numbers are not Numbers all.

They are Rotations in higher Dimensional Spaces.

About Pascal’s Triangle and Mount Meru

The String 1,3,3,1 of Clifford’s 3D Geometry is related to the 4th Level of Pascal’s Triangle. Level N of Pascal’s Triangle represents N-1-Dimensional Geometries.

The Sum of every level N of the Triangle is 2**N. This Number expresses the Number of Directions of the Geometric Structure of a Space with Dimension N.

A Point has 0 Direction, while a Line has 2 Directions, relative to its Center point, a Plane has 4 Directions, relative to its Center Point, and a Cube has 8 directions, relative to its Center point.

Pascal’s Triangle is also called the Binomial Expansion. This Expansion shows all the Combinations of two letters A and B in the function (A+B)**N. Level 1 of the Triangle is (A+B)**0 = 1  and level 2 is A x A + 2 A x B + B x B -> 1,2,1.

The Binomial Expansion converges to the Bell-Shaped Normal Distribution when N-> ∞.

The Diagonals of Pascal’s Triangle contain the Geometric Number Systems (Triangular Numbers, Pyramid Numbers, Pentatonal Numbers, ..) and the Golden Spiral of the Fibonacci Numbers.

Pascal’s Triangle is a Repository of all the Possible Magnitudes and their Components.

The Normal Distribution shows that the first level of the Triangle (the Tetraktys) is much more probable than the last levels.

The Hexagonal Numbers

The first four Levels of the Triangle of Pascal contain the Tetraktys of Pythagoras.

The Tetraktys  is an Ancient Vedic Mathematical Structure called the  Sri Yantra, Meru Prastara or Mount Meru.

About Numbers, Operations and the Klein Bottle

The Complex Numbers are not “Numbers” (Scalars) at all.

They are “Operations” (Movements) that can be applied to Magnitudes (Geometries) and Magnitudes are Combinations of the Simple Building Blocks of the Tetraktys, Points and Lines.

The Tao of Ancient China was not for nothing represented by a Flow of Water. According to the Ancient Chinese Mathematicians Every Thing Moves.  In the Beginning there was only Movement.

In the Beginning only the One was Moved but when the Duality was created the Two moved around each other never getting into contact to Avoid the Void.

When we look at the Numbers we now can see that they are the result of the Movements of  the first Diagonal of Pascals Triangle,  the 1′s (Points) or better the Powers of  the One: 1 **N (where N is a Dimension).

Even in the most simple Number System, the Unary Number System, Concatenation is an Operation, An Algorithm.

The Mathematician John Conway recently invented a new Number System called the Surreal Numbers that contains Every Number you can Imagine.

The Surreal Numbers are created out of the Void (ɸ)  by a simple Algorithm (Conway calls an Algorithm a Game) that describes Movements (Choices of Direction: Up, Down, Left, Right, ..)  that help you to Navigate in the N-Dimensional Number Space.

The Ancient Chinese Mathematicians played the same Game with the Numbers.

Algorithms were already known for a very long time by the Ancient Vedic Mathematicians. They called them Yantra’s.

Sri Yantra

Geometry is concerned with the Static Forms of Lines and Points but there are many other more “Curved” forms that are the result of  Rotating Expansion and Compression. These forms are researched by the modern version of Geometry called Topology.

The most interesting 4D Topological Structure is the Klein Bottle.  The Klein Bottle is  a combination of two Moebius Rings. It represents a Structure that is Closed in Itself.

It can be constructed by gluing both pairs of opposite edges of a Rectangle together giving one pair a Half-Twist. The Klein Bottle is highly related to the Ancient Art of Alchemy.

The movement of the Duality around the Void can be represented by a Moebius Ring the Symbol of Infinity ∞.

Later in this Blog we will see why the Number 8 is a Rotation of ∞ and the symbol of Number 8 is a combination of the symbol of the number 3 and its mirror.

First we will have a look at the Reciprocal Relation between Space and Time.

The Klein Bottle, The Universe Closed in Itself, the Basic Structure behind Alchemy.

About Dewey B. Larson, Velocity and Time

Dewey B. Larson (1898 – 1990) was an American Engineer who developed the Reciprocal System of Physical Theory (RST).

Larson believed that the failure to recognize that Motion is the most basic physical constituent of the universe has handicapped the progress of the traditional study of physics, which focuses on Forces.

The definition of Motion stems from the Equation of Velocity, v = ds/dt.

Instead of depending upon the change of the location of an object to define an arbitrary “quantum” of space per “quantum” of time, such as miles per hour, or meters per second, the RST assumes that the observed universal passage, or progression, of time is one aspect of a universal motion that necessarily must be accompanied by a universal “passage,” or progression, of space.

The Units of Time fill up the Units of Space. Space and Time are Duals.

Space is not-Time and Time is not-Space. Time is Non-Local, Cyclic and represented by the Rotating Imaginary Numbers. Space is Local, Linear and Represented by the Scalar Numbers. Space is the Vacuum and the Nothing and Time is the non-vacuum, the Every Thing, the Solids represented by the Cube of Space.

The Cube of Space is the structure behind the Tetraktys but also behind the Book of Genesis.

Our Reality contains two Reciprocal 3D-structures related to Space and Time. Space and Time are related by the Simple Formula N/N=1/1=1, the Formula of Diracs Delta Function.

We are able to perceive the Real 3D-Structure of Space. The 3D-Structure of Time is Imaginary. It is situated in the Imaginary Number Space of i.

Larson’s Cube, the Geometric Representation of the Octonion.

Larson, a Self Thought Genius like Grassmann, developed Geometric Algebra without knowing anything about Geometric Algebra but he also invented String Theory long before String Theory was invented.  The Mathematics of Larson is also the Mathematics of the Tetraktys of Pythagoras without even knowing anything about it.

The Periodic System of Larson

Larson was able to Calculate all the important Physical Numbers without any problem and was also able to Calculate Chemical Structures and Reactions.

The fourth line of Pascals Triangle and the Tetraktys contains 8 Directions in the Four Geometric Dimensions: 0, 1, 2, and 3.

Mathematicians are intrigued with this number 8, because they find it popping up unexpectedly in advanced mathematics.

In fact, expanding the Binomial Expansion to 8 dimensions just creates an inverse copy of these first Four Dimensions, and then the pattern just repeats itself with a half-twist and back from there, ad infinitum.

This is called Bott Periodicity discovered by the mathematician Raoul Bott (1923-2005).

The mathematician John Baez wrote an article in which he relates this 8-fold Periodicity to the Scalars (1), the Complex Numbers (2), the Quaternions (2×2), and the Octonions (2x2x2 = 2**3).

Bott Periodicity

The Universe of Numbers and Magnitudes  is Cyclic and Fractal.

Our own Reality, symbolized by the Tetraktys,  repeats itself in Higher Dimensions until Infinity.

The Tetrad, represents Completion, because it contains all its Previous Numbers, the 1, 2, 3, and itself, 4, in One Number, 10 = (The One) +  9 (= 3 (Trinity)x 3 (Trinity) = Tetraktys).

As you can see in the Picture above the Fractal Pattern of 8 contains two kinds of Trinities/Triangles, an Upside and a Downside (Rotated by 180 Degrees) Triangle. When you Rotate by 180 Degrees the 1 becomes -1 and 1 + -1 =0 is the Void.

The Star of David, the Symbol of Human Center, the Heart Chakra.

The Multi Dimensional Rotations of the Octonions always Come Back to Square 1/1=1, the One and keep Rotating around the Center, the Nothing,   Until Infinity.

About Triangular Numbers and Pascal’s Triangle

About the Relationship Between Geometry and Music

About the Game of the Surreal Numbers

About Larson and the Unification of Mathematics

The Collected Works of Dewey B Larson

About Ratio and Proportion by Euclid

A book of Augustus deMorgan about “The Connection between Number and Magnitude”

The text of the Fifth Book of Euclid

An Educational You Tube Channel called Insights in Mathematics

About the History of Geometric Algebra

Free Software to use Geometric Algebra

A Video that shows how the Platonic Solids are created out of the Trinity Numbers

All you want to know about Geometric Patterns

Monday, December 19th, 2011

This blog is about the Cycle of Nine implemented in the Digital Root or Modulus 9-Function. The Digital Root generates many Patterns that were used in Ancient Architectures.

One of the most important Digital Root Patterns is the Vedic Square. It is the Digital Root of the Multiplication Table of the numbers 1 to 9.

This Table contains the Harmonics of the Numbers 1 to 9. These Harmonics are highly related to the Harmonic Pattern behind the Cycles in our Universe.

The first part of this Blog is about the Digital Root. It contains the patterns that are behind the Cycle of Nine.

This part is very technical but it makes it possible to show that there is a deep structure  behind the Modulus-9.

This pattern has to do with just two numbers, 2 and 3. They generate the Spirals of Expansion and Compression of our Universe.

2 and 3 and their Sum 5  are also the Numbers behind the Harmonics of our Universe.

The Second Part is about the Vedic Square. It is called the Vedic Square because this Square is one of the most important tools in Ancient Vedic Mathematics.

Vedic Mathematics was used in many Ancient Cultures (China, Egypt, Greece) with different names. The Chinese art of Feng Shui was called Vaastu Shastra in India.

Pythagoras, trained in Egypt (Heliopolis),  used the same principles and used the same  Patterns the Ancient Vedic Scientists were using.

The last part is about the Game of Chess. This game is  just like many other Ancient Games a Simulator of the Game of the Universe.

This blog contains many links to other Blogs and Resources on the Internet. These references make it possible to dig deeper into this fascinating subject.

When  you divide a number X by a number N the Remainder of the division is called X Modulus N.  22 mod 7 = 1 because 22 = 3×7 + 1.

The Modulus-function N maps the Set of the Natural Numbers to the Numbers 0, 1, 2, ….,N-1.

One of the most famous and ancient Modulus-functions is called the Digital Root. The Digital Root is the Modulus 9 function.

Because 10 mod 9 = 1 every Power of 10 has a Modulus 9 of 1. Therefore (a10**X+ b10**Y+…) mod 9 = a + b, the Sum of the Digits of the Number. 62 mod 9 = 6+2 = 8.

Digital Roots have been recorded for thousands of years, formalized by Pythagoras in 530BC and even earlier in Indian Vedic Mathematics (Vaastu Shastra).

Digital Roots are used in Numerology. In Numerology Numbers have a Meaning.

In Gematria Letters and Words are transformed into Numbers which have a meaning.

In Ancient Languages like Hebrew Letters are also Numbers. Numerologists believe that Words with the same Digital Root have the same Meaning.

The numbers 0 to 9 of the Digital Root are the Points of the Tetraktys of Pythagoras.

The Tetraktys of Pythagoras

The Modulus 9 pattern contains 2 number groups (3,6, 9) and (1,2,4, 5,7,8).

Later we will see that the last group contains 2 subgroups (1,4,7) and (2,5,8).  Together with (3,6,9) we can map these 3 Triangels on the Modulus 9 Circle.

4 is the Middle of 1+7=8, 5 is the Middle of  2+8=10 =1 and 6 is the Middle of 3+9=12=3. 5 is also the Middle of the Middle.

The group (1,2,4,5,7,8 ) is called the Ring Z/9 in Mathematics. Z/9 is isomorphic with the Sequence 2**N mod 9 where N is positive and negative. The sequence 1,2,4,8,16(7),32(5),64(1),128 (2),256 (4),… repeats itself until infinity.

This Sequence is the Expansion and Compression Pattern of the Number 2.

The Ring Z/9 is part of the Tetraktys and forms a Hexagram. This Hexagram is a 2D-projection of the Cube of Space. When we combine the (3,6,9)-pattern with the Hexagon a (4×4) Triangle is created.

The number 2 is the Container, the Cube, inside the Tetraktys. That is the Reason why the Second letter in the Hebrew Alphabet Beth means Vessel or Container.

(3,6,9) is a Triangular Cycle that repeats itself until Infinity.  The Number 3, the Trinity, is the Mover of the Container of 2. This Rotation moves With and Against the Clock.

This is the reason why the 3th Letter of the Hebrew Alphabet, Gimel, means Camel. The Camel of Gimel carries the Water into the  2 Containers of Beth.

The Number-2-pattern contains 3 Binary Groups (called Polar Pairs) with a Sum of Nine (1,8), (2,7), (4,5). The Number-3-Pattern contains 2 Polar Pairs (3,6) and (0,9). The Polar Pairs represent the Lines of the Tetraktys.

(0,9) maps unto Itself and represents The Beginning and The End, The Now. (0,9) is a Point and a Line.

The Polar Pairs of the Z/9 create a Cyclic Pattern that contains two Squares, (1,2,4,0) and (5,7,8,0). Both of them Share the Zero, The Void.

The Sum of the Opposite Numbers of the Z/9, (4,8 = 12=3), (1,5 =6 ), (2,7=9) of the Tetraktys shows the 3,6,9-pattern again.

Lo Shu Magic Square

There are 8 Ternary Groups ((1,5,9), (1,6,8), (2,6,7), (2,5,8),(2,4,9),(3,4,8),(3,5,7),(4,5,6)) with a Sum of 15. This Ternary Group represents Triangels. All of them are part of the famous Lo Shu 3×3 Magic Square.

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

When we use the number 3 as a generator 3 Triangles are created (1,4,7), (2,5,8) and (3,6,9).

The 3 Triangles move With and Against the Clock ((1,4,7) and (7,4,1)).

It takes 3 rotations to get every Triangle back to its original position. (1,4,7) becomes (7,1,4) and (4,7,1). This means that there are 6 permutations of every Triangle.

Every addition of two Triangels produces another Triangle.   An Example:  (1,4,7) + (2,5,8) = (3,9,6).

When we create a Matrix to find all the combinations a new group of 9 transformations ((1,1,1),(2,2,2),(3,3,3),(4,4,4),(5,5,5),(6,6,6),(7,7,7),(8,8,8),(9,9,9)) appears. They are the Triangels that are a Line and a Point. An Example:  (1,4,7) + (1,1,1)=(2,5,8).

There are now (18 +9=27) x27 = 729=3**6 = 9**3 possibilities.

The same 27×27 Matrix appears when we Multiply the 3 Triangels. An Example:  (5,8,2)x(5,8,2)= (25,64,4)= (7,1,4) and (3,6,9)x(5,8,2)=(15,48,18)= (6,3,9).

Another interesting patterns  becomes visible when we look at the Opposite Numbers of 3 Triangels (1,5), (2,6), (3,7),(4,9) en (5,9) in the Picture above.  They recreate the Triangels. An Example: (5+9=5, 2+6=8 ,8+3=11=2).

About the Digital Root of the Golden Mean

The 27×27 Matrix pattern also emerges out of 24 repeating numbers (1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9) of the Digital Root of the Fibonacci Sequence (The Golden Ratio).

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

When we group the Golden Ratio pattern in 2′s (2×12) the Polar Pairs appear. The 12 pattern has  a Sum of 108 = 0 Modulus 9. 108 and 24 are related to the Gayatri Mantra.

1 1 2 3 5 8 4 3 7 1 8 9

8 8 7 6 4 1 5 6 2 8 1 9

When we  group the pattern of 24 numbers (3×8) of the Golden Ration into Trinities the Triangle Pattern appears again.

1   2  3  4                  -3 -2 -1 -4 (Pattern-number)

1   1  2  3  (7)           5   8   4   3   (2)

7   1  8  9  (7)          8   8   7   6   (2)

4   1  5  6  (7)          2   8   1    9   (2)

The Pattern of the Pattern is (1,2,3,4,-3,-2,-1,-4). The last part of the Pattern (-3,-2,-1,-4) can be transformed into the first part (1,2,3,4) by adding 4.

The Digital Sum of the first 3×4 numbers is 7 and the Digital Sum of the last 3×4 numbers is 2.

When we rearange the 24 cycle in 6 groups of 4 digits another pattern shows itself: (1,4,8,5), (1,3,8,6), (2,7,2,7),(3,1,6,8), (5,8,4,1), (8,9,1,9).

The pattern of the Golden Mean copied from the Linked Website

When we combine all the different rotations of the 3 Triangels a Cyclic Flow Pattern appears that looks like the Jitterbug of  Buckminster Fuller.

The Jitterbug is a 3D projection of the 4D 24-Cell (again 24!) also called the Hyperdiamond.

The 24-cell is self-dual and is the regular polytope with no analogue among the five Platonic solids of 3-space.

The 24-cell also called the Hyperdiamond

One of the Simple Structures of Numbers that contains a lot of patterns is the Vedic Square. The Vedic Square was called the Eight Mansions in China. The Vedic Square is the Digital Root of the Multiplication Table of the numbers 1 to 9.

The Multiplication Table is a subset of the 27×27 Matrix of the 3 Triangels.

The Multiplication Table contains the Harmonics of the Numbers 1 to 9.

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

The Vedic Square was used to build the Pyramids, create the Chinese I Ching, the Game of Chess, Dante Alighieri used it to structure his trilogy La Divina Commedia, the Sistine Chapel was build and the frescoes and symbols were arranged according to its concepts and the first chapter of Genesis was written and imbued with its numerous concepts graphic images.

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

Scholars and Artists discovered that the various lines of the Vedic Square could be used to direct a design. By selecting a line of numbers, and using a constant angle of rotation, various designs could be produced. These designs are visible in abstract Islamic Art.

The Patterns of Islamic Art are created out of the Vedic Square

The Vedic Square is a Symmetrical Structure because AxB=BxA. This is called the Associative Property of Multiplication.  The Square is a combination of Two Triangels and contains 45 distinctive numbers.

Vedic Square

The Vedic Square repeats itself until infinity when you extend the Square to a NxN Square.

The number pattern of the diagonal of the Vedic Square, 1,4,9,7,7,9,4,1,9,  is the Digital Root Pattern of the Square Roots. This patterns repeats itself until Infinity.

The Vedic Square contains the 5 Polar Pairs, the 8 Lo Shu Ternary Groups and the 3 Trinity Patterns ((1,4,7), (2,5,8), (3,6,9). It also contains the Star of David, The Zodiac, the Tree of Life and many other Mystic Patterns.

It is possible to transform the Vedic Square to the Lo Shu Magic Square.

The patterns of the Vedic Square Rotate. The End of a Horizontal and a Vertical Pattern connects with the Beginning of the Pattern. This means that the Vedic Square is a Torus.

This Torus is called the Rodin Torus. The Rodin Torus is a Coil that produces a Uniform Electro-Magnetic Field.

The Rodin Modulus 9 Number Torus

The 3-6-9 and 6-3-9 Cycle in the Vedic Square can be thought of as Clockwise and Counter-Clockwise, or as Electricity and Magnetism. They are transport-channels.

The ((3,6,9),(6,3,9)-Matrix divides the Vedic Square in 9 2×2 Squares.

The 9 2×2 Squares have a Sum of 9,18 and 27 which is 1×9,2×9 and 3×9. If we leave out the (3,6,9)-Matrix and divide by 9,  a 3×3 matrix results with 1,2,3 on the Outside  and a Cross of 2′s in the middle. This 3×3 matrix shows the Expansion of the 2 into the (1,2,3).

Patterns in the Vedic Square

The Rows and Colums of Ring Z/9 add up to 45. The Rows and Colums of the Number 3-Pattern add up to 54 which is a Mirror of 45. The (4,5)-pattern generates the Star of David and the Zodiac.

The Vastu-Mandala

The Game of Chess originated in India. It was passed on to the medieval West through the intermediary of the Persians and the Arabs.

The form of the Chess-Board corresponds to the Vastu-Mandala, the 9×9  diagram which also constitutes the basic lay-out of a temple or a city.

Hindu mythology has it that Vaastu Purusha was born of Lord Shiva’s sweat when he fought the deadly demon Andhakasura.

Vaastu Purusha himself became uncontrollable and destructive and the heavenly gods finally subjugated him and brought him down on earth with face down, with his face in the Northeast and his feet in the Southwest.

45 deities stayed there, 32 of them in the outer enclosure and 13 of them in the inner enclosure holding him in place at various points or locations on his body.

32 =64/2 and the Number of the 32 Paths of Wisdom of the oldest book of Hebrew Mysticms the Sepher Yesirah (the Book of Formation or Book of Creation, ספר יצירה).

64 is the Number of the I Tjing. 45 (5×9) is the Sum of the Lo Shu Magic Square and the Number of the Vedic Square.

All these Mystic Structures come from the same Source and are different Views on the same Pattern, the Tetraktys, the Triangular Numbers created by the Meru Prastara or Sri Yantra also known as the Pascal Triangle.

The Vastu Jain Symbol is a version of the Tetraktys

The Vastu Mandala is an expansion of  a Point (the Bindu) into the Line(2), The Trinity (3) and the Rotating (With the Clock and Against the Clock) and Expanding Square (4), represented by the Symbol of the Swastika. The Swastika is a Fractal Generating Pattern.

Every Point is a generator from which the Swastika-pattern generates a new Swastika. The 2×2 Square is transformed by the Swastika Pattern into the 4×4 and the 8×8 Square.

As you can see the Vastu Jain Symbol is an Indian Version of the Tetraktys of Pythagoras.

The Swastika contains the Four Points of the last line of the Tetraktys that are related to the Tethahedron.

The Borobudur represents Mountain Meru, Pascal's Triangle.

The Chess-Board symbolizes the Unfolding of Space by the Number-2-pattern and it synthesizes the Complementary Cycles of Sun and Moon.

The number 64, the sum of the Black & White (Yin/Yang) Squares on the Chess-board, is a divisor of the number 25920 (25920/64=405, 25920/9= 2880/9=320/5=72), which measures the Precession of the Equinoxes.

The Polar Pairs in the Modulo 9 Pattern are expressions of the Planets.

(1,8), the Castles, relates to the Planet Mars.

(2,7), the Bishops, relates to the Planet Venus. Venus is the Ruler of the Heart and the (2,7) is situated in the Middle of the Vedic Square.

When viewed from the Earth, the Planet Venus inscribes a near perfect five-pointed star (pentagram) around the sun every eight years. The points of a five-pointed star (pentagram) touch the circle of a pentacle every 72 degrees.  Likewise, many in Islam expect 72 virgins in heaven.

A full 360 degrees of procession takes 25,920 years, which is also seventy-two (72) 360-year cycles.

(3,6), the Knights relates to the Planet of the Messenger, Mercury. Mercury is Hermes, the Messenger God, with winged sandals. The moves of the Knights create a pattern that looks like the Swastika.

The (3,6)-number-lines are Transport-Channels (Gimel) as you can see in the Vedic Square and the Rodin Torus. The planet Mercury traces a Hexagram during its movement around the Zodiac.

(0,9) is the Planet Jupiter,  the Ruler of Modulus 9 who determinates the Rules of the Game. (0,9) is the Beginning and the End of the Game and is the cause of the Rotation of the Swastika related to the (3,6.9)-pattern.

The numbers 4 and 5 are the Moon (Queen) and the Sun (King). The Moon moves the quickest of all the planets, so does the Queen on the chessboard.

The Number 5 of the King is the Center of the 3×3 Lo Shu Magic Square and the Center of the Tetraktys.

The 8 Pawns represent  the number 2 and are connected to the Planet Saturn, the 2nd Son of the Central Sun and the Trinity (1+ 2 = 3). The Pawns start to move with 2 steps and later move 1 step. The 2 is the Center (The Son of the Sun) of the Trinity.

The 2 is also the Generator of the Expansion Pattern and the Polar Companion of the 7, the Center of the 3D-version of the Square, the Cube of Space.

The Pawn (2, Saturn) promotes into a Queen (Moon, 2×2) when he has reached the Other Side.

About the Multiplication Table of 9

About Genesis and the Vedic Square

About the Divina Comedia and the Vedic Square

About the Sistine Chapel and the Vedic Square

About the Tetraktys and the Lo Shu

About the Harmonics of the Universe

About the Jitterbug of Buckminster Fuller

A Simulation of the Jitterbug Pattern (CUBIC WONDER)

About Plato and the Sri Yantra

About Vastu Science and the Borobudur

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

Friday, December 2nd, 2011

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A Hexagram is a Two Dimensional Cube

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

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

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

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

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

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

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

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

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

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

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

The Sum of two Triangular Numbers is a Square

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

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

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

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

28 is a Hexagonal Number

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

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

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

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

Star Numbers are a Combination of Two Triangular Numbers

The Symbol of the Heart Chakra contains Two Triangles.

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

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

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

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

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

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

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

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

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

Triangle of Pascal

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

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

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

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

This pattern repeats itself all the time.

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

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

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

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

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

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

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

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

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

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

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

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

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

the Fibonacci Numbers as a Cube.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Life is not only about Me and the Other.

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

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

The Content of the Sepher Yesirah

About the Cube of Space and Psychology

About the Sepher Yesirah and the I Tjing

A correspondence table of the Cube of Space

About the Sri Yantra and Plato

About the Lo Shu and the I Tjing

All kind of strange relationships between Triangular Numbers

A website about Mystical Number Theory

About the Golden Spiral and Plato

About Pascal’s Triangle and the Normal Distribution

A complete course in elementary Number Theory

About the Psychology of the Cube of Space

About the Tetraktys and the Zodiac

About the Process Theory of Paul Young