Posts Tagged ‘China’

About Autarky

Thursday, October 11th, 2012

The amount of local cooperative energy-companies in the Netherlands (in 2012) is increasing. It all has to do with the decrease of the price of solar-sells but there is more at stake.

Local energy-companies are part of a movement aimed at Autarky. Autarky is the quality of being self-sufficient.

The word Autarky comes from Greek αὐτάρκεια (autárkeia), itself from αὐτός (autós) = self and ἀρκέω (arkéo) which in its passive form means be satisfied, content or “Out of my own Power“.

Fractal Human Distribution Network

Fractal Human Energy Distribution Network

The need to become self-sufficient has certainly to do with the current economic-, ecological- and energy crisis. People are afraid that the supply of vital energy and food will suddenly stop or become priceless. This is not the only reason.

Some people think we are wasting our money, food and energy by producing nondurable throw-away goods in which the packaging is sometimes more important than the content. They want to create durable products and services. This is not the only reason.

Some people don’t trust the suppliers of their food and energy, their politicians and/or the caretakers of their savings, the banks, anymore. They believe that their food is intentionally manipulated, their savings are wasted, their country is sold out to the rich and their environment is polluted for just one simple reason, Greed, the opposite of “Being Satisfied”.

The people of the autarkic movement want to recreate the necessary functions (bank, energy, food, work, healthcare, government) themselves in a cooperative way in their own living context.

Atmospheric Energy Diagram

Atmospheric Energy Diagram

The local cooperatives  are now joining on a higher (mostly country) level and are  in need of a shared (real-time)infrastructure. The current infrastructure does not support flows that move between many participants (so called peer-to-peer-networks).

The current system is a Hierarchic Top Down System containing Huge Highways that are connected to Big Production-facilities. This makes it impossible to consolidate, share and distribute resources peer-to-peer on a local level.

The Electric Highway

The Top Down System has created Global Markets where the local production is sold  to the highest bidder.

The Global Markets are the playing field for advanced speculators that make a lot of money. After the deal is made the local production is transported to a far away destination. That is the reason why the Top Down Systems are integrated in world-wide Logistic Systems.

If we want to coordinate all the local initiatives we need advanced systems that Up- and download, control, predict, connect, share, balance and distribute vital local resources on the short and long term.

The Panarchy Cycle of an Ecology

The Panarchy Cycle of an Ecology

Examples of these systems can be found in “crowd funding“, “virtual power plants”  and “treasury management” (balancing many small amounts of Money at the short and long term).

In Nature we find comparable (Fractal) Distribution-Systems in the Lungs, the Arteries, Riverdelta, the Weather Cycle and Ecologies.

All of these fysical Flow Systems, that map 1 Resource-> on Many Resources or Many Resources->on 1 Resource,  are based on the third Law of Thermodynamics called Constructal Law.

Virtual Power Plant

Virtual Power Plant

In the philosophy of the Stoics and Cynics of Ancient Greece the perfect human was not only arete (“all-around excellent”) but also independent of every need. Arete was associated with a full life as a citizen of a city.

Later Spinoza (“On Human Freedom“), Rousseau and David Thoureau (Walden) have developed their own autarkic philosophy.

Autarky was the ideal of a complete man who needed no one and no organization. This ideal can also be found in Zen Buddhism, Taoist practise, Early Christians (Nazarites), the Catholic Order of the Franciscans and the Protestant Quakers.

In reality it is almost impossible to be completely self-sufficient as a lonely human being. We need others when we are young, sick and old and we need others because we are specialized (“Talented”). Autarky is a possible aim for a small community.

Resource Balancing in Nature (Lungs, RiverDelta)

Resource Balancing in Nature (Lungs, RiverDelta)

The religious, cultural and political movements at the left and the right aimed at living the “good life” (“Living Slow“) are converging in  the movement of the Cultural Creatives. The Cultural Creatives are the new Elite that is moving to the AND-state (Left AND Right).

They don’t believe in the old Competitive Ideologies (Socialist, Capitalist, Religious, Liberal, Nationalist, Conservative, Globalist, etc..) but want  to focus on the Shared Collaborative Values between all Humans. We all need Love, Food and Shelter.

The Stoic Zeno shows the Doors to Truth and False

The Stoic Zeno shows the Doors to Truth and False

Autarky has always been a major political aim of countries. Autarky is about the possibility to control your own future and to become independent of other powers that could use your dependency to tell you what to do to get what you desperately want.

At this moment the shortage of fossil fuels, especially oil and gaz, is used to influence the policy of other countries and is the cause of local conflicts in the Middle East.

China is buying all kinds of vital materials to make sure their economy will not become dependent on others. If they will succeed and nobody doubts about that, they will become the economic controlling power of the world.

Reaching a state of local autarky, being satisfied with what your environment provides you, has become the aim of the wordwide sustainability movement.

Total autarky is possible when you want to live as a stoic or a survivalist. Most people accept a more simple living, move off-the-grid and build autonomous buildings that make use of  renewable energy.

Allotment Garden

Allotment Garden

Others start to farm in their own backyard, join an allotment garden or move back-to-the-land and practice sustainable agriculture.  To handle their surpluses and shortages independent  local markets are created that make use of  alternative currencies.

A strive for (regional) Autarky pushes the need to innovate to a higher level because not all regions contain the right materials, knowledge or experience to realize everything everybody needs. To become fully Autarkic the Production and Consumption Systems in the Region have to be based on simple concepts and parts that are available everywhere. Simple concepts are concepts that are easely copied.

The only way to combine Simple and Complex concepts is to use the Concept of Self-Reference (Fractals). The only way to make use of parts that are available everywhere is to make use of the Simple Building Blocks of Nature. A good example is the use of algea in food processing.

The concept of Autonomy not only plays an important role on the many levels of human (personal, organization, country) and animal society (Competitive vs Collaborative (Ants)). It is also a major subject in the research of (Second-OrderCybernetics (Control Theory).

Self Reproducing Systems

Self Reproducing Fractal Systems

Second Order Cybernetics is concerned with the Control of the System of Systems (the Meta-System). Hierarchy is not the solution to this problem because it leads to the logical problem of Infinite Regress (“Turtles on turtles on … all the way down”).

The Meta-System is controlling, reproducing and balancing itself by a Fractal Feedback-Cycle that moves Backwards (Consumption) and Forwards (Production).

The concept of Self-Reproduction is called Autopoiesis or Autocatalysis. The Design Philosophy of Cradle to Cradle is a good example of Autopoiesis.

A System is an Autarky if all its Levels are capable of Self-Reproduction.

If a human wants to become self-sufficient he has to accept and develop his own talent and find complementary people so he is able to create a Self-Reproducing Unit.

The smallest Self-Reproducing Unit in Human Society is the Family living in one House. This House has to produce more than enough Energy and the Garden has to produce more than enough food to support the Family.

Garden City (Culemborg, the Netherlands)

Garden City (Culemborg, the Netherlands)

If we cluster all these Self-Reproducing Families/Houses & Gardens in a small City the chances are high that the combined talents of the families will be able to supply every profession needed if the people of the city want to Live a Simple Life.

It is also possible that the surplusses and shortages of the families can be compensated by creating a Market on the level of the small City or one level higher, the Region.

An Autarkic Movement becomes an Aggressive Competitive Nationalistic Movement when the Citizens of the Country feel that their Country is plundered by outsiders and their Leaders promise to get it all back by expanding the Space to Live (“Lebensraum“) to facilitate everything that is needed.

This happened when Germany was preparing for War after the enormous reparation payments of 132 billion gold marks to the Winners of World War I.

The Autarkic Movement could also become a Peaceful Cooperative Social Movement like the Movement of Gandhi in India when the people accept the Good Life they already live and want to share the surpluses with others who are in a worse state.


Dutch Report about Local Energy Companies (in Dutch)

About the house that produces his own energy (Passive House)

The Complete Book of Self Sufficiency

About Close-Loops Systems

About Smart Grids

About Fractal Distribution Systems

Jerry Rifkin: About the Distributed Economy

The John Seymour School for Self Sufficiency

About Entrainment

About the EU Policy of Smart Specialization of Regions

About Darwin and Collaboration 

About Artificial Life

About Smart Technology

About the Thermodynamics of Living Systems

About Worldviews (How to find Complementary People)

A City is an Organism

About the Viable Systems Model of Stafford Beer

About Garden City

Gandhi’s view on Capitalism

The Autarkic Dreams of Germany


Over de Paradoxen van het Leven (in Dutch)

Tuesday, August 14th, 2012


Op  heb ik een verhaal verteld over Path of Change in het Dialoques House van ABN AMRO in Amsterdam ZO.

Hierbij de plaatjes van de presentatie. Als er vragen zijn stel ze gerust via het contact-formulier.

Paths of Change gaat over veranderen en is ontwikkeld door Will McWhinney.

Recent is er een boek “Faciliteren als tweede beroep” verschenen van Jan Lelie dat een eigen interpretatie van Paths of Change bevat. Mensen die meer over Paths of Change willen weten wordt aangeraden in ieder geval dit boek te lezen.

Het Egyptische Spel Mehen

Jan Lelie heeft Paths of Change (POC) toegepast op Faciliteren terwijl ik het vooral heb gebruikt bij het Ontwerpen van Systemen.

POC kan op alle terreinen worden toegepast die je maar kunt bedenken omdat het een Meta-Praxis is, een Theorie over de Praktijk van het Veranderen.

POC is in 1997 door Jan Lelie geintroduceerd door het delen van het boek Paths of Change. Ik was zo onder de indruk van dit boek dat ik onmiddellijk op bezoek ben gegaan bij Will McWhinney. Tot zijn dood zijn we (met vele anderen) bezig geweest om een verbetering van het boek te schrijven met als werktitel Grammars of Engagement (GoE). Het boek is helaas nooit gepubliceerd. GoE gaat over synchronisatie.


POC gaat er vanuit dat mensen een eigen onafhankelijke perspectief op de wereld hebben en dat het mogelijk is om die verschillende kijken te classificeren met behulp van vier zogenaamde wereldbeelden (“Worldview”). Veranderingen gaan altijd samen met veranderingen van wereldbeeld.

De oneindige varieteit aan menselijke kijken kan worden afgeleid door het toepassen van de wereldbeelden op zichzelf. Ieder mens leeft in zijn eigen universum maar er bestaat een fractaal patroon waardoor ze elkaar kunnen begrijpen.

Het patroon kan worden gevonden  door de regelmaat in de regelmaat (in de regelmaat…) te onderzoeken. Wat dan blijkt is dat die regelmaat een zelf-referentie is. Hetzelfde patroon blijft zich herhalen.

Veranderingen zijn Cyclisch en Fractaal. Dezelfde veranderingen komen altijd weer terug op iedere Schaal die er bestaat. Ze springen in de tijd van Schaal (bijv. Mens) naar Schaal (Organisatie) en zelfs de Schaalsprongen zijn weer onderdeel van een Cyclus.

Het is zoals de Sufi’s zeggen: Alles draait om Alles.

Alles draait om Alles

Alles draait om Alles


Hierdoor is Alles uiteindelijk in zichzelf gesloten en roteert rondom het centrum van het centrum (van het centrum…..), de Leegte.

De “mystieke” leegte heeft exact dezelfde eigenschappen als de wiskundige leegte die de lege verzameling wordt genoemd. Dat is niet zo raar als het klinkt als je beseft dat wat de Christenen in het Evangelie van Johannes de Logos noemen (“In den beginne was het Woord (Logos)”) dezelfde principes vertegenwoordigd als wat wij nu de Logica noemen.

De lege verzameling die de Leegte wordt genoemd staat voor alles wat nu nog onmogelijk is. In China wordt ze de Tao genoemd.

Alles is niet-Niets

Alles is niet Niets

De schepping, Alles wat  Is, is de ont-kenning van de Leegte. Die negatie ging gepaard met een enorme explosie in de vorm van een Dirac Delta-puls waardoor het Alles tot op heden permanent aan het expanderen is volgens de regel Space/Time, waarbij de tijd de inverse is van de ruimte. Alles spiraleert rondom het onmogelijke, het Niets. Als we alle schillen van de ui hebben afgenomen blijft er niets over.

Nothing is an opportunity to imagine something. Absolute nothing is the most powerful opportunity of all to imagine anything at all. Because there is really absolutely nothing, the contrast with even a flickering thought of something is enough to make that something seem real! (Law of Forms, Spencer Brown).

The Spiraling Spiral in Chartres Cathedral

De Spiralerende Spiraal in de Kathedraal van Chartres

Het betekent ook dat er altijd sprake is van een paradox, een onlogische situatie, die we steeds maar weer proberen op te lossen maar in zijn wezen onoplosbaar is.

Will McWhinney’s meest wijze woorden zijn dan ook “Our task is not to resolve the paradoxes, but to manage them”. Het heeft geen zin om de uiteindelijke logica achter de werkelijkheid te zoeken want die bestaat niet.

Decentralisatie roept Centralisatie op en een streven naar Menselijkheid tendeert uiteindelijk in On-Menselijkheid. De enige manier is om van alles het midden te vinden en van daar uit te gaan balanceren net als een koordanser. Vandaar de andere favoriete uitspraak van Will “There is only the Dance” (van Shiva).


Veranderingen lopen vast als het pad van de cyclus tegen het pad van een andere cyclus opbotst. Die botsing kan worden vermeden of het conflict kan worden opgelost door een Derde Wereldbeeld te introduceren (een By-Pass) of door een extra dimensie (een “bewustzijn of observator”) te introduceren waardoor ze boven (of onder) elkaar kunnen bewegen.

De vier wereldbeelden zijn inzichtelijk gemaakt in het Book of Kells


De wereldbeelden worden door Will McWhinney Unity, Sensory, Social en Mythic genoemd en zijn te vergelijken met de vier Archetypen van Jung die je kunt terugvinden in de persoonlijkheidstest MBTI. In deze test worden ze T(hinking, Unity), S(ensing, Sensory), F(eeling, Social) en I(ntuiting, Mythic) genoemd.

In MBTI worden er naast de 2-combinaties van STN & F ook nog Introvert & Extravert (I/E) en Perceiving/Judging (P/J) onderscheiden.

Introvert & Extravert slaan in het model van Will op de draairichting van de combinatie. Introvert is Naar-Binnen-Gericht (Van Sensory -> Unity -> Mythic -> Social -> Sensory) en Extravert is Naar-Buiten-Gericht (Sensory->Social->Mythic->Unity->Sensory).

P betekent Boven-het-Model (= de Observator/Bewustzijn/Abstractie/Monitor) en J is In-het-Model. Dit betekent dat er eigenlijk nog een Vijfde Wereldbeeld is maar het beeld bestaat buiten het 2-D-vlak van de vier anderen in de 3De dimensie.

Als je het projecteert in het 2D-vlak komt het in het midden uit op het Kruis-punt van de twee lijnen die de vier verbindt. In die zin staat het Kruis voor Centrum, Balans, Evenwicht en Bewustzijn.

Het grote verschil met de andere beelden is dat het net als het Oog van de Storm stil staat terwijl de rest van de paden ronddraaien. Mensen die problemen hebben met dit Wereldbeeld zijn onevenwichtig.

De Vijf Organen van de Acupunctuur passen perfect op de Vier Wereldbeelden. Het vijfde orgaan staat voor de relatie Geheel/Deel (Earth).


De wereldbeelden zijn ook te verbinden met Het Denken in Modellen (Unity, Waar-maken), het Sensory-Motor Systeem (Handelen en Waar-nemen, Sensory), de Emoties (Social, Waar-deren) en de Verbeelding (Mythic, Waar-om?).

Daarnaast kun je de wereldbeelden ook afbeelden naar IT-technologie (Smart Computing). In dat geval heeft Sensory te maken met Sensors & Proces-management, Unity met Analytics en natuurlijk Big Data, Mythic met User-Inter-Face & Action waarbij nu bijv. Augmented Reality en natuurlijk Games een grote rol spelen en Social met zoals de naam het al zegt Social Networks.

Die Sociale Netwerken kun je weer verdelen in vier onderdelen n.l. de Sociale Sociale netwerken (Friends & Family), de sociale netwerken die de interesse bevredigen (Mytisch Sociale Netwerken), de sociale netwerken waar je ervaring deelt (Sensory Sociale Netwerken, Community of Practice) en de Kenniscentra (Unity Sociale Networken). Hier zie je de “fractale” kracht van POC.

The map is not the territory.

Het is van groot belang om te beseffen dat het Model, de Waar-heid van POC  thuis hoort in de Unity-Mode, POC in de Praktijk van de Sensory-Mode kan worden toegepast, gewaardeerd wordt in de Social-Mode en In-zicht geeft in de Mythic-mode.

De ver-woording (= Unity) van POC die je nu leest is een voorbeeld van de Unity-Mode en niet van de andere Modes. Die andere Modes zul je zelf vorm moeten geven omdat je nu eenmaal een uniek (Mythic), sociaal (Social), bewegend & waarnemend (Sensory) en naar autonomie (Unity) strevend mens bent.

Je bent geen wandelend verhaal maar een fysiek lichaam, je emoties zijn geen woorden maar die voel je en wat je ziet wordt door je verbeelding in beelden omgezet. De buitenwereld is geen plaatje dat je bekijkt maar iets waar je met je fysieke lichaam aan deelneemt.

Vindt je eigen beperkingen:

Door een MBTI test te doen (druk hier) kun je een indruk krijgen van je eigen perceptie van de  wereldbeelden die je hanteert (Let op: “die perceptie kan niet kloppen!”) en wat veel belangrijker is welke wereldbeelden je niet kunt zien (voelen, meten, vaststellen). Die vormen je zwakke plek of bias. Geloof je niet in een Bias dan maak je in ieder geval niet gebruik van het Mythische wereldbeeld omdat je daar “gelooft” (en het dus niet zeker weet). Zeker weten doe je als je Unity bent.

Als je dit verhaal tot op heden niet practisch vindt en voorbeelden mist dan ben je in ieder geval Sensory. Als je plaatjes mist en diagrammen en snel overzicht wilt hebben ben je Mythic en als je graag wilt weten wie POC ook gebruikt en aanbeveelt ben je Social.


Als je de wereld als een geheel wilt zien, ervaren, waarnemen of waarderen zul je andere mensen moeten zoeken die beschikken over een complementaire kijk  en via de gedeelde wereldbeelden moeten gaan communiceren. Hierbij moet je beseffen dat je die anderen volledig zult moeten vertrouwen in de kijk die jij niet hebt.

De meeste mensen kunnen twee wereldbeelden hanteren en eenvoudig tussen die wereldbeelden switchen zonder dat ze dat merken.

Als je maar 1 wereldbeeld aankunt ben je volgens de Psychiatrie geestelijk in de war. Mensen die uitsluitend het Wereldbeeld Unity gebruiken zijn Paranoide en een unieke orientatie op Mythic wordt Histrionisch genoemd. In dat laatste geval speel je permanent een (toneel)spel en geloof je je eigen leugens omdat je volledig in je eigen fantasie leeft.

De trickster gelooft in zijn eigen leugens

De trickster gelooft in zijn eigen leugens

in POC worden de vier wereldbeelden gecombineerd in 4 1-combinaties, 6 2-combinaties, 4 3-combinaties en 3 vier-combinatie die allemaal ook nog een keer met de klok of tegen de klok kunnen ronddraaien. In dit opzicht kunnen ze allemaal worden gezien als een cyclus.

De zes 2-combinaties kun je Ontwerpen/Inspireren, Analyseren/Toetsen, Evalueren/Alloceren, Brainstormen/Faciliteren, Beinvloeden/Meningvormen en Uitvinden/Ondernemen noemen. Will McWhinney noemt ze Spellen (“Games”). De combinatie Social/Sensory noemt hij bijvoorbeeld het Spel van de Markt.

Het bijzondere van POC is dat de vier wereldbeelden met het bewustzijn in het midden afzonderlijk weer kunnen worden verdeeld in vier delen met behulp van dezelfde wereldbeelden waardoor er in eerste instantie 16 mogelijkheden ontstaan waarvan er 12 verschillend zijn en vier weer gelijk zijn aan de oorspronkelijke wereldbeelden. UnityXUnity is nu eenmaal gelijk aan Unity.


Het toepassen van POC op zichzelf laat zien dat POC fractaal (zelf-referend) is. De zelf-referentie kan eindeloos worden voortgezet met als enige kanttekening dat het bepaald ogenblik erg moeilijk wordt om de combinaties te benoemen.

De expansie naar 2**8=64 is een heel bijzondere en wordt toegepast in de I Tjing. Daarnaast zie je hem ook terug in spellen zoals het Schaakspel.

Na de 64 herhaalt alles zich en die herhaling kun je zien in het getal 8 dat een eigenlijk een Moebius-ring voorstelt. 8 = 2×4 en door het koppelen van twee ronddraaiende 4-hoeken op 1 punt (het Hart) komen we op de bekende 3-1-3 (=7) structuur die zich toont in de Zeven Chakra’s.

De Moebius Ring, het getal 8, toont zich in de Panarchy Theorie.

In wiskundig oogpunt heeft POC erg veel verband met de z.g. Driehoek van Pascal, die heel lang geleden de berg Meru werd genoemd.

De basis-fractal van POC is niet 2×2 maar tweevoudig of beter gezegd drie-voudig omdat het onmogelijk is om een verdeling te maken zonder de deling (“het onderscheid”) zelf ook mee te nemen ook al is het eerste onderscheid  ”geen onderscheid”.


De eerste Trinity  (Yin/Yang + Midden) kan op vele manieren worden weergegeven. Ik prefereer de Uitgaande (“Expansie”, Extravert) en de Ingaande (“compressie”, Introvert) Beweging die in combinatie het “ademen” van de werkelijkheid simuleert. De derde (“het Midden”, Rust, de Leegte) is het moment waarop de uitgaande beweging weer omdraait in een ingaande beweging.

In zijn wezen is ieder mens een unieke fractale expansie van Path of Change dus van het Ademende Heelal.

Er zijn aanwijzingen dat we op dit moment in de 42e expansie zitten. De expansies vinden ook weer volgens een regelmaat plaats. Die regelmaat wordt  beschreven door de Bronze Mean.

De Bronze Mean komt voort uit een abstractie van de Gulden Snede ook wel de reeks van Fibonacci genoemd. Het bijzondere is dat de generator van de Bronze Mean wordt gestuurd door de Trinity.

De berg Meru


Volgens Will McWhinney spelen mensen met elkaar een spel waarbij ze soms gevangen zitten in een gezamenlijke cyclus zoals de “drama triangle“.

Het meest intigrerende spel is een combinatie van Mythic & Social (“Het waarderen van ideeen”) (Brainstorm, Explosie), (“het collectieve onderbewustzijn activeren”) (Scheppen, Implosie)). Dit is het spel dat spellen maakt, het spel dat kinderen spelen, het spel van de magier en de  trickster en het oneindige spel (“the infinite game“) omdat het niet wordt gespeeld om te winnen maar omdat het voortgaan van het spelen op zich het mooiste is wat de mens, als mede-schepper kan doen.

Een hele bekende 2-cyclus is de wetenschapscyclus waarbij theorien (Unity) worden getoetst aan de werkelijkheid (Sensory) en andersom.

Door de enorme waarde (= Social) die er in onze tijd aan wetenschap (=objectief) wordt toegekend lopen we het risico dat de weg terug (het aanpassen van de theorie aan de praktijk) niet plaatsvindt waardoor onze werkelijkheid een theoretische constructie wordt (een norm, regel) waar we ons aan te houden hebben.

Een veranderingspad werkt als er geen sprake is van een conflict van de wereldbeelden. Een conflict ontstaat als twee paden verschillende kanten oplopen en daardoor botsen.

Een voorbeeld is het boven beschreven conflict tussen praktijk en theorie maar het kan nog vervelender worden als de andere wereldbeelden (Social = Emoties en Mythic = Verbeelding) ook worden gedomineerd door het Unity-wereldbeeld.

Hierdoor worden Emoties vervormd tot Emotionele Intelligentie en wordt de Verbeelding gediskwalificeerd als een Illusie of Fantasie.


In de presentatie toon ik een klein aantal van de enorme hoeveelheid toepassingen van POC.


Het eerste deel, de Logica van de Schepping, laat zien dat de vier wereldbeelden eigenlijk bestaan uit een combinatie van slechts twee beelden (Expansie & Compressie (niet-Expansie)) die de lege verzameling gemeen hebben.

Het ene beeld is een negatie (ont-kenning) van het andere beeld. Ze worden in de praktijk aangeduid met bijv. Yang en Yin, mannelijk en niet-mannelijk (=vrouwelijk), orde en niet-orde (chaos), licht en niet-licht (donker), koud en niet-koud (warm) etc etc.

De twee andere wereldbeelden zijn combinaties van de Yin en de Yang via EN (Mythic, Yin EN Yang) en OF (Social, Yin OF Yang).



De vier kijken op de wereld worden in het Boeddhisme het Tetralemma genoemd.

Het Tetralemma wordt gebruikt om de “vreemde” wereld van de Quantum-mechanica te verklaren.

Met behulp van  de twee-heid “Naar buiten gaan” (Extravert, Expansie) en “Naar binnen gaan” (Introvert, Compressie) en de logische operatoren EN en OF kun je eindeloos veel 2-combinaties maken waarvan het aantal een macht van twee is.

Een mooi voorbeeld van een betekenisvolle binaire combinatiereeks is de I Tjing die 64 (2**8) combinatie bevat. Het getal 64 is een heel bijzonder getal omdat na de 64 de cyclus zich weer herhaalt. Het symbool van het getal 8 is niet voor niets de Moebius Ring.

De Cyclus van 2-machten ook wel de Bron genoemd.

In de I Tjing zijn alle combinaties voorzien van een symbool en een kort verhaaltje dat de betekenis van dit symbool uitlegt. Duidelijk zal zijn dat men hierbij gebruik heeft gemaakt van de uitingen van het combinatie-spel in de Natuur en de Mens.

In het Mythic wereldbeeld dat wordt verbonden met Ideeen, Scheppen en het Verbeelden worden twee tegenstellingen (Appel en Niet-Appel, Goed en niet-Goed = Kwaad)  opgeheven door naar een hoger niveau van abstractie (Fruit, Ethiek) te springen. Die fusie geeft erg veel energie net als de combinatie van materie en anti-materie.

Mensen met een Social Wereldbeeld moeten steeds een keuze maken tussen de twee tegenstellingen omdat ze anderen te-vreden willen houden. Ze durven het conflict niet aan dat voortvloeit uit de contraverse tussen de twee tegendelen (bv Orde & Chaos).

Een waarde is een karakteristieke manier waarop mensen een keuze maken. Omdat POC fractaal (zelf-refererend) is bestaan er weer vier verschillende manieren om een keuze te maken, waarbij de vierde manier een combinatie van hetzelfde (hier Social/Social) is.  In dat laatste geval kiezen mensen door weer nieuwe keuzes te gaan formuleren.

Doordat mensen een keuze maken (en hun opties niet open houden) wordt de wereld die ze beschouwen steeds kleiner (gefragmenteerd, gespecialiseerd). Als de twee-deling overlapt (wat meestal het geval is) wordt de wereld ook steeds vager en conflicteuzer omdat mensen een claim gaan leggen op elkaars gebied. Ze zijn niet instaat om het conflict op te lossen omdat alles met alles lijkt samen te hangen. Er zijn geen duidelijke grenzen meer.


Het tweede deel gaat over de vier wereldbeelden in samenhang met het weer-systeem. Hier wordt ook de vier-combinatie-cyclus zichtbaar. Hier zie je dat Social en Mythic nodig zijn om de voortgaande expansie en/of  compressie een halt toe te roepen zodat het geheel niet weer opgaat in de Leegte die het Niet-Al is. Hierdoor ontstaat de Levensgolf (de Svara).

gravity waves

gravity waves


Het derde deel “Het Rad van Avontuur” genoemd, behandelt oude vormen van Path of Change (Indianen (Medicine Wheel), China(Sheng-Cyclus, Lo Shu) en het Buddhisme). Die staan allemaal uitgebreid beschreven in dit document.


Deel vier gaat over Interpersonal Theory, waarna wordt getoond dat de twee gekoppelde vier-combinatie-cycli overeenstemmen met het aloude concept van de chakra en dat het symbool van het Hart-Chakra (twee gekoppelde driehoeken) hetzelfde toont.


Het laatste plaatje in dit onderdeel gaat over de Egyptische Godin (= concept) Ma’at, die ook de bestuurder was van de toegang tot het dodenrijk (het pad van reincarnatie, het rad van avontuur). Het Egyptische concept Ma’at laat zien dat je uit het Rad van Avontuur kunt stappen als je de twee driehoeken van de Geest en het Lichaam in evenwicht brengt in het Hart dat dan zo licht weegt als de Veer van Ma’at. Ma’at werd door de Grieken Sophia (Wijsheid) genoemd.


In deel vijf worden een aantal toepassingen van POC getoond in het bedrijfsleven en het laatste deel gaat over de relatie tussen POC en tijd. Dit laatste deel staat ook hier beschreven.

In het deel over het bedrijfsleven toont POC dat de bedrijfskundigen vergeten zijn om het pad rond te laten lopen in een bedrijf waardoor er geen sprake is van feed-back. De beroemde waardeketen van Porter is eigenlijk een waardencyclus en ketenomkering is niets anders dan consumeren en consumeren is niets anders dan de omgekeerde rotatie van produceren.

Produceren is een vorm van Compressie (Versimpelen van de Praktijk) en Consumeren een vorm van Expansie die elkaar eigenlijk zouden moeten opheffen.

In dit laatste geval zijn we vergeten om de rechtsdraaiende cyclus van de productie te koppelen met de linksdraaiende cyclus van de consumptie waardoor het opheffen van de productie door de consumptie niet meer evenwichtig plaatsvindt. Productie en consumptie zijn een prachtig voorbeeld van boven beschreven complementaire wereldbeelden die samen de Leegte vormen.


Over de wiskunde van het bewustzijn

Over de theorie van de Cycli

Over de vier wereldbeelden

Over Logica

About Number and Magnitude

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.

File:Dirac function approximation.gif

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.

About Ratio’s and Magnitudes

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.

angle between vectors in 2 dimentions

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”).

screw theory 2

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.

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.

About the Bott Periodicity

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

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

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

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

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 the Tetraktys (1)

About the Tetraktys (2)

About Triangular Numbers and Pascal’s Triangle

About the Empty Set

About the Relationship Between Geometry and Music

About the Trinity

About the Game of the Surreal Numbers

About Larson and the Unification of Mathematics

The Collected Works of Dewey B Larson

About Number and Magnitude

About Ratio and Proportion

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

About the Sri Yantra

About Geometric Algebra

Free Software to use Geometric Algebra

About Clifford Algebra

About Yantra’s

About Movement

About Topology

About the Digital Root Patterns

About the Heart Chakra

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

All you want to know about Geometric Patterns

About the Vedic Square

Monday, December 19th, 2011

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

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

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

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

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

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

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

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

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

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

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

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

About the Digital Root

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

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

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

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

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

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

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

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

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

The Tetraktys of Pythagoras

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

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

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

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

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

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

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

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

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

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

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

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

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

Lo Shu Magic Square

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

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

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

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

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

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

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

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

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

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

About the Digital Root of the Golden Mean

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

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

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

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

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

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

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

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

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

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

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

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

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

The pattern of the Golden Mean copied from the Linked Website

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

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

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

The 24-cell also called the Hyperdiamond

About the Vedic Square

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

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

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

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

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

File:Michelino DanteAndHisPoem.jpg

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

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

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

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

Vedic Square

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

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

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

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

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

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

The Rodin Modulus 9 Number Torus

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

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

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

Patterns in the Vedic Square

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

About Indian Vastu Science

The Vastu-Mandala

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

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

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

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

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

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

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

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

The Vastu Jain Symbol is a version of the Tetraktys

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

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

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

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

The Borobudur represents Mountain Meru, Pascal's Triangle.

About the Game of Chess

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


About the Multiplication Table of 9

About Genesis and the Vedic Square

About the Divina Comedia and the Vedic Square

About the Sistine Chapel and the Vedic Square

About the Tetraktys

About the Trinity

About the Vedic Square

About the Tetraktys and the Lo Shu

About the Lo Shu

About the Harmonics of the Universe

About the Hyperdiamond

About the Void

About Harmonics and Entrainment

About Good and Bad Vibrations

About the number 24

About the Jitterbug of Buckminster Fuller

A Simulation of the Jitterbug Pattern (CUBIC WONDER)

About the Game of Chess

About Plato and the Sri Yantra

About the Rodin Torus

About Vastu Science

About Gematria

About Vastu Science and the Borobudur

About Mystical Number Theory and Pascal’s Triangle

Friday, December 2nd, 2011

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

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

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

About Mystical Number-Patterns

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A Hexagram is a Two Dimensional Cube

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

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

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

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

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

Star of David in The Israeli Art Genesis-2

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

About the Triangular Numbers

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

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

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

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

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

The Sum of two Triangular Numbers is a Square

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

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

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

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

28 is a Hexagonal Number

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

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

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

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

Star Numbers are a Combination of Two Triangular Numbers

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

The Symbol of the Heart Chakra contains Two Triangles.

About Pascal’s Triangle

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

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

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

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

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

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

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

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

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

Triangle of Pascal

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

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

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

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

This pattern repeats itself all the time.

File:Yanghui triangle.gif

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

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

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

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

File:Pyramid of 35 spheres animation.gif

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

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

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

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

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

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

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

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

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

the Fibonacci Numbers as a Cube.

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

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

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

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

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

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

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

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

The Triangle of Pascal Modulo 3

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

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

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

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

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

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

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

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

Life is not only about Me and the Other.

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

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


The Content of the Sepher Yesirah

About the Sepher Yesirah

About the Cube of Space

About the Tetractys

About the Cube of Space and Psychology

About the Sepher Yesirah and the I Tjing

A correspondence table of the Cube of Space

About Bereshit

About Genesis

About Patterns in the Bible

About Saturn

About the Trinity

About the Sri Yantra and Plato

About the Lo Shu and the I Tjing

All kind of strange relationships between Triangular Numbers

A website about Mystical Number Theory

About the Figurative Numbers

About Combining the Combinations

About the Golden Spiral and Plato

About the Logic of Creation

About Pascal’s Triangle and the Normal Distribution

A complete course in elementary Number Theory

About the Psychology of the Cube of Space

About the Tetraktys and the Zodiac

About the Process Theory of Paul Young

About the Theory of Dewey B. Larsson

Mysteries of the Equilateral Triangle

About Visual Patterns in Number Theory

About Pascal’s Triangle and Cell Division

About Saturn, the Son of the Sun

Sunday, October 30th, 2011

Early astronomical traditions identify the “Primeval Sun” as the planet Saturn.

Saturn was identified with Osiris in Egypt and Shiva in India.  The Babylonians, the founders of Astronomy, called Saturn the “Light of Heaven, the Sun-God Shamash (or Šamaš).

Tacitus records the Jews as worshipping the planet Saturn, Shabbatai,  as their god. In Plato’s Timaeus, the word for the planet Saturn is Helios, the “Sun” god. Popular Greek traditions identified Saturn as Kronos (Father Time). At that time Saturn ruled “over the Pole“.

The Pole was seen as the Entrance to the Other World. The Shaman using his Light Body had to Climb Jacobs Ladder to get there.

In Sankrit, Suryaputrah,  ”Son of the Sun”  is the name for the planet Saturn. The Medieval Alchemists called Saturn the “Best Sun”.

The Kabbalah divides the Universe  into Ten Spheres, or Sephiroth, which have Planetary and other Correspondences. The Sphere of Saturn is known as Binah (“Understanding“) and the terrestrial sphere, known as Malkuth, is said to have its foundation in Binah.

Long ago Saturn exercised the supreme power on Earth, his reign being remembered as the Golden Age, his time revered, a time which Man longed to return to. At this moment the Cycle is  ruled by Kali. The Mother Goddess Kali represents the Moon in Saturn.

Semitic civilizations referred to the god Saturn as “El”. El was the source of the Great Flood. El was represented by a Black Cube. The Kaaba (“the Cube of Space”) is a cube-shaped building in Mecca, Saudi Arabia, and is the most sacred site in Islam. The building predates Islam, and, according to Islamic tradition, the first building at the site was built by Abraham.

In Ancient History the World was ruled by the Female and  the Mother Goddess. The Vulva or the Yoni (Circle), of the Ancient Mother Goddess was always combined with a Black Stone (with a Point), the Symbol of the Phallus (Lingham) of the Male Creator. The God and the Godddess created the Universe in a short moment of Extreme Extacy.

The Pelgrims have to encircle the Black Stone of El Seven Times just like the Rings of Saturn encircle the Center.

About the Number Seven

The Blog “About the Sum of Things” is about the so called Seal of Saturn and the related 3×3 Magic Square of Saturn also called the Lo Shu in China.

Saturn is the Seventh Planet and Saturday (Sabbath) the Seventh Day of the Week.

In the Blog it is shown that the 3×3 Magic Square generates Two Cyclic Systems. One System, the Wheel of Karma  is a Fractal Torus and is related to the Expansion and Compression of the Universe and the number 2. The Fractal Torus contains Seven Levels associated with the Chakra’s, the Seven Heavens, The Seven Tones and the Seven Colors. The Tower of Bable and the Egyptian Pyramids are a Simulator of the Torus.

The other is a Cyclic Vortex System, the Cycle of Death, related to the number 3 and the Triangel.

The Number Seven is the Repeating Factor of the Torus System. After Seven Steps a new Fractal Level of the Expansion/Compression-Cycle is reached. On the Seventh Day a new Stage in the Creation Process of our Universe is started.

The Torus looks like a Yoni and as you can see the Lingham of Shiva is Inside the Torus. This is the reason why Saturn is associated with Binah and Shakti, the Female Part of Shiva. It also explains why Saturn is related to Fertility.

The Sequence of the Trinity (A-U-M) controls the Creation/Destruction-Cycle of our Universe.  This is the Cycle of Shiva, the Creator and Destructor.

Shiva and Shakti are the Circle (Torus, Yoni) and the Lingham (the Point, Singularity, Lingham). They are the 0 (or the 9) and the 1.

The only thing you need to Create our Universe is de Tao (0), the One, the Two and the Three and they will Generate Everything.

Why is Saturn the Son of the Sun?

There are two explanations. The first one is a Physical Explanation related to Plasma Physics. In the stage of the Golden Age our Solar System contained only Three Planets and Saturn was the Best Sun. Plasma Physics is able to explain many ancient Symbols.

The Second explanation goes back to the Numbers. The Magic Square of Saturn is the First Magic Square that can be constructed. 2×2 is not existent and 1×1 is a Point. The Magic Square of the Sun is the 6×6 Magic Square with Sum 666! and Constant 111. The 3×3 Square can be extracted (is a Son) out of the 6×6 Square.


The Demiurg with the Torus and the Zodiac

About  Kronos, Father Time

The Greek called Saturn Kronos, Father Time.

Most of the Souls are not aware of Cycling Cycle of the Fractal Torus of the Number Two of Duality,  the Wheel of Karma. They stay in the same Place and Their reincarnations are a Merry Go Round.

Some Souls move around the Torus and experience many experiences in the Seven Universus.

All of them are Kept in de Rotating Matrix of the Demiurg.

If you really want to be Free you have move out of Duality into the Cycle of the Divine Trinity (Shiva).

When you want to leave the Rotating Cycle (Torus) of everyday Life and move into a higher dimension (more possibilities, more perspectives), you have to Twist Yourself and Merge with Yourself.

You Twist when you Cross the Cycle of Life and Enter the Cycle of Death of the Trinity. When you join this Cycle you leave the Seven Heavens and move to the Void in the Middle of the Death Cycle.

The Void, the Kingdom,  Cleans every thing and makes every thing New.

You Merge with Yourself when you integrate the Inside and the Outside, the Up and the Down, the Male and the Female.

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

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


About the Sum of Things

About the Cube of Space

About the Myth of Saturn

About the Mother Goddess

About the Dance of Shiva

About Saturn and the Sun

About the Seven Heavens

About the Great Flood

About the Golden Age of Saturn

About the Torus and the Vortex

About Time

About the Twisted Universe

About Alchemy

About the Polar Myth

About the Sum of Things

Thursday, September 22nd, 2011

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


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

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

The Chinese Pollen Path

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

The Seal of Saturn or the Dance of Yu

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

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

The Lo Shu and the Well

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

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

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

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

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

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

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

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

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

Lo Shu Torus

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

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






















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

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

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

About the Sum of Things

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

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

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

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

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

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

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


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

About the I Tjing

The Calculations  behind the Number of Things

The Relationship between the Lo Shu and the I Tjing

About the Lo Shu Torus

About Saturn, the Son of the Sun

About the Trickster

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

The Sistine Chapel (Rome) and the Seal of Saturn

About the Relationship between the Lo Shu and Music

About the History of the Lo Shu

About the Wheel of Eternal Order and the Wheel of Time

About the Relationship between the Lo Shu and Chinese Alchemy

About the Great Flood

About the Orion Myth

About Plato, Pythagoras and the Lo Shu

About the Nothingness

About the Trinity

About the Heart Chakra

About the Sri Yantra

About the Lo Shu Magic Square and the Chinese Domino Game

About the Mirror Universe

About Acupuncture

About Tit for Tat

About the Prisoners Dilemma

About the Bronze Mean

About the Sheng Cycle

Everything You want to Know about Sacred Geometry

About the Influence of the Lo Shu on Recent Art

The Lo Shu Square with Imaginary Numbers

About the Law of KLeiber

Friday, October 29th, 2010

A power law relates one variable to another raised to a constant power. The general form takes y = xa, where y and x are variables, and a is a constant exponent.

A power law exhibits the property of scale invariance. When you multiply the Scale (x) with a factor b the function (y =  baxa) does not change its Shape.

In 1817 Goethe wrote his book ”Zur Morphologie“. This book was the start of a new science called Morphology, the Science of the Shapes.

In his book Goethe describes the so called Uhrplant, the Primal Plant, which is based on the shape of the Leaf. Goethe believed that every Plant was a Leaf within a Leaf within a Leaf.

At the time of Goethe the concept of the fractal was not known. It was developed in 1975 by Benois Mandelbrot (“The Fractal Geometry of Nature”).

A fractal is a self-similar structure. It’s shape repeats itself on every level of expansion.  Some fractals are scale-invariant.

Scale Invariance in the Leaf

The scale invariant fractal structure of the Leaf

About Kleiber’s Law

Metabolism is the process by which your body converts what you eat and drink into energy. During this complex biochemical process, calories in food and beverages are combined with oxygen to release the energy your body needs to function.

The oldest and best known Power Law is Kleiber’s law devised by the Swiss-American zoologist Max Kleiber in 1932.

Kleiber’s Law, MR = W3/4, describes the relationship of the metabolic rate (MR) to the biomass W, raised to an exponent.

Kleiber’s law means that a cat’s metabolic rate is not a hundred or 21.5 times greater than a mouse’s, but about 31.6 — 100 to the three-quarter power.

This relationship seems to hold across the animal kingdom  and it has since been extended all the way down to single-celled organisms, and possibly within the cells themselves to the internal structures called mitochondria, the cellular powerplants, that turn nutrients into energy. Mitochondria have many features in common with bacteria.

The law of Kleiber is  also applicable to super-organisms” like ant colonies, Cities and Eco-Systems.

Because of Economies of Scale, larger and more complex forms of organisms need less energy for each individual cell. They grow and reproduce more slowly and they live longer.

Kleiber's law

Kleiber's law

About Fractals and Kleiber’s Law

Kleiber’s law can be explained from a general model that describes how essentialmaterials are transported through Space-Filling Fractal Networks of Branching Tubes.

Goethe was right. The leaf is a one of the fundamental structures of Nature.

The factor 3/4 (3/3+1)  is a consequence of the fact that a Fractal Structure has to incorporate a not-Fractal Structure, 3-Dimensional Space.

One of the Bilateria: The Octopus

One of our fellow Bilaterians: The Octopus

About Bilateria

About 590 million years ago, the Central Nervous System (CNS), the Brain, appeared.

The organisms with a CNS (including the Humans), the Bilateria, are able to Act and React to a Possible Harmful Stimulus.

The CNS of the Bilateria is  a result of  an Increase in Competition between the Life Forms that came out of the Continuing Fusion of  the Cooperative Life Forms, the Bacteria. The first step in this proces was the Tube of the Sponge.

The fundamental Bilaterian Body Shape is a Tube running from Mouth to Anus, and a second Tube called the Notochord, with an especially large Sphere at the front, called the “Brain“.

The Bilateria have Five Body Spheres (1) the Brain; (2) the Spinal Cord; (3) the Heart and Lungs; (4) the Digestive Organs and Kidneys; (5) the Bladder and Reproductive organs.


About the Tiny Spheres of the Lungs

When we look at the Lungs, one of the Fractal Tube-like branching-structures of the human organism, we can see that the Branches end in Nodes called the alveoli (“little cavities“). The end-nodes of the branching system are tiny Spheres.


The Alveoli use another basic structure of Nature: The Sphere

In the tiny Spheres the Exchange takes place of Carbon Dioxide and Oxygen between the Lungs and the Blood-Vessels. Each human lung contains about 600 million alveoli.

Water diffuses from the alveoli cells into the alveoli so that they are constantly moist.  Oxygen dissolves in this water before diffusing through the cells into the blood.

The Oxygen-rich blood returns to the Heart via the pulmonary veins to be pumped back into circulation. The Carbon Dioxide is pumped out by the Lungs.

In the many million Small Spherical End-Nodes Two Circulatory Systems, the Cardiovascular System (Heart) and Pulmonary System (Lungs), are Connected.

The Big Structures of Nature are able to Scale because the Connection-Points of the Networks are Very Small.

They are reusable on Every Scale that is Bigger than the Scale of the Connection Points.

We will see that all the other Fractal Systems in our Body are based on the Same Principle.



About the Tube of the Digestive System

The Tube contains the Digestive System. It breaks-down larger food molecules into smaller ones that can be absorbed into the blood stream. This happens in the Small Intestine in the vili and microvilli.

The villi (“shaggy hair“)  are tiny, finger-like projections that are approximately 0.5-1mm in length. The microvilli are mechanosensors and have a lot in common with the flagellum (“the roter“) of a bacterium. Microvilli appear in many places in the body. They are also of importance on the cell surface of white blood cells, as they aid in the migration of white blood cells.

The Tube of our Digestive System is highly similar to the Tube of the Sponge, the first fusion of the Bacteria in a  more efficient metabolic structure.

The Output part of the Tube is called the Large Intestine (Colon). Its function is to absorb water from the remaining indigestible food matter, and then to pass Useless Waste Material from the body. The large intestine houses over 700 species of bacteria that perform a variety of functions.


The Liver Fractal

About the Chemical Factory of the Liver Cell.

The Liver is  a Fractal Branching System that detoxifies harmful substances absorbed via the Small Intestines.  It’s basic structure is the Liver Cell.

The Chemical reactions in the liver cells produces a lot of waste heat. This is carried round the body in the blood and warms less active regions.

The Liver regulates the amount of Blood Sugar, Lipids, Amino Acids.  The Liver is the Storage House of  Blood, Iron, Vitamine A, D, B12  and Clycogen, the Source of Energy of the Body.

The Liver produces Bile that is stored in the Gall Bladder. Bile is used to dissolve fat.

White Blood Cell

The White Blood Cell

About the Spleen and the Immune System

Another Fractal Branching System, the Lymphatic System ( “The Immune System“, Spleen),  maintains the health of the body by protecting it from invasions by harmful pathogens, such as bacteria, viruses, fungi, and parasites. These pathogens are the cause of many diseases, so it is necessary to detect and eliminate them rapidly.

The Lymphatic System is connected to the Blood System and produces the White Blood Cells (Lymphocites).  The Lymphocites are the Basic Unit of the Human Defense System. They look like Bacteria.

A lymphocite fits with a pathogen

A lymphocite fits with a pathogen

The surface of a lymphocyte is covered with a large number of identical receptors. Recognition and destruction occurs when the receptors of the lymphocyte fit like a key into the surface of the pathogen.

The Immune System is a Highly Adaptive System. It is able to generate new types of Lymphocites out of a Library of DNA-components.



About the Kidneys and the Bladder

The Kidneys serve the body as a natural filter of the blood, and remove wastes which are diverted to the urinary bladder. The Nephron is the basic structural and functional unit of the kidney. In humans, a normal kidney contains 800,000 to 1.5 million nephrons. Nephrons are wave-like structures. Its chief function is to regulate the concentration of water and soluble substances like sodium salts by filtering the blood.

About the Electro Magnetic System

The last  Fractal Branching System, the Meridians, is a Fast Electro-Magnetic Channel that connects to Organs to the Spine and the CNS.

The Ancient Chinese Scientists believed that this System pumps its Energy, the Chi Force, out of the Earth Magnetic Field.

The Ancient Egyptian Scientists believed the same thing. They named  the Four Cavities the Four Suns of Horus and the Spine, the Djed Pillar (Dj means “Snake”).

The Five Organs of Acupuncture

The Five Systems of Acupuncture

About the number Five

The number Five (4 +1 ) plays an important role in Ancient Chinese Medicine.

The Central Fractal System, the One (1), the Fire System, the Blood System, with its Center the Heart,  contains Four (4) Links to the Other Systems.

The Four Systems are combinations of Two Forces, Expansion and Compression.

The Two Forces came out of One Force (“The Void“).

Two Systems contain the same combination (Expansion x Expansion (Wood, Gall Bladder, Liver, Three Heater), Compression x Compression (Metal, Lungs)).

Two Systems combine Expansion and Compression in a different order (Expansion x Compression, Compression x Expansion). They create a Wave (Water, Bladder, Kidney) or Spiral -like (Earth, Immune System) structures.

The Five Spheres of the Body are incorporated in one Super Protective Sphere, the Coelum, the Skin.

In the Human Embryo the First step of Division of the Cells is between the Coelum,the Multi-Cellar Body, and the Brain, the CNS. In the first step the One was divided into the Two, the Actor and the Monitor.

The CNS  “mirrors” the activities of  the Body and acts as a “predictive simulator”.

The Shape of the CNS is a mirror (Up-Side-Down) of the Shape of the Multic-Cellular Body.

The Bilateria are a Fuse of two Organisms in which one organism changed into the Body and the other changed into the Brain.

These two structures are always competing in the Human Being.

The Brain is Looking Up at the Sky. The Body is Looking Down at Mother Earth, it’s Creator.

The Brain and the Body are United in  the Heart, the Balancer of Body and Soul and the Keeper of the Rhythm (The Pericardium).


Sponge, The Tube

About Bacteria

The Body Shape of the Tube was inherited from the first Multi-Cellular Organism, the Sponge. The Sponge is a  Static Cluster of Amoeba, free moving bacteria, that are propelled by their flagellum (a Roter).


The Basic Building Block of the Body, the Bacterium

Bacteria are small chemical factories that are able to share and combine their production processes by exchanging DNA.

With their flagellum the Bacteria Explore the environment to find the chemical food they need. When they have found food the Circulation of the Flagellum moves into the Opposite Direction.

The bacterium integrates incoming chemical signals during a few-second period of its travels, and adjusts its direction of advance accordingly.

The integration is achieved through temporary chemical modifications to molecules located in the bounding membrane, which transfer nutrients to the cytoplasm, and also through changes to certain other molecules in the interior of the cell. Such integration is essentially a short-term memory mechanism.

Bacteria exchange their DNA

Bacteria exchange their DNA

At a certain moment the Bacteria got together because the Sponge is a much more Efficient Metabolic Structure than a group of Free Moving Bacteria.

Strangely enough the DNA of the Sponge already contains the complete Bleuprint of the Humans.

Scientists now believe that most species on earth are a result of a loss of DNA from the original Bleuprint that was created when the Bacteria finally fused into one Organism.

About the Nano-Level

Recently scientists have found Electric Fields as strong as 15 million volts per meter in the Nano-Parts of the Cells. These fields are as strong as those produced in lightning bolts.

It’s not clear what causes these strong fields or what they might mean but they could account for a until now unknown (by Science) or well-known (by the Old Scientists, Chi) Source of  Bodily Energy.

The nano-parts of the Cell could be the very very very small universal building blocks of Nature.

What has Happened?

In the beginning the Chemical Soup generated Self-Reproducing Chemical Factories.

These Factories combined into bigger Factories by exchanging and combining their factory-designs.

Every time when the Factories fused they became more Efficient, Bigger, Stronger, Older and therefore more Competitive.

When the Factories became more competitive they had to Protect themselves against the other Factories.

To Protect themselves the Multi-Cellular Systems started to Sense in Many Directions.

In the first step a Nerve-Net was created.

In the second step one part of the Organism turned Upside-Down and became a Specialized Predictive System (the CNS).

This part fused with the Bodily Part creating one Organism that contained Two Organisms.

The Bilateria, the Organisms with Two Complementary Parts, Body and Mind, were born.

Some of the Bilateria called Humans fused in Social Structures and the Social Structures fused and fused and became more efficient, bigger, stronger, older and therefore more competitive.

In due time they will Rule the Earth and will start to create bigger and bigger systems until they will rule the complete Universe.

To make this possible the Humans need every other part of Nature to Sustain their Growth. This will certainly result in a huge Collapse of their own Eco-System.

The main reason this is happening is that the Humans forgot to Copy the Designs of Mother Nature.

The Brain (Thinking and Sensing, Left Brain), Looking Up at the Sky,  believed it could do a much Better Job than it’s counterpart the Multi-Cellular Body (Emotions and Imagination, Right Brain), Looking Down at the Earth, the Source of All Creation.

Kleiber’s Law shows that we are now using 122 times more Energy per Person than we really would need if we would Scale in the right way.

What can we learn from Kleiber’s Law?

All organisms including the humans depend for their maintenance and reproduction on the close integration of numerous subunits.

These components need to beserviced in a relatively `democratic’ and efficient fashionto supply energy, remove waste  and regulate activity.

Natural Selection solved this problem byevolving hierarchical fractal-like branching networks, whichdistribute Energy, Information and Materials between Big Reservoirs (“Lungs”) and Small Connection Points (“Alveoli”) of other Circulatory Systems (“Blood”).

This Fractal Bottum-Up approach is highly effective and efficient.

The not-fractal, Top-Down,  machinery designed by the Humans is Scaled with a factor 1 so we can improve a lot by copying Mother Nature.


About Goethe and Morphology

About the Heart Chakra or Why the Heart connects the Brain and the Body

How to use the Heart to find Balance

About the Heart and Ethics

Why Humans look a lot like Bacteria

About Kleibers Law and the Rain Forest

About the Five Worldviews

A movie about Fractal Structures in Nature

About the Fusion of Water in the Cell

About the Number Five in Chinese Medicine

Why Humans are part of Super Organisms

How the Humans evolved out of the Bacteria

About the Left and the Right Brain

About the Void

How the Heart synchronizes the Body

About the Fast Transmisson Channel in the Body

About Kleiber’s Law

A general model to explain Kleiber’s Law

About Kleiber’s Law and the Growth of Cities

About the Immune System

About Bacteria

About the Law of Kleiber and Ant Colonies

The terrestrial evolution of metabolism and life

A movie about the origin of vertebrates