Posts Tagged ‘Pascal’

Over de Paradoxen van het Leven (in Dutch)

Tuesday, August 14th, 2012

INLEIDING

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.

THEORIE

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

VERKLARING MIDDELS DE LOGICA

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).

VERANDERPADEN

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 EN DE PERSOONLIJKHEID

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).

VERGELIJKBARE (META-)MODELLEN

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.

OMDAT MENSEN VERSCHILLEN HEBBEN ZE ANDEREN NODIG OM HET GEHEEL TE KENNEN.

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.

PATH OF CHANGE KAN OP ZICHZELF WORDEN TOEGEPAST

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 ESSENTIE VAN PATH OF CHANGE IS MEER-DIMENSIONAAL ADEMEN.

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

HET LEVEN IS EEN SPEL

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.

TOEPASSING

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

LOGICA VAN DE SCHEPPING

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).

Tetralemma

Tetralemma

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 WEERSYSTEEM

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

OUDE MODELLEN

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.

PSYCHOLOGIE

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.

EGYPTISCHE GODEN

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.

BEDRIJVIGHEID

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.

LINKS

Over de wiskunde van het bewustzijn

Over de theorie van de Cycli

Over de vier wereldbeelden

Over Logica

About Perspective

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

Medieval Perspective

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

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

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

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

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

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

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

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

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

The Future is Expanding Space with Infinite Potential.

Time moves behind Space and Scales Space.

About Perspective Drawing

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

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

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

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

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

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

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

Perspective Machine designed by Albrecht Durer

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

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

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

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

IMAGE: Leonardo's perspectograph

The Perspectograph of Leonardo Da Vinci

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

About Projective Geometry

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

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

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

Two projections of the same object.

Different Points of View

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

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

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

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

Pappus' Theorem

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

Desargues' theorem

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

Pascals Theorem

About the Four Points of the Cross-Ratio

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

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

The Cross-Ratio

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

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

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

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

ratios.jpg

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In Eucludian Geometry A Ruler contains a Fixed Scale

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

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

alchemy

The Expectation looks through a Small Hole.

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

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

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

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

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

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

About Space measured by Time and Space measured by Space

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

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

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

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

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

At that Time Space was measured by Time.

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

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

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

From that Time on Space was measured by Space.

Why the Future contains Infinite Possibilities

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

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

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

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

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

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

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

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

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

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

Moving Movement is the Essence of our Reality.

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

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

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

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

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

golden mean spiral2

The Spiraling Spiral of Time

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

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

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

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

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

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

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

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

Let’s Move.

LINKS

About Pythagoras and Heliopolis (Egypt)

About Pythagoras

Paul Dirac: About Physics and Projective Geometry

About the Middle Ages

About Expanding Space and Not-Euclidian Geometry

About the Renaissance

About Resistance and Mass

About the Mathematics of Perspective

About the Ether

About Clean Space

About Projective Geometry

Pictures of the Projective Plane

About Geometric Algebra

About As Within, So Without

About the Reciprocal Theory of Dewey B. Larson

A Video about Mobius Transformations

About Projective Geometry and our Senses

About Projective Geometry and Geometric Algebra

About the Cube of Space

About Time and Paranormal Experiences

About Geometry

A Textbook about Metric Geometry

About the Four Points of View

About the Void

Why the Future is Open Space

About Alchemy and the Klein Bottle

About the Human Sensory System

Stuart Kaufmann: About the Adjacent Possible

About 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”.

blades

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.

add-bivectors

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.

SriYantra

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.

LarsonsScalarCube.jpg

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.

wom_image3.jpg

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.

LINKS

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.

LINKS

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.

LINKS

The Content of the Sepher Yesirah

About the Sepher Yesirah

About the Cube of Space

About the Tetractys

About the Cube of Space and Psychology

About the Sepher Yesirah and the I Tjing

A correspondence table of the Cube of Space

About Bereshit

About Genesis

About Patterns in the Bible

About Saturn

About the Trinity

About the Sri Yantra and Plato

About the Lo Shu and the I Tjing

All kind of strange relationships between Triangular Numbers

A website about Mystical Number Theory

About the Figurative Numbers

About Combining the Combinations

About the Golden Spiral and Plato

About the Logic of Creation

About Pascal’s Triangle and the Normal Distribution

A complete course in elementary Number Theory

About the Psychology of the Cube of Space

About the Tetraktys and the Zodiac

About the Process Theory of Paul Young

About the Theory of Dewey B. Larsson

Mysteries of the Equilateral Triangle

About Visual Patterns in Number Theory

About Pascal’s Triangle and Cell Division