Het magische vierkant
Een magisch vierkant is een rooster waarin getallen zo zijn geordend dat de rijen en kolommen elk het zelfde getal opleveren als ze worden opgeteld. Dit getal wordt de constante genoemd. De afmetingen van een magische vierkant noemen we de orde, welke aan gegeven wordt met N. Met de orde kun je de constante berekenen met behulp van de volgende formule: 1/2(N3+N)
Hieronder zie je het magisch vierkant van de orde 3. De constante is 15.
Een magisch vierkant wordt standaard genoemd als zij de getallen 1 t/m N2 bevat. Er bestaan ook magische vierkanten waarin speciale getallen zijn verwerkt zoals bijvoorbeeld de getallen van je geboortedatum.
Bij een panmagisch vierkant is ook de som van de diagonalen gelijk aan de constante en bij een volkomen perfect magisch vierkant is de constante tevens terug te vinden in elke 2x2 deelvierkant.
De geschiedenis van het magische vierkant gaat terug tot voor onze jaartelling. Volgens de Chinese overlevering had de mythische keizer Yu een visioen waarin hij aan de oever van de rivier Lo een schilpad zag. Op het schild ontwaarde hij een rooster van 3 bij 3 waarin de getallen 1 t/m 9 waren verwerkt in de vorm van punten. Tot zijn verbazing ontdekte hij dat de som van de rijen, kolommen en diagonalen dezelfde uitkomst hadden. Het werd geïnterpreteerd als een bovennatuurlijk teken van orde en balans in het universum en zou daarom magische krachten bezitten. Dit meest elementaire magische vierkant werd de Lo shu genoemd, wat “kaart van de rivier” zou betekenen. Vanaf die tijd speelde de Lo shu een belangrijke rol in het Chinese denken. Zij diende als inspiratie bron voor het bouwen van tempels met negen ruimtes. De posities van de getallen werden verwerkt in de Daoist dans en houdingen in de tai chi. Daarnaast werd ze gebruikt in toekomstvoorspellingen, talismannen en Feng Shui. Voor de Chinezen staan de even getallen staan voor Yin, de aarde, het vrouwelijke aspect, en de oneven getallen voor yang, de hemel, het mannelijke aspect.
Pas na de 13de eeuw maakten de Chinezen magische vierkanten van een grotere orde. Dit heeft waarschijnlijk te maken met het feit dat de Lo Shu gezien werd als een magisch object en niets iets waarop gestudeerd kon worden uit nieuwsgierigheid.
Het eerste magische vierkant van de orde 4 duikt op in India omstreeks de eerste eeuw na onze jaartelling en werd gemaakt door de wiskundige Nagarajuna. In Jaina geschriften die uit de 12de eeuw dateren is de eerste perfect magische vierkant gevonden. Jaina is een religie en filosofie. Het is niet bekend welke rol het magische vierkant in deze religie speelde.
De eerste vierkanten van de orde 5 en 6 zijn gevonden in Arabische geschriften van rond de eerste eeuwwisseling. Waarschijnlijk hadden zij de kennis opgedaan van de Indiërs daar zij delen van India een tijd hebben beheerst. De wiskundige Al- Buni werkte intensief met magische vierkanten en geloofde in de bovennatuurlijke eigenschappen die zij zouden bezitten. Bronnen vermelden dat magische vierkanten werden gebruikt om astrologische berekeningen te maken en voorspellingen te doen.
Via de Arabieren bereikt de kennis over de magische vierkanten Europa. De Griek Manual Moschopoulos schrijft in 1300 echter alleen over de wiskundige eigenschappen.
Pas anderhalve eeuw later worden de magische eigenschappen gedocumenteerd door de Italiaan Marsilio Ficino die medeoprichter was van een magische school. De meest gedetailleerde beschrijvingen over de magische eigenschappen van de vierkanten en hoe ze te construeren kunnen we nu nog lezen in het werk van Agrippa.
Hij legt hierin verbanden tussen de magische vierkanten en de planeetenergieën.
In 1514 maakte Albrecht Durer zijn beroemde houtsnede “Melancholia” waar een magisch vierkant in de orde van 4 een onderdeel is. Volgens Agrippa heeft dit vierkant correspondenties met jupiter. Er is een figuur te zien welke we als archetype van de denker terug zien in vele andere kunstwerken. Het figuur heeft een depressieve uitdrukking want het kan de oplossing niet vinden. Het magische vierkant van Jupiter brengt de energie van die planeet en zal helpen de depressie te verdrijven.
De constructie en correspondenties van de magische vierkanten.
In de theorie van de sympathetische magie draait alles om het ordenen van objecten, kleuren, geluiden etc. die dezelfde energie dragen. Door deze bij elkaar te brengen kun je die energie versterken en richten op een bepaald doel. In de middeleeuwen zochten magiërs en occultisten naar correspondenties die pasten bij de (toen) bekende 7 planeten: de Zon, Maan, Mercurius, Venus, Mars, Jupiter en Saturnus.
De magische vierkanten werden als rekenkundige archetypes van de planeten beschouwd. De orde en de constante bepaalde onder welke invloed een bepaald magisch vierkant stond.
Wie de grondlegger is geweest van de toewijzing van getallen aan de planeten is onduidelijk. Veel bronnen verwijzen naar Agrippa, maar ook de Arabieren en de Babyloniers worden genoemd. De laatst genoemden geloofden dat de aarde het middelpunt van de kosmos was waar de 7 planeten omheen draaiden en wel in de volgende orde: Maan, Mercurius, Venus, Zon, Mars, Jupiter, Saturnus. Zij gaven de planeet die het verst van de aarde verwijderd was het kleinste magische vierkant. Saturnus kreeg de orde 3 toegewezen, Jupiter orde 4, enz. Opvallend in dit systeem is het feit dat de Zon aan de orde 6 is gekoppeld. Hiermee loopt de theorie in de pas met een heel oud numerologisch systeem uit Soemerië. In deze cultuur werd de zon vereerd en zij hadden een zesvoudig telsysteem waarin werd vast gelegd dat de dag 24 uur duurde en de zon opkwam om 6 uur en onderging om 18 uur. Daarnaast legde zij voor het eerst vast dat een cirkel uit 360 graden bestaat. Pythagoras noemde in zijn getallentheorie de 6 het eerste perfecte nummer. Met deze feiten is het te begrijpen dat het magische vierkant in de orde van 6 met een constante van 666 een diepe indruk maakte op de grondleggers van de westerse magie.
Om je magische intentie kracht te geven kunnen magische vierkanten worden ingezet als talisman. Agrippa beschrijft hoe een liefdes talisman kan worden gemaakt door het magische vierkant van Venus (orde 7) te graveren op een plaatje koper.
Daarnaast kan het magische vierkant gebruikt worden om magische tekens (sigils) te maken.
Hiervoor kies je een trefwoord wat je magische intentie beschrijft. Het woord zet je om in cijfers zoals dat in de numerologie wordt gedaan. Daarna kies je het magische vierkant dat de gewenste energie belichaamt. Hierover leg je een overtrekpapiertje en trek je een lijn tussen de cijfers van je trefwoord.
Als voorbeeld: Mijn naam Willie bestaat uit de cijfers 593395. Stel ik wil een sigil van mijn geboorteplaneet, de maan. Hiervoor kies ik het magische vierkant in de orde van 9.
Op het overtrekpapiertje trek ik dus een lijn van 5 naar 9 naar 3, nog een keer naar 3, weer naar 5 en weer terug naar 9.
Er bestaan verschillende algoritmes en geometrische formules om zelf magische vierkanten te maken. Ik bespreek alleen de formules die ik zelf onder de knie heb.
Voor vierkanten van de oneven orde:
Teken het raamwerk voor de gewenste orde. Plaats de 1 in het midden van de bovenste rij. Plaats de volgende cijfers diagonaal naar rechts behalve:
*Als de bovenste rij bereikt is. Het volgende nummer wordt dan geplaatst op de onderste rij alsof deze boven de bovenste rij staat. (tip: zie het vierkant als een cilinder waarbij de boven en onderkant elkaar raken)
*Als de meest rechtse rij bereikt is plaats je het volgende cijfer in de meest linkse kolom alsof deze naast de rechtse staat. (zie het vierkant nu als een cilinder waar de zijkanten elkaar raken)
*Als je een vakje bereikt wat al bezet is ga je verder onder het laatste getal.
Voor vierkanten in de orde van vier en meervouden hiervan:
Teken het raamwerk voor de orde van 4. Plaats hierin de cijfers 1 t/m 16 van links naar rechts en van onder naar boven. Maak een tweede raamwerk waarin je de cijfers plaats maar dan van rechts naar links startend van onder naar boven. Trek de diagonalen on het eerste vierkant. Alle getallen die door het diagonaal worden geraakt verwissel je met de getallen die in het tweede vierkant staan. Denk eraan dat als je met meervouden van vier werkt je moet werken met subdiagonalen. In een magisch vierkant van 8 zitten 2 diagonalen en 8 subdiagonalen.
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