Teoría de la cuarta dimensión: el hipercubo

Martes, 16 de marzo de 2010

Hipercubo o teseracto

Si para los físicos la cuarta dimensión está representada por el tiempo, en el mundo de las matemáticas esas dos palabras tienen una connotación muy distinta. La geometría euclidiana prevé una o más dimensiones por encima de nuestro mundo en 3D en forma de teoría matemática. Esto es, aparte de las tres dimensiones conocidas (altura, anchura y profundidad), se supone como mínimo otra dimensión más, la cuarta en discordia, que cumpliría perfectamente las propiedades cartesianas, es decir, sería perpendicular a las otras tres y partiría de un origen de coordenadas común.

Nosotros no podemos siquiera imaginar cómo sería el mundo en 4D, porque estamos recluidos en un universo de sólo tres dimensiones. Si partimos de un sistema de coordenadas en el espacio 3D, tenemos un eje X que representa la anchura, un eje Y que es la altura y un eje Z que representa la profundidad. La cuarta dimensión vendría simbolizada por un hipotético eje W que, como decíamos antes, sería perpendicular a X, Y y Z. Algo que con nuestra mente es imposible comprenderlo porque pensamos y vemos en tres dimensiones. Pero ello no quiere decir que no exista una cuarta dimensión, sino simplemente que no podemos verla.

Imagina por un momento que no eres una persona, eres un cuadrado habitante de Planilandia, la genial novela de Edwin A. Abbott. Los cuadrados viven en el plano y sólo tienen dos dimensiones (ancho y largo), no conocen nada más. Un día, aparece volando por el espacio 3D (que tú no comprendes) un cubo y te saluda. Como él está por encima de ti y no concibes el concepto de altura, prácticamente creerás que te estás volviendo majareta.

Para hacerse visible, el cubo decide posarse sobre tu plano 2D. En ese momento tú verás surgir de repente un cuadrado a tu lado y te darás un susto inmenso, pues no sabes por dónde ni de dónde ha aparecido. Después de hablar unos minutos con él, le preguntas a ver por qué demonios se llama a sí mismo cubo, cuando lo que ves es un simple cuadrado como tú. Él te intentará explicar que lo que estás visualizando es una proyección de su figura 3D sobre tu mundo en dos dimensiones, e intentará proyectarse de manera inclinada, atravesando tu suelo, para ver si te haces una idea de su forma real. La imagen que tú verás en ese momento será la siguiente.

Cubo proyectado en un plano

No comprendes nada. Lo que antes era un cuadrado ahora se ha transformado en dos cuadrados que se cruzan y que tienen líneas inclinadas que unen sus vértices. No sabes mirar más allá del plano.

Vuelve al mundo real; eres una persona otra vez. De la misma forma que el cuadrado nunca podría imaginarse cómo es un cubo porque desconoce lo que es la tercera dimensión, nosotros no podemos comprender los elementos que componen la cuarta. Sabemos perfectamente que la proyección de un cubo sobre un plano genera una figura como la anterior, con unas líneas más largas y otras más cortas y ángulos diferentes. Sin embargo, podemos afirmar con total seguridad que en un cubo todas sus aristas son de igual longitud y todos sus ángulos son rectos (90º) porque vivimos en un mundo tridimensional y podemos palpar un cubo.

De la misma manera, geométricamente se puede demostrar la existencia de los llamados hipercubos (o teseractos) en cuatro dimensiones. Desde que vi, hace ya años, la segunda parte de la película Cube (Cube 2: Hypercube), los hipercubos se han metido en mi cabeza y ya no los puedo sacar de ahí. Es, probablemente, la figura más hermosa desde el punto de vista matemático y más quimérica desde el de la vida real. Por cierto, la primera de Cube es una de mis obras maestras cinematográficas de culto; no así tanto la segunda y la tercera.

La imagen que encabeza este post es la de un hipercubo. Bueno, técnicamente es la proyección de un hipercubo sobre nuestro espacio tridimensional. Aunque si queremos rizar más el rizo, como lo estás viendo en una imagen plana en la pantalla de tu ordenador, diríamos que es la proyección plana 2D de la proyección espacial 3D de un hipercubo en 4D. Vaya lío, madre mía.

Si cogemos un cuadrado y lo proyectamos en el espacio, dándole altura, conseguimos un cubo. Si proyectamos un cubo hacia el espacio de cuatro dimensiones conseguimos un hipercubo. El teseracto se considera un cubo desfasado en el tiempo, es decir, cada instante de tiempo por el cual se movió pero todos ellos juntos. Por supuesto no podemos ver un teseracto en la cuarta dimensión, porque vivimos solo en tres con la limitación mental de nuestros cerebros.

Al igual que le pasaba al cuadrado de Planilandia, vemos unas líneas más largas que otras y ángulos diversos en su proyección, sin embargo el hibercubo tiene todas sus aristas iguales y todos sus ángulos rectos. Además lo que hay entre el cubo interior y el cubo exterior son más cubos, todos ellos tan rectos y cúbicos como los tridimensionales. El teseracto tiene 16 vértices, 32 aristas, 24 caras y 8 células (éstas serían el equivalente a los cubos tridimensionales).

Lo que sí podemos hacer es desdoblar el teseracto tetradimensional para trasladarlo a nuestro espacio tridimensional. De la misma manera que somos capaces de cortar las aristas de un cubo y desdoblarlo en un plano bidimensional obteniendo una suerte de cruz latina formada por seis cuadrados, si hacemos lo mismo con un hipercubo el resultado es una cruz 3D de ocho cubos (células).

Cubo e hipercubo desdoblados

Lo que ya no podemos imaginar es ese engendro doblado sobre sí mismo, por la incapacidad ya comentada de comprender un espacio en cuatro dimensiones. Si tienes la mente muy abierta y necesitas un buen dolor de cabeza, la siguiente figura muestra el orden en el que deben coincidir las caras de los distintos cubos para conseguir el teseracto. Por supuesto, teniendo en cuenta que todas las aristas deben medir lo mismo y que todos los ángulos deben ser rectos. Algo imposible para nuestras cabezas de chorlito tridimensional.

Orden para doblar un hipercubo

De la misma forma en que existen figuras geométricas en un supuesto mundo de cuatro dimensiones, ¿quién nos puede a nosotros asegurar que no existan otro tipo de manifestaciones “inteligentes” en él? Llámalo ente, alma, energía, mente o como quieras. Es posible que en un espacio tetradimensional haya entidades que sólo algunos de entre nosotros podemos ver bajo determinadas circunstancias y cuando se proyectan sobre nuestro espacio 3D. Es posible que los llamados fantasmas no sean más que seres 4D que están siempre ahí pero que no podemos ver más que cuando invaden nuestro espacio y que nos pegan terribles sustos como al pobre cuadrado cuando el cubo aparece de sopetón en su plano.

En fin, ese tema se lo dejamos a Iker Jiménez y a su parienta y nosotros nos quedamos con la versión matemática y geométrica del asunto que es mucho más factible.

Hasta el propio Dalí, que estaba un poco zumbado del bolo (sólo hay que ver sus cuadros), dibujó un hipercubo en una de sus obras, Corpus hipercubicus.

Corpus hipercubicus (Dalí)

Para terminar, os dejo un vídeo en el que el conocido astrónomo y divulgador científico Carl Sagan explica todo lo anterior de una forma más gráfica que un servidor. Vale la pena echarle un vistazo e invertir 7 minutos de nuestro tiempo para intentar comprender un poco mejor lo que la cuarta dimensión representa podría representar.

 

http://www.teknoplof.com/2010/03/16/teoria-de-la-cuarta-dimension-el-hipercubo/

Airports

• September 29th, 2011
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This same Anubis statue was also ritually ferried past the City of London…

AFP photo

…and the Statue of Liberty in New York harbor.

AFP photo

These scenes remind me of Charon ferrying souls of the newly deceased into hell.

Luca Giordano (1686)

The disturbing imagery at DIA doesn’t end there unfortunately. Leo Tanguma was commissioned to paint some murals at DIA. Here are two parts of “Peace and Harmony with Nature”:

The letter in the lower right corner is an actual letter written by a Hama Herchenberg, 14 years old, that died December 18, 1943 in the Auschwitz Concentration camp.

Here is the masonic corner stone marker that claims to have been laid by the New World Airport Commission, which doesn’t actually exist.

There is much more about this symbolism here, here, and here.

As part of the master plan update, the airport announced selection of Parson’s transportation group to design a new hotel, rail station and two bridges leading into the main terminal. Santiago Calatrava has been selected as the architect for the project. -Source

Calatrava designed Sundial Bridge and the PATH Transportation Hall.

World Trade Center

• September 3rd, 2011
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Calatrava’s Transportation Hub replaces a temporary PATH station that had to be built after the underground system was destroyed on 9/11. The temporary station put two uptown services into operation: Newark–33rd Street (red) and Hoboken–33rd Street (blue).

The underground part of the new PATH hall will look like this:

The RISING special says, “PATH Hall sits directly below Memorial Plaza and consists of a spine with 25 vertebrae connecting 275 extending ribs.”

Fellow researcher Mark Gray points out that Swans have 25 cervical vertebrae. And 275 / 25 = 11

In the name of security, Santiago Calatrava’s bird has grown a beak. Its ribs have doubled in number and its wings have lost their interstices of glass…. The main transit hall, between Church and Greenwich Streets, will almost certainly lose some of its delicate quality, while gaining structural expressiveness. It may now evoke a slender stegosaurus more than it does a bird. -Source

Most researchers now support the view that birds are a group of theropod dinosaurs. The Transportation Hub looks to me like a monumental reptilian eye. Check out the Reptilian Agenda for another deep dark rabbit hole.

The Wedge of Light

Daniel Liebeskind won the competition to be the master plan architect for the reconstruction of the World Trade Center site. Liebeskind’s buildings include the Jewish Museum in Berlin, the Danish Jewish Museum in Copenhagen, the Contemporary Jewish Museum in San Francisco, the Wohl Centre in Israel, and the Imperial War Museum North in the UK.

Liebeskind’s concept for WTC2, called the “wedge of light,” states that no shadows shall be cast on the memorial park on September 11th. Lord Foster designed the top of the building like a wedge with four rhombi at the top.

Speaking of the Pleiades, shadows, and Lords:

Seek him that maketh the seven stars and Orion, and turneth the shadow of death into the morning, and maketh the day dark with night: that calleth for the waters of the sea, and poureth them out upon the face of the earth: The LORD is his name. -Amos 5:8

Rendering copyright Silverstein Properties

The Transportation Hall was also supposedly designed so at the exact time the second tower collapsed, every Sept. 11th at 10:28am, the sun enters the transportation hub illuminating the building with light.

However Eli Attia, an architect who has been a frequent critic of the rebuilding process, published a study showing that 40 percent to 99 percent of the Wedge of Light would be in shadow cast by the Millenium Hilton Hotel during the times when Mr. Libeskind had said it would be free of shadows.

Mr. Libeskind won the enthusiastic support of both Gov. George E. Pataki and Mayor Michael R. Bloomberg with his design, and Mr. Pataki ensured Mr. Libeskind’s selection for the ground zero commission when he overruled a committee of directors of the development corporation, which had recommended the selection of the other finalist, a group of architects known as Think. -Source

This situation reminds me of how I.M. Pei was selected for the Louvre Pyramid design without a competition and then how Mitterand stepped in and selected Johann Otto von Spreckelsen to be the architect of the La Grande Arche even though he wasn’t winning the national design competition for the megaproject. Sometimes competitions are a necessary sham when secret architecture has long been in the pipeline.

“Richard Meier, the eminent architect who participated in the ground zero design competition — and who therefore lost to Mr. Libeskind — disagreed [with the fact that shadows fall on the wedge of light]. ”How could you not take it literally?” he asked. ”He didn’t say that the sun would be reflected off buildings and that illumination would somehow create a wedge of light.’

Terence Riley, the chief curator for the department of architecture and design at the Museum of Modern Art, took a historical view. ”Given the long history of solar orientation in architectural design, I don’t suppose it is unreasonable for people to have assumed that the Wedge of Light would be illuminated by direct sunlight on Sept. 11,” he said. -Source

Light is obviously very important to this mega-ritual in the making. Mark Gray discovered one more timely piece of the puzzle:

On 9-11-11 at sunset, the famous “Tribute in Light” will return to the skies above New York City for the night. The tribute is comprised of “88” 7,000-watt xenon light bulbs, arranged in 48-foot squares inspired by the shape and orientation of the Twin Towers. The illuminated memorial reaches four miles into the sky and is visible from as far as 30 miles away. The two arrays cast the strongest shaft of light ever projected from Earth into the night sky. -Source

Image courtesy D L under the Creative Commons Attribution 2.0 Generic license

The Tribute in Light ought to get the light bearer’s attention, don’t you think?

Grant Chamberlin pointed out in a comment on this post that the 88 light bulbs in the Tribute in Light might be significant. This is another dark rabbit hole, but if we are to be whole and healthy we must have the courage to look into our shadows.

The 8s stand for the eighth letter of the alphabet (H), with ‘HH’ standing for “Heil Hitler”. -Source

The phrase “Heil Hitler” was inspired by a statement, 88 words in length, from Volume 1, Chapter 8 of Adolf Hitler’s Mein Kampf:

What we must fight for is to safeguard the existence and reproduction of our race and our people, the sustenance of our children and the purity of our blood, the freedom and independence of the fatherland, so that our people may mature for the fulfillment of the mission allotted it by the creator of the universe. Every thought and every idea, every doctrine and all knowledge, must serve this purpose. And everything must be examined from this point of view and used or rejected according to its utility.

Eighty-eight is used as code among Neo-Nazis to identify each other. -Source

Another such illuminist monument was called the Cathedral of Light:

Image courtesy Bundesarchiv, Bild 183-1982-1130-502 / CC-BY-SA

The cathedral of light was a main aesthetic feature of the Nuremberg Rallies. It was the brainchild of Albert Speer, who was commissioned by Adolf Hitler to design and organise the Nuremberg Parade Grounds for the annual celebrations. It is still considered amongst Speer’s most important works.

The effect was a brilliant one, both from within the design and on the outside. -Source

The primary aspect of the Nuremberg Rallies was to strengthen the personality cult of Adolf Hitler, portraying Hitler as Germany’s saviour, chosen by providence. The gathered masses listened to the Führer’s speeches, swore loyalty and marched before him. Representing the Volksgemeinschaft as a whole, the rallies served to demonstrate the might of the German people.

The visitors of the rallies by their own free will were subordinate to the discipline and order in which they should be reborn as a new people. -Source

The Cathedral of Christ the Light also resonates with this post:

Image courtesy Skier Dude under the Creative Commons Attribution-Share Alike 3.0 Unported license.

The worship space in Christ the Light is a vesica pisces shape, the shape formed by the intersection of two circles [symbol of the female generative organs]. The walls are composed of overlapping panels of wood and glass rising skyward to form the vault, much like the scales of a fish.

The Catholic Cathedral Corporation of the East Bay instituted a design competition for Christ the Light. Various designs were judged and the corporation announced Santiago Calatrava (architect of the PATH Transportation Hall and Sundial Bridge) as the winner. Calatrava’s design for Christ the Light was chosen before a site was appropriated for the project.

However, by the time a site was chosen, Calatrava’s design fell out of favor and instead replaced by a design of competition runner-up Craig W. Hartman, FAIA of Skidmore, Owings and Merrill (SOM). -Source

SOM designed WTC1 (the Freedom Tower) and the new WTC7.

Consider these provocative coincidences:

• 20 April 1292 Jaques de Molay became the Grand Master of the Knights Templar
• 20 April 1889 Adolph Hitler was born.
• 20 April 1999 Colombine massacre happened.
• 20 April 2008 Pope Benedict XVI visited Ground Zero at the destroyed WTC.
• 20 April 2010 Deepwater Horizon explosion and oil spill occurred

Joseph Ratzinger joined the Hitler Youth in 1941 when, according to him and his supporters, it became compulsory for all German boys. Millions of Germans were in a position similar to that of Joseph Ratzinger and his family, so why spend so much time focusing on him? Because he is no longer merely Joseph Ratzinger, or even a Catholic Cardinal — he is now Pope Benedict XVI. None of the other Germans who joined the Hitler Youth, were part of the military in Nazi Germany, lived near a concentration camp, and watched Jews being rounded up for death camps has ever become pope. -Source

Pope Benedict prayed for “eternal light and peace” at ground zero on a black square pool on the 111th day of the year (2008 was a leap year).

On September 11th there are 111 days remaining until the end of the year. The original Twin Towers had 111 floors if you count the roof. See my post on 110-111.

Mark Gray posted this on Facebook:

See my post on the Freedom Tower for more on the black cube and its encoding in the 9/11 Memorial.

Image from Star Trek The Next Generation

Resistance is anything but futile. Remember the Borg and the overt Nazis were defeated.

William Henry’s research goes into some fascinating areas I haven’t touched on and I’m excited to be interviewed by him tomorrow. I’ll post the link on my Interviews page when it’s online. Here is William’s fascinating take on the mysteries of 9/11/11.

On the flipside Mark Gray has made some great discoveries relating to the positive side of 88.

Mark Gray and William Henry wrote the book Freedom’s Gate together about the lost symbols in the US Capitol. I highly recommend it.

Image courtesy Mark Gray

The distance from the center of the Alexander Hamilton U.S. Customs House (built over the location of the original Fort Amsterdam) to the center of the Freedom Tower is exactly 1000 meters.

The distance from the corner of Reflecting Absence (former WTC North Tower) to the center of the Customs House is 888 meters.

The elliptical glass (see Ellipses) in the rotunda has 88 rays.

Image courtesy Mark Gray

Mark makes the following brilliant comparison, calling the Customs House Manhattan Gate:

Let’s also not forget the Wedge of Light itself (WTC2) has 88 floors and WTC1, WTC4, and WTC7 have numerical resonances with the keyboard.

Are we being played?

In addition there are 88 constellations in the sky:

Image courtesy Mark Gray

Eighty-eight (88) symbolizes fortune and good luck since the word 8 sounds similar to the word Fā (发, which implies 发财, or wealth, in Mandarin). The number 8 is considered to be the luckiest number of all in Chinese culture and prices in Chinese supermarkets can often be found containing many 8′s The Chinese government has even been auctioning auto license plates containing many 8s for tens of thousands of dollars. The 2008 Beijing Olympics opened on 8/8/08 at 8 p.m.. -Source

Thank you Mark for the reminder that in duality every symbol has an opposite. Every coin has two sides. The wise look at both of them.

This just in…

Jan Thulstrup uncovered an amazing connection in Psalm 119 which is the longest chapter in the Bible.

119 is the mirror reflection of 911.

Psalm 119 has 176 verses. 176 = 88 + 88. 176 is an octagonal number.

Psalm 119 is divided into twenty-two stanzas, one stanza for each letter of the Hebrew alphabet. Each stanza begins with the corresponding letter of the Hebrew alphabet.

Each stanza has 8 verses, composed as octaves (from the Latin octo meaning “eight”) in music.

Here a Jew reads Psalm 119 at the Wailing Wall in Jerusalem:

Image courtesy Brian Jeffery Beggerly under the Creative Commons Attribution 2.0 Generic license.

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