CRUZ DE CRISTO / STAUROS / TAURO / PLEYADES / CLAVE GRIALICA - Secreto Masonico

Hypersphere[edit]

Stereographic projection of a Clifford torus: the set of points (cos(a), sin(a), cos(b), sin(b)), which is a subset of the 3-sphere.

Main article: Hypersphere

The set of points in Euclidean 4-space having the same distance R from a fixed point P₀ forms a hypersurface known as a 3-sphere. The hyper-volume of the enclosed space is:

mathbf V = egin{matrix} frac{1}{2} end{matrix} pi^2 R^4

This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity where R is substituted by function R(t) with t meaning the cosmological age of the universe. Growing or shrinking R with time means expanding or collapsing universe, depending on the mass density inside.^[12]

Cognition[edit]

Research using virtual reality finds that humans in spite of living in a three-dimensional world can without special practice make spatial judgments based on the length of, and angle between, line segments embedded in four-dimensional space.^[13] The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience in 4D virtual environments."^[13] In another study,^[14] the ability of humans to orient themselves in 2D, 3D and 4D mazes has been tested. Each maze consisted of four path segments of random length and connected with orthogonal random bends, but without branches or loops (i.e. actually labyrinths). The graphical interface was based on John McIntosh's free 4D Maze game.^[15] The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point. The researchers found that some of the participants were able to mentally integrate their path after some practice in 4D (the lower-dimensional cases were for comparison and for the participants to learn the method).

Dimensional analogy[edit]

A net of a tesseract

To understand the nature of four-dimensional space, a device called dimensional analogy is commonly employed. Dimensional analogy is the study of how (n − 1) dimensions relate to n dimensions, and then inferring how n dimensions would relate to (n + 1) dimensions.^[16]

Dimensional analogy was used by Edwin Abbott Abbott in the book Flatland, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a three-dimensional being has seemingly god-like powers, such as ability to remove objects from a safe without breaking it open (by moving them across the third dimension), to see everything that from the two-dimensional perspective is enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension.

By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from our three-dimensional perspective. Rudy Rucker illustrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.

Cross-sections[edit]

As a three-dimensional object passes through a two-dimensional plane, a two-dimensional being would only see a cross-section of the three-dimensional object. For example, if a spherical balloon passed through a sheet of paper, a being on the paper would see first a single point, then a circle gradually growing larger, then smaller again until it shrank to a point and then disappeared. Similarly, if a four-dimensional object passed through three dimensions, we would see a three-dimensional cross-section of the four-dimensional object—for example, a hypersphere would appear first as a point, then as a growing sphere, with the sphere then shrinking to a single point and then disappearing.^[17] This means of visualizing aspects of the fourth dimension was used in the novel Flatland and also in several works of Charles Howard Hinton.^[18]

Projections[edit]

A useful application of dimensional analogy in visualizing the fourth dimension is in projection. A projection is a way for representing an n-dimensional object in n − 1 dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places and things are represented in two dimensions by projecting the objects onto a flat surface. When this is done, depth is removed and replaced with indirect information. The retina of the eye is also a two-dimensional array of receptors but the brain is able to perceive the nature of three-dimensional objects by inference from indirect information (such as shading, foreshortening, binocular vision, etc.). Artists often use perspective to give an illusion of three-dimensional depth to two-dimensional pictures.

Similarly, objects in the fourth dimension can be mathematically projected to the familiar 3 dimensions, where they can be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.

The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection from four dimensions produces similar foreshortening effects. By applying dimensional analogy, one may infer four-dimensional "depth" from these effects.

As an illustration of this principle, the following sequence of images compares various views of the 3-dimensional cube with analogous projections of the 4-dimensional tesseract into three-dimensional space.

Cube	Tesseract	Description
		The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the cell-first perspective projection, shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube. Note that the other 5 faces of the cube are not seen here. They are obscured by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell.
		The image on the left shows the same cube viewed edge-on. The analogous viewpoint of a tesseract is the face-first perspective projection, shown on the right. Just as the edge-first projection of the cube consists of two trapezoids, the face-first projection of the tesseract consists of two frustums. The nearest edge of the cube in this viewpoint is the one lying between the red and green faces. Likewise, the nearest face of the tesseract is the one lying between the red and green cells.
		On the left is the cube viewed corner-first. This is analogous to the edge-first perspective projection of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3 deltoids surrounding a vertex, the tesseract's edge-first projection consists of 3 hexahedral volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, so the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet.
		A different analogy may be drawn between the edge-first projection of the tesseract and the edge-first projection of the cube. The cube's edge-first projection has two trapezoids surrounding an edge, while the tesseract has three hexahedral volumes surrounding an edge.
		On the left is the cube viewed corner-first. The vertex-first perspective projection of the tesseract is shown on the right. The cube's vertex-first projection has three tetragons surrounding a vertex, but the tesseract's vertex-first projection has four hexahedral volumes surrounding a vertex. Just as the nearest corner of the cube is the one lying at the center of the image, so the nearest vertex of the tesseract lies not on boundary of the projected volume, but at its center inside, where all four cells meet. Note that only three faces of the cube's 6 faces can be seen here, because the other 3 lie behind these three faces, on the opposite side of the cube. Similarly, only 4 of the tesseract's 8 cells can be seen here; the remaining 4 lie behind these 4 in the fourth direction, on the far side of the tesseract.

Shadows[edit]

A concept closely related to projection is the casting of shadows.

If a light is shone on a three dimensional object, a two-dimensional shadow is cast. By dimensional analogy, light shone on a two-dimensional object in a two-dimensional world would cast a one-dimensional shadow, and light on a one-dimensional object in a one-dimensional world would cast a zero-dimensional shadow, that is, a point of non-light. Going the other way, one may infer that light shone on a four-dimensional object in a four-dimensional world would cast a three-dimensional shadow.

If the wireframe of a cube is lit from above, the resulting shadow is a square within a square with the corresponding corners connected. Similarly, if the wireframe of a tesseract were lit from “above” (in the fourth dimension), its shadow would be that of a three-dimensional cube within another three-dimensional cube. (Note that, technically, the visual representation shown here is actually a two-dimensional shadow of the three-dimensional shadow of the four-dimensional wireframe figure.)

Bounding volumes[edit]

Dimensional analogy also helps in inferring basic properties of objects in higher dimensions. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces. By applying dimensional analogy, one may infer that a four-dimensional cube, known as a tesseract, is bounded by three-dimensional volumes. And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project to volumes in the image, not merely two-dimensional surfaces.

Visual scope[edit]

Being three-dimensional, we are only able to see the world with our eyes in two dimensions. A four-dimensional being would be able to see the world in three dimensions. For example, it would be able to see all six sides of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as we can see the interior of a square on a piece of paper. It would be able to see all points in 3-dimensional space simultaneously, including the inner structure of solid objects and things obscured from our three-dimensional viewpoint. Our brains receive images in the second dimension and use reasoning to help us "picture" three-dimensional objects. Just as a four-dimensional creature would probably receive multiple three-dimensional pictures.^{[citation needed]}

Limitations[edit]

Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the circumference of a circle $C = 2pi r$ and the surface area of a sphere: $A = 4pi r^2$ . One might be tempted to suppose that the surface volume of a hypersphere is $V=6pi r^3$ , or perhaps $V=8pi r^3$ , but either of these would be wrong. The correct formula is $V = 2pi^2 r^3$ .^[10]

Índice

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Características destacables[editar]

Constelación Tauro AlltheSky.com

Aldebarán (α Tau), laestrella más brillante de laconstelación, es unagigante roja de primera magnitud; forma unsistema binario con una tenue enana roja. Elnath(β Tau), segunda estrella más brillante, forma los cuernos del toro junto a ζ Tauri; esta última es una de las estrellas Be más estudiadas. Dos estrellas variables en la constelación, T Tauri y RV Tauri, son los prototipos de sendos grupos de estrellas que llevan sus nombres.

Tauro también contiene dos de los cúmulos abiertos más conocidos del firmamento: lasPléyades, donde cabe señalar a Merope (23 Tau), y las Híades, cuya estrella principal es γ Tauri.

La Nebulosa del Cangrejo (M1) se localiza en esta constelación. Es un resto de supernova proveniente de la explosión de una supernova en 1054 (SN 1054). La nebulosa fue observada por vez primera en 1731 por John Bevis.

Estrellas principales[editar]

Las Pleyades.

α Tauri (Aldebarán), estrella gigante rojo-anaranjada de magnitud0,87. Tiene una tenue compañera de magnitud 13,5 a 28 segundos de arco.
β Tauri (Elnath), gigante blanco-azulada de magnitud 1,68.
γ Tauri (Hyadum I), gigante naranja miembro del cúmulo de las Híadesde magnitud 3,65.
δ Tauri, en realidad tres estrellas, todas miembros de las Híades. δ¹Tauri (Hyadum II) es una estrella similar a Hyadum I.
ε Tauri (Ain), gigante amarillo-naranja miembro de las Híades con magnitud 3,53 y en donde se ha descubierto un planeta extrasolar.
ζ Tauri, estrella binaria de magnitud 2,97 situada en el cuerno sur del toro.
η Tauri (Alcíone), la estrella más brillante del cúmulo de las Pléyades y la tercera de la constelación con magnitud 2,85, es una joven estrella blanco-azulada.
θ Tauri, denominación de Bayer compartida por dos estrellas distintas, θ¹ Tauri y θ² Tauri; de color naranja y blanco, respectivamente, se hallan separadas 5,6 minutos de arco y forman parte de las Híades.
κ Tauri, estrella doble visual cuyas componentes, de color blanco, están separadas 5,6 minutos de arco.
λ Tauri, sistema estelar triple y variable eclipsante.
ν Tauri, estrella blanca de la secuencia principal de magnitud 3,90.
ξ Tauri, sistema estelar cuádruple de magnitud 3,73.
ο Tauri y π Tauri, gigantes amarillas de magnitud 3,62 y 4,69 respectivamente.
ρ Tauri, variable Delta Scuti de magnitud 4,66.
ω² Tauri, estrella Am de magnitud 4,93.
n Tauri (109 Tauri), gigante amarilla de magnitud 4,96.
Merope (23 Tauri), estrella de las Pléyades que ilumina la nebulosa de reflexión NGC 1435.
10 Tauri, estrella amarilla situada a 45 años luz con un posible disco circunestelar a su alrededor.
30 Tauri (e Tauri), joven estrella binaria de magnitud 5,06 con una edad estimada inferior a 28 millones de años.
39 Tauri, enana amarilla algo más fría que el Sol de magnitud 5,90.
79 Tauri, estrella de las Híades rodeada por un disco de polvo circunestelar.
90 Tauri (c Tauri), estrella blanca de magnitud 4,27 que forma parte de las Híades.
111 Tauri, enana amarilla a 48 años luz del Sistema Solar.
139 Tauri, gigante luminosa azul de magnitud 4,82.
T Tauri, estrella variable prototipo de una clase de estrellas en proceso de formación y que aún no han entrado en lasecuencia principal.
Y Tauri, estrella de carbono y variable semirregular cuyo brillo varía entre magnitud 6,5 y 9,2 en un período de 241,5 días.
RV Tauri, prototipo de estrellas variables intermedias entre las cefeidas y las variables Mira.
RZ Tauri, binaria eclipsante de contacto en donde se produce transferencia de masa entre las dos estrellas.

Imagen de DG Tauri obtenida con elobservatorio de rayos X Chandra.

UX Tauri, sistema triple formado por tres estrellas T Tauri; alrededor de la más brillante se ha descubierto un disco protoplanetario.
AA Tauri, estrella T Tauri que posee un disco protoplanetario en el que se han detectado moléculas orgánicas.
CD Tauri, binaria eclipsante de magnitud 6,77.
CE Tauri (119 Tauri), supergigante roja, una de las estrellas más grandes que se conocen.
DG Tauri, estrella T Tauri en donde se han observado dos chorros de rayos X.
EQ Tauri, binaria de contacto de magnitud 11,20.
IK Tauri (NML Tauri), una de las estrellas de la rama asintótica gigante más estudiadas; es una variable Mira cuyo brillo oscila entre magnitud 10,8 y 16,5.
HD 37124, estrella amarilla con tres planetas conocidos.
Gliese 176 (HD 285968), enana roja con un planeta extrasolar de tipo Súper-Tierra.
CoKu Tauri/4, joven estrella T Tauri de 1 millón de años de edad.
Gliese 169, enana naranja distante 37,5 años luz.
HL Tau 76 (V411 Tauri), enana blanca pulsante, variable del tipo ZZ Ceti.

Objetos profundos notables en el cielo[editar]

M1 (Nebulosa del Cangrejo).

M45, un cúmulo abierto de estrellas situado al norte de la constelación y conocido como las Pléyades. Dentro de él se encuentra la Nebulosa de Merope (NGC 1435).

Las Híades, un cúmulo abierto de estrellas que contribuye a formar la "V" en el cielo que forma el inicio de la cabeza del toro. Se encuentra a 161 años luz de distancia. Consta de 80 estrellas situadas en una esfera de 12 pársecs de diámetro.

M1 o Nebulosa del Cangrejo, formada por los restos de una supernova cuya explosión, observada en la Tierra el 4 de julio de 1054, fue lo bastante brillante para verse de día. Es mencionada en crónicas chinas y en la poesía amerindia. En su interior se encuentra un púlsar —estrella de neutrones muy densa que gira a gran velocidad— denominado Pulsar del Cangrejo. Tiene aproximadamente 25 km de diámetro y rota a razón de 30 veces por segundo.

NGC 1514, nebulosa planetaria a unos 600 años luz.

Mitología[editar]

Constelación de Tauro "El Toro" .

En la mitología griega, Tauro es la forma de toro que el dios Zeus adoptó para seducir a Europa, una mítica princesa fenicia. Tuvieron tres hijos: Minos, el legendario rey deCreta, Sarpedón y Radamantis.

Otra versión nos dice que fue la bestia que envió la diosa Hera para acabar con Orión.

Según otra versión, Tauro es Ío convertida en vaca por su amante Zeus para evitar que Hera se enterase de sus amores.

Enlaces externos[editar]

Wikimedia Commons alberga contenido multimedia sobre Tauro.

Eratóstenes: Catasterismos (Καταστερισμοί).
- 14: Toro (ταῦρος; Taurus): el toro en el que se transformó Zeus para raptar aEuropa, o bien Ío transformada en vaca.
- Texto latino en Google Books; facsímil electrónico.
- Texto griego en Internet Archive; facsímil electrónico.