Octacube (sculpture)

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Inviato: 27/06/2015 02:33

Octacube (sculpture)

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This article is about the steel sculpture. For other uses of the word, see Octacube.

The Octacube and its designer, Adrian Ocneanu

The Octacube is a large, steel sculpture of a mathematical object: the 24-cell or "octacube". Because a real 24-cell is four-dimensional, the artwork is actually a projection into the three-dimensional world. Octacube has very high intrinsic symmetry, which matches features in chemistry (molecular symmetry) and physics (quantum field theory).

The sculpture was designed by Adrian Ocneanu, a mathematics professor at Pennsylvania State University. The university's machine shop spent over a year completing the intricate metal-work. Octacube was funded by an alumna in memory of her husband, Kermit Anderson, who died in the September 11 attacks. The sculpture is displayed in the lobby of Penn State's math department.

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Artwork[edit]

Octacube metal skeleton is about 6 feet (2 meters) in all three dimensions. It is a complex arrangement of unpainted, tri-cornered flanges. The base is a 3-foot (1 meter) high granite block, with some engraving.^[1]

The artwork was designed by Adrian Ocneanu, a Penn State mathematics professor. He supplied the specifications for the sculpture's 96 triangular pieces of stainless steel and for their assembly. Fabrication was done by Penn State's machine shop, led by Jerry Anderson. The work took over a year, involving bending and welding as well as cutting. Discussing the construction, Ocneanu said:^[1]

It's very hard to make 12 steel sheets meet perfectly—and conformally—at each of the 23 vertices, with no trace of welding left. The people who built it are really world-class experts and perfectionists—artists in steel.

Because of the reflective metal at different angles, the appearance is pleasantly strange. In some cases, the mirror-like surfaces create an illusion of transparency by showing reflections from unexpected sides of the structure. The sculpture's mathematician creator commented:^[1]

When I saw the actual sculpture, I had quite a shock. I never imagined the play of light on the surfaces. There are subtle optical effects that you can feel but can't quite put your finger on.

Interpretation[edit]

Regular shapes[edit]

The Platonic solids are three-dimensional shapes with special, high, symmetry. They are the next step up in dimension from the two-dimensional regular polygons (squares, equilateral triangles, etc.). The five Platonic solids are the tetrahedron (4 faces), cube (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedron (20 faces). They have been known since the time of the Ancient Greeks and valued for their aesthetic appeal and philosophical, even mystical, import. (See also the Timaeus, a dialogue of Plato.)

The Platonic solids

Tetrahedron	Cube	Octahedron	Dodecahedron	Icosahedron

In higher dimensions, the counterparts of the Platonic solids are the regular polytopes. These shapes were first described in the mid-19th century by a Swiss mathematician, Ludwig Schläfli. In four dimensions, there are six of them: the pentachoron (5-cell), tesseract (8-cell), hexadecachoron (16-cell), octacube (24-cell), hecatonicosachoron (120-cell), and the hexacosichoron (600-cell).

The 24-cell consists of 24 octahedrons, joined in 4-dimensional space. The 24-cell's vertex figure (the 3-D shape formed when a 4-D corner is cut off) is a cube. Despite its suggestive name, the octacube is the not the 4-D analog of either the octahedron or the cube. In fact, it is the only one of the six 4-D regular polytopes that lacks a corresponding Platonic solid.^{[note 1]}

Attempts to picture the 24-cell

Schlegel diagram	4-dimensional rotation

Projections[edit]

Stereographic projection of the Earth

Ocneanu explains the conceptual challenge in working in the fourth dimension:^[1] "Although mathematicians can work with a fourth dimension abstractly by adding a fourth coordinate to the three that we use to describe a point in space, a fourth spatial dimension is difficult to visualize."

Although it is impossible to see or make 4-dimensional objects, it is possible to map them into lower dimensions to get some impressions of them. An analogy for converting the 4-D 24-cell into its 3-D sculpture is cartographic projection, where the surface of the 3-D Earth (or a globe) is reduced to a flat 2-D plane (a portable map). This is done either with light 'casting a shadow' from the globe onto the map or with some mathematical transformation. Many different types of map projection exist: the familiar rectangular Mercator (used for navigation), the circular gnomonic (first projection invented), and several others. All of them have limitations in that they show some features in a distorted manner—'you can't flatten an orange peel without damaging it'—but they are useful visual aids and convenient references.

Stereographic projection of a 24-cell

In the same manner that the exterior of the Earth is a 2-D skin (bent into the third dimension), the exterior of a 4-dimensionsal shape is a 3-D space (but folded through hyperspace, the fourth dimension). However, just as the surface of Earth's globe can not be mapped onto a plane without some distortions, neither can the exterior 3-D shape of the 24-cell 4-D hyper-shape. In the image on the right a 24-cell is shown projected into space as a 3-D object (and then the image is a 2-D rendering of it, with perspective to aid the eye). Some of the distortions:

Curving edge lines: these are straight in four dimensions, but the projection into a lower dimension makes them appear to curve (similar effects occur when mapping the Earth).
It is necessary to use semi-transparent faces because of the complexity of the object, so the many "boxes" (octahedral cells) are seen.
Only 23 cells are clearly seen. The 24th cell is the "outside in", the whole exterior space around the object as seen in three dimensions.

To map the 24-cell, Ocneanu uses a related projection which he calls windowed radial stereographic projection. As with the stereographic projection, there are curved lines shown in 3-D space. Instead of using semitransparent surfaces, "windows" are cut into the faces of the cells so that interior cells can be seen. Also, only 23 vertices are physically present. The 24th vertice "occurs at infinity" because of the projection; what one sees is the 8 legs and arms of the sculpture diverging outwards from the center of the 3-D sculpture.^[1]

Symmetry[edit]

Octahedral symmetry diagram showing mirror planes as great circles (6 red, 3 blue). Rotation axes are also shown: 2-fold (pink diamonds), 3-fold (red triangles), and four-fold (blue squares).

The Octacube sculpture has very high symmetry. The stainless steel structure has the same amount of symmetry as a cube or an octahedron. The artwork can be visualized as related to a cube: the arms and legs of the structure extend to the corners. Imagining an octahedron is more difficult; it involves thinking of the faces of the visualized cube forming the corners of an octahedron. The cube and octahedron have the same amount and type of symmetry: octahedral symmetry, called O_h (order 48) in mathematical notation. Some, but not all, of the symmetry elements are

3 different four-fold rotation axes (one through each pair of opposing faces of the visualized cube): up/down, in/out and left/right as seen in the photograph
4 different three-fold rotation axes (one through each pair of opposing corners of the cube [along each of the opposing arm/leg pairs])
6 different two-fold rotation axes (one through the midpoint of each opposing edge of the visualized cube)
9 mirror planes that bisect the visualized cube
- 3 that cut it top/bottom, left/right and front/back. These mirrors represent its reflective dihedral subsymmetry D_2h, order 8 (a subordinate symmetry of any object with octahedral symmetry)
- 6 that go along the diagonals of opposing faces of the visualized cube (these go along double sets of arm-leg pairs). These mirrors represent its reflective tetrahedral subsymmetry T_d, order 24 (a subordinate symmetry of any object with octahedral symmetry).

Science allusions[edit]

Many molecules have the same symmetry as the Octacube sculpture. The organic molecule, cubane (C₈H₈) is one example. The arms and legs of the sculpture are similar to the outward projecting hydrogen atoms. Sulfur hexafluoride (or any molecule with exact octahedral molecular geometry) also shares the same symmetry although the resemblance is not as similar.

Molecules with the same symmetry

Cubane	Sulfur hexafluoride

The Octacube also shows parallels to concepts in theoretical physics. Creator Ocneanu researches mathematical aspects of quantum field theory (QFT). The subject has been described by a Fields medal winner, Ed Witten, as the most difficult area in physics.^[2] Part of Ocneanu's work is to build theoretical, and even physical, models of the symmetry features in QFT. Ocneanu cites the relationship of the inner and outer halves of the structure as analogous to the relationship of spin 1/2 particles (e.g. electrons) and spin 1 particles (e.g. photons).^[1]

Memorial[edit]

Octacube was commissioned and funded by Jill Anderson, a 1965 PSU math grad, in memory of her husband, Kermit, another 1965 math grad, who was killed in the 9-11 terrorist attacks.^[1] Summarizing the memorial, Anderson said:^[1]

I hope that the sculpture will encourage students, faculty, administrators, alumnae, and friends to ponder and appreciate the wonderful world of mathematics. I also hope that all who view the sculpture will begin to grasp the sobering fact that everyone is vulnerable to something terrible happening to them and that we all must learn to live one day at a time, making the very best of what has been given to us. It would be great if everyone who views the Octacube walks away with the feeling that being kind to others is a good way to live.

Anderson also funded a math scholarship in Kermit's name, at the same time the sculpture project went forward.^[1]

Reception[edit]

An article on the sculpture appeared in the March 2006 issue of Playboy.^[3] A more complete explanation of the sculpture, including how it came to be made, how its construction was funded and its role in mathematics and physics, has been made available by Penn State.^[1] In addition, Ocneanu has provided his own commentary.^[4]

Four-dimensional space

3D projection of a tesseract undergoing a simple rotation in four dimensional space.

In mathematics, four-dimensional space ("4D") is a geometric space with four dimensions. It typically is more specifically four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space. It has been studied by mathematicians and philosophers for over two centuries, both for its own interest and for the insights it offered into mathematics and related fields.

Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular a vector with four elements (a 4-tuple) can be used to represent a position in four-dimensional space. The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.

In modern physics, space and time are unified in a four-dimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Spacetime is not a Euclidean space.

History[edit]

See also: n-dimensional space § History

Lagrange wrote in his Mécanique analytique (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a four-dimensional space — three of dimensions of space, and one of time.^[1] In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,^[2] and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work was not published until after his death.^[3] Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 Habilitationsschrift, Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he considered a "point" to be any sequence of coordinates (x₁, ..., x_n). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.

An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R.

One of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension?; published in the Dublin University magazine.^[4] He coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension.^[5]^[6] In 1886 Victor Schlegel described^[7] his method of visualizing four-dimensional objects with Schlegel diagrams.

In 1908, Hermann Minkowski presented a paper^[8] consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.^[9] But the geometry of spacetime, being non-Euclidean, is profoundly different from that popularised by Hinton. The study of Minkowski space required new mathematics quite different from that of four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973 H. S. M. Coxeter felt compelled to write:

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H. G. Wells in The Time Machine, has led such authors as John William Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.

—H. S. M. Coxeter, Regular Polytopes^[10]

Vectors[edit]

Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example, a general point might have position vector a, equal to

mathbf{a} = egin{pmatrix} a_1 a_2 a_3 a_4 end{pmatrix}.

This can be written in terms of the four standard basis vectors (e₁, e₂, e₃, e₄), given by

mathbf{e}_1 = egin{pmatrix} 1 0 0 0 end{pmatrix}; mathbf{e}_2 = egin{pmatrix} 0 1 0 0 end{pmatrix}; mathbf{e}_3 = egin{pmatrix} 0 0 1 0 end{pmatrix}; mathbf{e}_4 = egin{pmatrix} 0 0 0 1 end{pmatrix},

so the general vector a is

mathbf{a} = a_1mathbf{e}_1 + a_2mathbf{e}_2 + a_3mathbf{e}_3 + a_4mathbf{e}_4.

Vectors add, subtract and scale as in three dimensions.

The dot product of Euclidean three-dimensional space generalizes to four dimensions as

mathbf{a} cdot mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.

It can be used to calculate the norm or length of a vector,

left| mathbf{a} ight| = sqrt{mathbf{a} cdot mathbf{a} } = sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2},

and calculate or define the angle between two vectors as

heta = arccos{frac{mathbf{a} cdot mathbf{b}}{left|mathbf{a} ight| left|mathbf{b} ight|}}.

Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate pairing different from the dot product:

mathbf{a} cdot mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4.

As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing $b_4$ actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.

The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:

egin{align} mathbf{a} wedge mathbf{b} = (a_1b_2 - a_2b_1)mathbf{e}_{12} + (a_1b_3 - a_3b_1)mathbf{e}_{13} + (a_1b_4 - a_4b_1)mathbf{e}_{14} + (a_2b_3 - a_3b_2)mathbf{e}_{23} + (a_2b_4 - a_4b_2)mathbf{e}_{24} + (a_3b_4 - a_4b_3)mathbf{e}_{34}. end{align}

This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e₁₂, e₁₃, e₁₄, e₂₃, e₂₄, e₃₄). They can be used to generate rotations in four dimensions.

Orthogonality and vocabulary[edit]

In the familiar 3-dimensional space in which we live there are three coordinate axes — usually labeled x, y, and z — with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude. Lengths measured along these axes can be called height, width, and depth.

Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively. A position along the w axis can be called spissitude, as coined by Henry More.

Geometry[edit]

The geometry of 4-dimensional space is much more complex than that of 3-dimensional space, due to the extra degree of freedom.

Just as in 3 dimensions there are polyhedra made of two dimensional polygons, in 4 dimensions there are 4-polytopes made of polyhedra. In 3 dimensions there are 5 regular polyhedra known as the Platonic solids. In 4 dimensions there are 6 convex regular 4-polytopes, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform 4-polytopes, analogous to the 13 semi-regular Archimedean solids in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.

Regular polytopes in four dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A₄, [3,3,3]	B₄, [4,3,3]		F₄, [3,4,3]	H₄, [5,3,3]
5-cell	tesseract	16-cell	24-cell	120-cell	600-cell

In 3 dimensions, a circle may be extruded to form a cylinder. In 4 dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps", known as a spherinder), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder). The Cartesian product of two circles may be taken to obtain a duocylinder. All three can "roll" in 4-dimensional space, each with its own properties.

In 3 dimensions, curves can form knots but surfaces cannot (unless they are self-intersecting). In 4 dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction, but 2-dimensional surfaces can form non-trivial, non-self-intersecting knots in 4-dimensional space.^[11] Because these surfaces are 2-dimensional, they can form much more complex knots than strings in 3-dimensional space can. The Klein bottle is an example of such a knotted surface .^{[citation needed]} Another such surface is the real projective plane.^{[citation needed]}

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Inviato: 27/06/2015 02:49

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]

martes, 20 de marzo de 2012

La última cena de los científicos

De izquierda a derecha: Galileo Galilei, Marie Curie, J. Robert Oppenheimer, Isaac Newton, Louis Pasteur, Stephen Hawking, Albert Einstein, Carl Sagan, Thomas Edison, Aristoteles, Neil deGrasse Tyson, Richard Dawkins y Charles Darwin.

Do you notice the symbolic 8 rays within the circle? Symbolic of the star Sirius.

LA MISMA PIRAMIDE DEL VATICANO, AL ESTAR DISEÑADA EN FUNCION A SIRIO, Y A SU NEXO CON EL ESTE, OESTE, NORTE Y SUR, OSEA LAS CUATRO ORIENTACIONES, ES UNA REFERENCIA A LA "MAQUINA DEL TIEMPO". POR ESA CAUSA ESTA ADENTRO DE LA CERRADURA Y TIENE RELACION CON LA LLAVE DE DAVID "EXPERIMENTO FILADELFIA".

2. Isaías 22:22: Y pondré la LLAVE de la casa de David sobre su hombro; y abrirá, y nadie cerrará; cerrará, y nadie abrirá.

3. Mateo 16:19: Y a ti te daré las LLAVEs del reino de los cielos; y todo lo que atares en la tierra será atado en los cielos; y todo lo que desatares en la tierra será desatado en los cielos.

4. Lucas 11:52: ¡Ay de vosotros, intérpretes de la ley! porque habéis quitado la LLAVE de la ciencia; vosotros mismos no entrasteis, y a los que entraban se lo impedisteis.

5. Apocalipsis 1:18: y el que vivo, y estuve muerto; mas he aquí que vivo por los siglos de los siglos, amén. Y tengo las LLAVEs de la muerte y del Hades.

6. Apocalipsis 3:7: Escribe al ángel de la iglesia en Filadelfia: Esto dice el Santo, el Verdadero, el que tiene la LLAVE de David, el que abre y ninguno cierra, y cierra y ninguno abre:

BARILOCHENSE6999

Sent: 24/09/2015 11:19

Edredón señalado ocho de la estrella reloj redondo grande

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Punto Deco de la estrella del azul y del oro Reloj Cuadrado

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Reloj de pared de la estrella de 8 puntos

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Azul de lujo de Brown Reloj Redondo Grande

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fleur de lis= escorpion= octavo signo zodiacal Azul de lujo de Brown Reloj Redondo Grande

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Símbolo del Pentagram - estrella Cinco-Acentuada Reloj Redondo Grande

Message 73 of 75 on the subject

Sent: 24/09/2015 11:27

cubo=19= 235 ciclos lunares= saturno

19 años=235 ciclos lunares

las matematicas son exactas para el todopoderoso

Hexágonos y estrellas Seis-acentuadas Reloj Redondo Mediano

Message 74 of 75 on the subject

Sent: 24/09/2015 11:28

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Arte abstracto de la estrella de oro reloj redondo grande

Messaggio 8 di 14 di questo argomento

http://forum.andrewgough.co.uk/viewtopic.php?f=1&t=3596&start=1600

Inviato: 03/10/2015 05:18

BARILOCHENSE6999

Sent: 03/10/2015 02:05

Dome of the Rock

The floor plan of the Dome of the Rock and some construction lines; possible source of inspiration for Templar constructions

and the cross pathee

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https://en.wikipedia.org/wiki/Centered_octagonal_number

Sent: 03/10/2015 01:52

Centered octagonal number

From Wikipedia, the free encyclopedia

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A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.^[1] The centered octagonal numbers are the same as the odd square numbers.^[2] Thus, the nth centered octagonal number is given by the formula

(2n-1)^2 = 4n^2-4n+1.

The first few centered octagonal numbers are^[2]

1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089.

Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.^[2]

References[edit]

Jump up ^ Teo, Boon K.; Sloane, N. J. A. (1985), "Magic numbers in polygonal and polyhedral clusters" (PDF), Inorganic Chemistry 24: 4545–4558, doi:10.1021/ic00220a025 .
^ Jump up to: ^a ^b ^c "Sloane's A016754 : Odd squares: (2n+1)^2. Also centered octagonal numbers.", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

The New Jerusalem Diagram – Introduction

“The New Jerusalem Diagram” is the name given by the scholar of ancient lore, John Michell [The original explanations in John Michell's books, especially "The Dimensions of Paradise", Thames and Hudson, 1986] to a geometrical construction that allows the “squaring of the circle” and the division of the circle to 28 equal sectors (a zodiac wheel) and the ceremonial positioning of the twelve tribes (or constellations) in a pattern of four camps at the four sides of the square.

John Michell has reconstructed the geometric pattern of the ‘heavenly city’ which is the template of the New Jerusalem of Revelation 21.

He discovered that this diagram served as the design (blueprint) for many sacred sites, including the Stonehenge, The Great Pyramid, St. Mary’s Chapel at Glastonbury and the City of Magnesia in Plato’s The Laws.

The Great Pyramid, Earth and Moon fit the New Jerusalem Diagram

It turns out that also the Nazca glyph called “Mandala” aka “Sun-Star-Cross” perfectly matches the New Jerusalem Diagram.

Superimposed image of NJ Diagram and Mandala glyph near Nazca, Peru

It is no surprise that the NJ diagram matches blueprint for the The Dome of the Rock, a shrine located on the Temple Mount in the Old City of Jerusalem.

The wheel of 28 divisions is common to various esoteric systems and serves as representation for the schematic month: 4 weeks = 28 days. The solar calendar of the Dead Sea scrolls sect was of 364 days, which is a multiple of 28 ( 28×13 = 364). In many Sufi diagrams, including in the ceiling of the dome at the El-Aqsa Mosque* (not the Dome of the Rock) at the Temple Mount, there appears the motif of the wheel made of 28 divisions.

Al-Aqsa. The resplendent ceiling of the Al-Aqsa Mosque on the Temple Mount in Jerusalem. The mosque, one of Islam’s most holy sites, sits near the Dome of the Rock and above the Marwani Mosque, aka Solomon’s Stables.
Image Source/Credit.

The NJ diagram is also hinted and enfolded (according to Micheell), in the description of the holy city, The New Jerusalem, whose descent from heaven concludes the Christian Bible.

Below we present 3-D version of the New Jerusalem Diagram.

Is it possible the 3D NJ diagram was used as a blueprint for the Crown Jewels?… Perhaps this is a case of a “secret hidden in plain site”?

The collective term Crown Jewels denotes the regalia and vestments worn by the sovereign of the United Kingdom during the coronation ceremony and at other state functions. The term refers to the following objects: the crowns, sceptres (with either the cross or the dove), orbs, swords, rings, spurs, colobium sindonis, dalmatic, armill, and the royal robe or pall, as well as several other objects connected with the ceremony itself. The Crown Jewels have a religious and sacral connotation. Their hidden symbolism seems to be related to the sacred geometry of the New Jerusalem Diagram.

The Construction of the New Jerusalem Diagram

The construction of this diagram starts with the attempt to “square the circle” by drawing a circle and a square that have the same circumference.

So in building this sacred diagram, one starts with a circle with a diameter of 22 units, and blocks it with a square of 22 x 22 units (like the number of the Hebrew letters, with which supposedly “heaven and earth were made”).
Then one draws from the same centre an additional circle whose diameter is 28 units – “the perfect number” that is already the number of the letters in the first verse of genesis and marks the circle of the seasons. The circle with diameter 28 has circumference of Pi x 28, which is as noted 22/7 x 28 = 88 – just the circumference of the square with edges of 22 and circumference thus 2 x 22 = 88.

The circle and the square have the same circumference, and they intersect in 8 points, where the square juts out of the circle in 4 vertices.

The building of the complete pattern continues with the drawing of 12 circles with diameter of 6 units – the number of letters in the first word of Genesis – Bereshit – as the first “perfect number”. The centers of these twelve circles are situated on the outer big circle, whose diameter is 28, and they touch (are tangent to) the inner circle, whose diameter is 22. Four of these circles are situated according to the points were the square is tangent to the inner circle, and the other 8 circles are placed according to the 8 points of intersection between the square and the outer circle.

There are two amazing consequences of this construction:

This construction squares the circle
It divides the circle into 28 equal segments
It reflects ratio 22:7 which is good approximation of the “Pi”
It reflects golden ratio 1.618
It reflects the ratio of 22:6, which issues theoretical-geometrical considerations is also the exact real ratio between the diameter of the earth and the diameter of the moon!

There are many more features of the NJ diagram which will be presented in our upcoming articles (e.g. amazing connection with Teotihuacan).

Messaggio 9 di 14 di questo argomento

Inviato: 05/10/2015 14:16

Message 337 of 350 on the subject

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Resultado de imagen para Raphael KEY 528 2012 2012forum.com

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6. Apocalipsis 3:7: Escribe al ángel de la iglesia en Filadelfia: Esto dice el Santo, el Verdadero, el que tiene la LLAVE de David, el que abre y ninguno cierra, y cierra y ninguno abre:

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Inviato: 05/10/2015 15:20

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http://www.taringa.net/posts/info/18448000/Cuarta-dimension.html

Inviato: 20/10/2015 15:53

El término cuarta dimensión aparece en diversos contextos como la física, las matemáticas y la ciencia ficción. En cada contexto el significado es diferente:

En física, se hace referencia a la cuarta dimensión al hablar del tiempo, principalmente desde el planteamiento de la Teoría de la Relatividad.
En matemática, el concepto aparece asociado o bien a espacios euclídeos de más de tres dimensiones o, más generalmente, a espacios localmente euclídeos o 4-variedades diferenciables.

cuatro

Hipercubo de 4 dimensiones espaciales girando, tal como se vería proyectado en el espacio tridimensional.

El interés en las dimensiones más altas alcanzó su clímax entre 1870 y 1920.1 En esos años se convirtió en tema frecuente en la literatura fantástica, el arte e incluso algunas teorías científicas. La cuarta dimensión, entendida como dimensión espacial adicional (no como dimensión temporal, como en la teoría de la relatividad) apareció en las obras literarias de Oscar Wilde, Fiódor Dostoyevski, Marcel Proust, H. G. Wells y Joseph Conrad, inspiró algunas obras musicales de Alexander Scriabin, Edgar Varèse y George Antheil y algunas obras plásticas de Pablo Picasso y Marcel Duchamp influyendo en el desarrollo del cubismo. Incluso personajes tan diversos como el psicólogo William James, la escritora Gertrude Stein o el socialista revolucionario Vladimir Lenin se interesaron en el tema.

Igualmente los matemáticos habían estado interesados en el tema al tratar de generalizar los conceptos de la geometría euclídea tridimensional. El matemático Charles L. Dodgson, que enseñó en la Universidad de Oxford, deleitó a generaciones de escolares escribiendo libros, bajo el pseudónimo de Lewis Carroll, que incorporaban algunas ideas sobre la cuarta dimensión. Desde el punto de vista académico, el estudio general de la geometría de la cuarta dimensión en gran parte resultado de los trabajos de Bernhard Riemann. Charles Howard Hinton, matemático y escritor de ciencia ficción británico, acuñó muchos neologismos para describir elementos en la cuarta dimensión. De acuerdo con el Oxford English Dictionary, fue el primero en emplear la palabra tesseract en su libro Una nueva era del pensamiento. También inventó las palabras “kata” (del griego “abajo”) y “ana” (del griego “arriba”) para describir las dos direcciones opuestas en la cuarta dimensión, equivalentes a derecha-izquierda, arriba-abajo, y adelante-atrás.

Los trabajos matemáticos sobre geometrías multidimensionales y geometrías no euclídeas habían sido considerado por los físicos como simples abstracciones matemáticas hasta que Henri Poincaré probó que el grupo de transformaciones de Lorentz que dejaban invariantes las ecuaciones del electromagnetismo podían ser interpretadas como "rotaciones" en un espacio de cuatro dimensiones. Más tarde, los trabajos de Einstein y la interpretación geométrica de estos por parte de Hermann Minkowski llevaron a la aceptación de la cuarta dimensión como una descripción necesaria para explicar los hechos observados relacionados con el electromagnetismo. Sin embargo, aquí la "cuarta dimensión" no era un lugar separado del espacio tridimensional (como en varias de las obras de ficción de la época) ni tampoco una dimensión espacial análoga a las otras tres dimensiones espaciales, sino una dimensión temporal que sólo puede recorrerse hacia el futuro. En la teoría general de la relatividad el campo gravitatorio es explicado como un efecto geométrico de la curvatura de un espacio-tiempo de cuatro dimensiones.

Más tarde, la teoría de Kaluza-Klein propuso que no sólo el campo gravitatorio podía ser interpretado de forma más sencilla como curvatura de un "espacio" de más de tres dimensiones, sino que si se introducía una nueva dimensión espacial enrollada o «compactificada», también el campo electromagnético podía ser interpretado como un efecto geométrico de la curvatura de dimensiones superiores. Así, la Kaluza proponía una teoría de campo unificado del electromagnetismo y la gravedad en un espacio-tiempo de cinco dimensiones, con una dimensión temporal, tres dimensiones espaciales extendidas y una dimensión espacial «compactificada» adicional, que, debido a su condición de compactificada, no era directamente visible pero su efecto era perceptible en forma de campo electromagnético.

Un ángulo recto se describe como un cuarto de una revolución. La Geometría Cartesiana escoge direcciones ortogonales arbitrariamente a través del espacio, lo que significa que cada dirección está en ángulo recto con las demás. Las 3 dimensiones ortogonales del espacio se conocen como altitud, longitud y latitud. La Cuarta Dimensión por lo tanto es la dirección en el espacio con ángulo recto a las 3 direcciones observables.

Un vector espacial es un conjunto de vectores, los cuales podemos imaginarlos como flechas, que proviene de un simple lugar llamado origen (vectores geométricos), que apuntan a otros lugares.

Un punto es un objeto de cero dimensiones. No tiene extensión en el espacio ni propiedades, como una flecha pero sin longitud. Este vector es llamado el vector cero y es el más simple vector espacial.

Una línea es un objeto unidimensional. Si escogemos un cierto vector distinto a cero en una cierta dirección, este vector tiene cierta longitud definida. Ese vector tiene una cabeza en un cierto punto en espacio y una cola en el origen. Si pensamos en estirar que ese vector así sea dos veces su largo, tres veces, etcétera y uniformemente, tomando todas las longitudes posibles (incluso la longitud cero, conseguir el vector cero), conseguiremos una sola línea con una sola dimensión: La de la longitud. Todos los vectores que describen puntos en esta línea serían paralelos. Aunque para visualizar la línea es necesario que ésta tenga un ancho mínimo, sin embargo, una línea de 1D no la tendría.

Un plano es un objeto de dos dimensiones. Tiene longitud y anchura pero no profundidad - algo como una hoja de papel, o más exactamente algo como las imágenes en un televisor común. El pensamiento en un plano en términos de vectores puede ser un poco más desafiante. Si pensamos en tomar un vector y lo movemos de modo que su cola esté tocando la cabeza del primero y esté formando un vector con su cola en el origen y la cabeza en la cabeza del segundo vector colocado de nuevo, tenemos una manera razonable de hablar de vectores de adición. Si tenemos dos vectores que no sean paralelos, podemos hablar de todos los puntos que podemos alcanzar por o solamente el estirar o ningunos de los vectores, y, agregando estos vectores juntos, estos puntos forman un plano.

El espacio, como lo percibimos, es tridimensional. Podemos pensar en poner una línea junto con un plano. Estas líneas son como un emparedado. Para conseguir a un cierto punto en espacio, podemos imaginarnos el viajar encima de la línea y después el movernos a través del plano al punto. Entonces tenemos tres vectores a pensar alrededor, uno a viajar una cierta distancia encima de la línea y dos para conseguir a un cierto punto en espacio.

Para tres rectas ortogonales en el espacio tridimensional (x, y y z) existe una cuarta, normal al espacio, ortogonal a estas tres rectas, que forma un eje p. ej. w.

El producto vectorial es la determinante de una matriz 4×4, donde una de las filas (o columnas) son los vectores unitarios h, i, j y k y las demás (filas o columnas respectivamente) están formadas por las componentes de tres vectores cuadradimensionales cualesquiera, este producto nos dará un cuarto vector perpendicular a estos tres mismos.

La Geometría euclidiana prevé una mayor variedad de formas para existir que en tres dimensiones. Los poliedros tridimensionales son recintos espaciales hechos de caras de dos dimensiones conectadas, los policronos cuadridimensionales son recintos del espacio cuadridimensional hechos de poliedros tridimensionales. Donde en tres dimensiones, hay exactamente cinco poliedros regulares, o los sólidos platónicos, que pueden existir, seis policronos regulares existen en la cuarta dimensión. Cinco de los seis se pueden interpretar como extensiones naturales de los sólidos platónicos, así como el cubo, un sólido platónico, es una extensión del cuadrado de dos dimensiones. El pentachoron está hecho de 5 tetraedros para las caras y 10 caras triangulares, y es el análogo cuadridimensional del tetraedro. El teseracto, o el hipercubo, se compone de 8 caras cúbicas y de 24 cuadrados, y es el politopo cuadridimensional medido. Los teseractos se doblan, la 16-celdas, son el equivalente del octaedro, pues son ambos politopos de cruz. Los politopos de 120 celdas y los de 600 celdas se doblan de igual modo, y son análogos al dodecaedro y al icosaedro, respectivamente. El de 24 celdas es un policrono regular único y que no tiene ningún equivalente tridimensional. Apenas pues la esfera, o 2-esfera, es una superficie de dos dimensiones curvada compuesta de todos los puntos equidistantes de un punto central dado, en un espacio tridimensional, la 3-esfera, una clase de hiperesfera, es el espacio que contiene todos los puntos equidistantes a un punto central dado, en un espacio cuadridimensional. Cada sección transversal tridimensional de un 3-esfera es un 2-esfera.

cuarta

La analogía dimensional se usa frecuentemente para comprender el salto de una dimensión (en este caso, la tercera dimensión) a una más alta (cuarta dimensión). La analogía dimensional consiste en resolver un problema en n + 1 dimensiones relacionándolo primero con un problema análogo de (n - 1) dimensión, vale decir, "una dimensión menos". E igualmente debe analizarse el caso de cómo se relaciona el problema en n con el de (n + 1) dimensiones, es decir, "una más".

Edwin Abbott Abbott en su libro Planilandia (Flatland) escribe sobre un "ser cuadrado" que vive en un mundo de dos dimensiones, como la superficie de un pedazo de papel. Este "cuadrado" se enfrenta a experimentos de un ser tridimensional. El ser tridimensional es percibido por el "cuadrado" como un ser aparentemente divino, ya que puede poner y quitar objetos de una caja fuerte sin romperla ni abrirla (moviéndolos a través de su tercera dimensión), ver todo desde de la perspectiva de dos dimensiones sea incluido detrás de las paredes (puesto que ve "sobre" Planilandia), y totalmente invisible para los habitantes de Planilandia, puesto que está "arriba" y una dimensión por arriba de las dos dimensiones en las que el cuadrado está atrapado. No obstante, el ser tridimensional podría manifestarse en el mundo de dos dimensiones, pero sólo parcialmente, si fuera una esfera, aparecería como una secuencia de círculos sucesivos "que cambian de tamaño" (intersecciones de la esfera con el plano de dos dimensiones). Aplicando analogía dimensional, uno puede deducir que el ser cuadridimensional sería capaz de hazañas similares de nuestra perspectiva tridimensional.

Rudy Rucker demuestra esto en su novela "Spaceland", en la cual el protagonista encuentra los seres cuadridimensionales que demuestran tales energías. Un uso útil de la analogía dimensional en visualizar la cuarta dimensión está en la proyección. Una proyección es una manera para representar un objeto (n+1)-dimensional en la n-dimensión. Por ejemplo, las pantallas de computadora son de dos dimensiones, y todas las fotografías de objetos tridimensionales son representadas en dos dimensiones puesto que la información de la tercera dimensión (o de la profundidad) no puede ser representada por la pantalla (si el observador se mueve, aleje o acerque, la imagen no cambiará). En este caso, la profundidad se quita y se substituye por la información indirecta. La retina del ojo es un arsenal de dos dimensiones de receptores pero puede permitir que el cerebro perciba la naturaleza de objetos tridimensionales usando la información indirecta (como la perspectiva, el sombreado, visión binocular, etc.).

La perspectiva del uso de los artistas da profundidad tridimensional a los cuadros de dos dimensiones. Asimismo, los objetos en la cuarta dimensión se pueden proyectar matemáticamente a las familiares tres dimensiones, donde pueden entonces ser examinados más convenientemente. En este caso, la "retina de un ojo cuadridimensional" tendría un arsenal de receptores tridimensionales. El ser hipotético con tal ojo percibiría la naturaleza de objetos cuadridimensionales usando la información indirecta contenida en las imágenes que recibe en su retina. La proyección de la perspectiva a partir de cuatro dimensiones produce efectos similares como en el caso tridimensional, tal como la perspectiva.

Esto agrega "profundidad cuadridimensional" a estos cuadros tridimensionales. La analogía dimensional también ayuda a entender tales proyecciones. Por ejemplo, los objetos de dos dimensiones son limitados por límites unidimensionales: un cuadrado es limitado por cuatro bordes o líneas. Los objetos tridimensionales son limitados por superficies de bidimensionales: un cubo es limitado por 6 cuadrados. Aplicando analogía dimensional, uno puede deducir que un cubo cuadridimensional, conocido como teseracto, es limitado por los volúmenes tridimensionales.

Y de hecho, éste es el caso matemáticamente: el teseracto es limitado por 8 cubos. Saber esto es indispensable para entender cómo interpretar una proyección tridimensional del teseracto. Los límites del teseracto proyectan a los volúmenes en la imagen, superficies no simplemente de dos dimensiones. Esto ayuda a entender las características de dichas dimensiones que de otra manera sólo confundirían. De igual manera, el concepto de sombras puede ayudarnos mejor a entender la teoría de cuatro dimensiones. Si usted proyectara una luz sobre objeto tridimensional, éste proyectaría una sombra de dos dimensiones. Por lo tanto la luz en un objeto de dos dimensiones echaría una sombra unidimensional (en un mundo de dos dimensiones), y la luz en un objeto unidimensional en un mundo unidimensional echaría una sombra cero-dimensional, es decir, un punto de la no-luz. Esta idea se puede utilizar en la otra dirección; la luz en un objeto cuadridimensional proyectaría una sombra tridimensional. Como ejemplo, la sombra de un cubo transparente, proyectaría una sombra sobre el papel, de dos cuadrados, unidos por sus vértices con 4 segmentos.

Semejantemente, si era un cubo cuadridimensional iluminado con luz de 4 dimensiones, su sombra sería la de un cubo tridimensional dentro de otro cubo tridimensional. Siendo tridimensionales podemos solamente ver el mundo con nuestros ojos en dos dimensiones; el ser cuadridimensional consideraría el mundo en tres. Así podría, por ejemplo, ver los seis lados de una caja opaca simultáneamente. No solamente eso; también podría ver lo que hay al interior de la caja, como en Planilandia, en donde la esfera ve objetos en el mundo de dos dimensiones y todo dentro de ellos simultáneamente. Análogo, un espectador cuadridimensional vería todos los puntos en nuestro espacio tridimensional simultáneamente, incluyendo la estructura interna de objetos sólidos y de cosas obscurecidos de nuestro punto de vista.

Albert Einstein en su célebre teoría de 1905 de la relatividad especial habló por primera vez del tiempo como una cuarta dimensión y como algo indispensable para ubicar un objeto en el espacio y en un momento determinado. El tiempo en la teoría de la relatividad no es una dimensión espacial más, ya que fijado un punto del espacio-tiempo éste puede ser no alcanzable desde nuestra posición actual, hecho que difiere de la concepción usual de dimensión espacial. Aunque inicialmente se interpretó el tiempo como una "dimensión" matemática necesaria para ubicar un evento u objeto, en la teoría de la relatividad general el tiempo es tratado como una dimensión geométrica más, aunque los objetos materiales no puedan seguir una trayectoria completamente arbitraria a lo largo del tiempo (como por ejemplo "dar la vuelta" y viajar al pasado). La necesidad del tiempo dentro de la teoría de la relatividad es necesaria por dos motivos:

En primer lugar, los objetos no sólo se mueven a través del espacio sino que también lo hacen a través del tiempo, es decir su coordenada temporal aumenta continuamente, por lo que hubo la necesidad de hablar del tiempo ligado al espacio como la cuarta dimensión (en inglés spacetime, en castellano espacio-tiempo). Además el ritmo de avance en la dimensión temporal depende del estado de movimiento del observador, produciéndose una dilatación temporal efectiva para los observadores más rápidos en relación al tiempo medido por un observador estacionario.
En segundo lugar, el carácter intrínseco del espacio-tiempo y su cuatridimensionalidad requiere un modo conceptualmente diferente de tratar la geometría del universo, puesto que una cuarta dimensión implica un espacio plano (bidimensional) que se curva en la teoría de la relatividad general por la acción de la gravedad de la materia originándose la curvatura del espacio-tiempo.
Finalmente cabe añadir que algunas teorías físicas como la teoría de Kaluza-Klein y las teoría de supercuerdas, en sus varias versiones, añaden a las tres dimensiones físicas espaciales entre 1 y 9 dimensiones espaciales adicionales, de tipo compacto; además de la dimensión temporal.

Messaggio 12 di 14 di questo argomento

Inviato: 20/10/2015 16:44

es.walyou.com
2 Alcancia cubo Rubik
521 × 423 - 24k - jpg

es.wikipedia.org
El interior de un cubo de
232 × 199 - 28k - jpg

solostocks.com
Cubo Mágico
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del Rubik - Cubo Magico -
450 × 338 - 28k - jpg

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EL MISMO "CUBO MAGICO" ES UN HIPERCUBO (HAY UN CUBO MAS CHICO EN EL CENTRO DEL MISMO)

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Sent: 20/10/2015 12:57

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Message 15 of 15 on the subject

Sent: 20/10/2015 13:33

2 AL CUBO=8

3 AL CUBO=27

4 AL CUBO=64

5 AL CUBO=125

5 AL CUBO-CUATRO AL CUBO=5*5*5-4*4*4=125-64=61 (EXAGONAL)

4 AL CUBO-TRES AL CUBO=4*4*4-3*3*3=37 ("LLAVE DE DAVID")

3 AL CUBO-DOS AL CUBO=3*3*3-2*2*2=19 (EXAGONAL)

2 AL CUBO-UNO AL CUBO=2*2*2-1*1*1=7 (EXAGONAL)

LAS MATEMATICAS SON EXACTAS

APARENTEMENTE, EN ESTE CONTEXTO, LA PUERTA DE LA "CUARTA DIMENCION", EN EL CONTEXTO A LA RELACION DE QUE EL EXAGONO, TAMBIEN ES UN CUBO, EL OCTOGONO, SERIA LA PUERTA DEL HIPERCUBO, OSEA, INSISTO, LA CUARTA DIMENCION.

LAS MATEMATICAS SERIA:

2*2*2*2-1*1*1*1=15

CONSIDERANDO TAMBIEN AL CERO, ES UNA REFERENCIA AL 16, CODIFICADO EN EL MISMO VATICANO, EN EL PLANETA VENUS, EN EL SISTEMA DIGITAL, ETC,ETC.

Messaggio 13 di 14 di questo argomento

http://forum.andrewgough.co.uk/viewtopic.php?f=1&t=3996&start=375

Inviato: 21/10/2015 17:35

Sauniere had a mon-key called Mela Latin apple

Venus/Mary blue apples and it's pentagonal cycle

_________________
E.T.A.E

Messaggio 14 di 14 di questo argomento