Esoteric Other Worlds

Exploring Mysticism, and Parapsychology.This blog is also an attempt to promote awareness of a Modern Universal Paradigm known as Multi-Dimensional Science. It offers a "Scientific" testable Hypothesis for a more "objective" understanding of claimed Psychic and Spiritual Phenomena. A link to this subject should be found onsite or alternatively easily traceable via a word search. Please note that the Internet articles here may not reflect the views of the Blogger.

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Wednesday, 20 November 2013

Topological Dimensions

To some extent, topology may have great relevance to my evolving project along with the Hilbert Space. http://www.p2pfoundation.net/Multi-Dimensional_Science

RS

topology, branch of mathematics, sometimes referred to as “rubber sheet geometry,” in which two objects are considered equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking while disallowing tearing apart or gluing together parts. The main topics of interest in topology are the properties that remain unchanged by such continuous deformations. Topology, while similar to geometry, differs from geometry in that geometrically equivalent objects often share numerically measured quantities, such as lengths or angles, while topologically equivalent objects resemble each other in a more qualitative sense.

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The area of topology dealing with abstract objects is referred to as general, or point-set, topology. General topology overlaps with another important area of topology called algebraic topology. These areas of specialization form the two major subdisciplines of topology that developed during its relatively modern history.

Basic concepts of general topology

Simply connected

In some cases, the objects considered in topology are ordinary objects residing in three- (or lower-) dimensional space. For example, a simple loop in a plane and the boundary edge of a square in a plane are topologically equivalent, as may be observed by imagining the loop as a rubber band that can be stretched to fit tightly around the square. On the other hand, the surface of a sphere is not topologically equivalent to a torus, the surface of a solid doughnut ring. To see this, note that any small loop lying on a fixed sphere may be continuously shrunk, while being kept on the sphere, to any arbitrarily small diameter. An object possessing this property is said to be simply connected, and the property of being simply connected is indeed a property retained under a continuous deformation. However, some loops on a torus cannot be shrunk, as shown in the figure.
Many results of topology involve objects as simple as those mentioned above. The importance of topology as a branch of mathematics, however, arises from its more general consideration of objects contained in higher-dimensional spaces or even abstract objects that are sets of elements of a very general nature. To facilitate this generalization, the notion of topological equivalence must be clarified.

Topological equivalence

The motions associated with a continuous deformation from one object to another occur in the context of some surrounding space, called the ambient space of the deformation. When a continuous deformation from one object to another can be performed in a particular ambient space, the two objects are said to be isotopic with respect to that space. For example, consider an object that consists of a circle and an isolated point inside the circle. Let a second object consist of a circle and an isolated point outside the circle, but in the same plane as the circle. In a two-dimensional ambient space these two objects cannot be continuously deformed into each other because it would require cutting the circles open to allow the isolated points to pass through. However, if three-dimensional space serves as the ambient space, a continuous deformation can be performed—simply lift the isolated point out of the plane and reinsert it on the other side of the circle to accomplish the task. Thus, these two objects are isotopic with respect to three-dimensional space, but they are not isotopic with respect to two-dimensional space.

The notion of objects being isotopic with respect to a larger ambient space provides a definition of extrinsic topological equivalence, in the sense that the space in which the objects are embedded plays a role. The example above motivates some interesting and entertaining extensions. One might imagine a pebble trapped inside a spherical shell. In three-dimensional space the pebble cannot be removed without cutting a hole through the shell, but by adding an abstract fourth dimension it can be removed without any such surgery. Similarly, a closed loop of rope that is tied as a trefoil, or overhand, knot (see figure) in three-dimensional space can be untied in an abstract four-dimensional space.

Homeomorphism

An intrinsic definition of topological equivalence (independent of any larger ambient space) involves a special type of function known as a homeomorphism. A function h is a homeomorphism, and objects X and Y are said to be homeomorphic, if and only if the function satisfies the following conditions.

(1) h is a one-to-one correspondence between the elements of X and Y;
(2) h is continuous: nearby points of X are mapped to nearby points of Y and distant points of X are mapped to distant points of Y—in other words, “neighbourhoods” are preserved;
(3) there exists a continuous inverse function h⁻¹: thus, h⁻¹h(x) = x for all x ∊ X and hh⁻¹(y) = y for all y ∊ Y—in other words, there exists a function that “undoes” (is the inverse of) the homeomorphism, so that for any x in X or any y in Y the original value can be restored by combining the two functions in the proper order.

The notion of two objects being homeomorphic provides the definition of intrinsic topological equivalence and is the generally accepted meaning of topological equivalence. Two objects that are isotopic in some ambient space must also be homeomorphic. Thus, extrinsic topological equivalence implies intrinsic topological equivalence.

Topological structure

In its most general setting, topology involves objects that are abstract sets of elements. To discuss properties such as continuity of functions between such abstract sets, some additional structure must be imposed on them.

Topological space

One of the most basic structural concepts in topology is to turn a set X into a topological space by specifying a collection of subsets T of X. Such a collection must satisfy three axioms: (1) the set X itself and the empty set are members of T, (2) the intersection of any finite number of sets in T is in T, and (3) the union of any collection of sets in T is in T. The sets in T are called open sets and T is called a topology on X. For example, the real number line becomes a topological space when its topology is specified as the collection of all possible unions of open intervals—such as (−5, 2), (1/2, π), (0, √2), …. (An analogous process produces a topology on a metric space.) Other examples of topologies on sets occur purely in terms of set theory. For example, the collection of all subsets of a set X is called the discrete topology on X, and the collection consisting only of the empty set and X itself forms the indiscrete, or trivial, topology on X. A given topological space gives rise to other related topological spaces. For example, a subset A of a topological space X inherits a topology, called the relative topology, from X when the open sets of A are taken to be the intersections of A with open sets of X. The tremendous variety of topological spaces provides a rich source of examples to motivate general theorems, as well as counterexamples to demonstrate false conjectures. Moreover, the generality of the axioms for a topological space permit mathematicians to view many sorts of mathematical structures, such as collections of functions in analysis, as topological spaces and thereby explain associated phenomena in new ways.

A topological space may also be defined by an alternative set of axioms involving closed sets, which are complements of open sets. In early consideration of topological ideas, especially for objects in n-dimensional Euclidean space, closed sets had arisen naturally in the investigation of convergence of infinite sequences (see infinite series). It is often convenient or useful to assume extra axioms for a topology in order to establish results that hold for a significant class of topological spaces but not for all topological spaces. One such axiom requires that two distinct points should belong to disjoint open sets. A topological space satisfying this axiom has come to be called a Hausdorff space.

Continuity

An important attribute of general topological spaces is the ease of defining continuity of functions. A function f mapping a topological space X into a topological space Y is defined to be continuous if, for each open set V of Y, the subset of X consisting of all points p for which f(p) belongs to V is an open set of X. Another version of this definition is easier to visualize, as shown in the figure. A function f from a topological space X to a topological space Y is continuous at p ∊ X if, for any neighbourhood V of f(p), there exists a neighbourhood U of p such that f(U) ⊆ V. These definitions provide important generalizations of the usual notion of continuity studied in analysis and also allow for a straightforward generalization of the notion of homeomorphism to the case of general topological spaces. Thus, for general topological spaces, invariant properties are those preserved by homeomorphisms.

Algebraic topology

The idea of associating algebraic objects or structures with topological spaces arose early in the history of topology. The basic incentive in this regard was to find topological invariants associated with different structures. The simplest example is the Euler characteristic, which is a number associated with a surface. In 1750 the Swiss mathematician Leonhard Euler proved the polyhedral formula V – E + F = 2, or Euler characteristic, which relates the numbers V and E of vertices and edges, respectively, of a network that divides the surface of a polyhedron (being topologically equivalent to a sphere) into F simply connected faces. This simple formula motivated many topological results once it was generalized to the analogous Euler-Poincaré characteristic χ = V – E + F = 2 – 2g for similar networks on the surface of a g-holed torus. Two homeomorphic surfaces will have the same Euler-Poincaré characteristic, and so two surfaces with different Euler-Poincaré characteristics cannot be topologically equivalent. However, the primary algebraic objects used in algebraic topology are more intricate and include such structures as abstract groups, vector spaces, and sequences of groups. Moreover, the language of algebraic topology has been enhanced by the introduction of category theory, in which very general mappings translate topological spaces and continuous functions between them to the associated algebraic objects and their natural mappings, which are called homomorphisms.

Fundamental group

A very basic algebraic structure called the fundamental group of a topological space was among the algebraic ideas studied by the French mathematician Henri Poincaré in the late 19th century. This group essentially consists of curves in the space that are combined by an operation arising in a geometric way. While this group was well understood even in the early days of algebraic topology for compact two-dimensional surfaces, some questions related to it still remain unanswered, especially for certain compact manifolds, which generalize surfaces to higher dimensions.

The most famous of these questions, called the Poincaré conjecture, asks if a compact three-dimensional manifold with trivial fundamental group is necessarily homeomorphic to the three-dimensional sphere (the set of points in four-dimensional space that are equidistant from the origin), as is known to be true for the two-dimensional case. Much research in algebraic topology has been related in some way to this conjecture since it was posed by Poincaré in 1904. One such research effort concerned a conjecture on the geometrization of three-dimensional manifolds that was posed in the 1970s by the American mathematician William Thurston. Thurston’s conjecture implies the Poincaré conjecture, and in recognition of his work toward proving these conjectures, the Russian mathematician Grigori Perelman was awarded a Fields Medal at the 2006 International Congress of Mathematicians.
The fundamental group is the first of what are known as the homotopy groups of a topological space. These groups, as well as another class of groups called homology groups, are actually invariant under mappings called homotopy retracts, which include homeomorphisms. Homotopy theory and homology theory are among the many specializations within algebraic topology.

Differential topology

Many tools of algebraic topology are well-suited to the study of manifolds. In the field of differential topology an additional structure involving “smoothness,” in the sense of differentiability (see analysis: Formal definition of the derivative), is imposed on manifolds. Since early investigation in topology grew from problems in analysis, many of the first ideas of algebraic topology involved notions of smoothness. Results from differential topology and geometry have found application in modern physics.

Knot theory

Another branch of algebraic topology that is involved in the study of three-dimensional manifolds is knot theory, the study of the ways in which knotted copies of a circle can be embedded in three-dimensional space. Knot theory, which dates back to the late 19th century, gained increased attention in the last two decades of the 20th century when its potential applications in physics, chemistry, and biomedical engineering were recognized.

History of topology

Mathematicians associate the emergence of topology as a distinct field of mathematics with the 1895 publication of Analysis Situs by the Frenchman Henri Poincaré, although many topological ideas had found their way into mathematics during the previous century and a half. The Latin phrase analysis situs may be translated as “analysis of position” and is similar to the phrase geometria situs, meaning “geometry of position,” used in 1735 by the Swiss mathematician Leonhard Euler to describe his solution to the Königsberg bridge problem. Euler’s work on this problem also is cited as the beginning of graph theory, the study of networks of vertices connected by edges, which shares many ideas with topology.

During the 19th century two distinct movements developed that would ultimately produce the sibling specializations of algebraic topology and general topology. The first was characterized by attempts to understand the topological aspects of surfacelike objects that arise by combining elementary shapes, such as polygons or polyhedra. One early contributor to combinatorial topology, as this subject was eventually called, was the German mathematician Johann Listing, who published Vorstudien zur Topologie (1847; “Introductory Studies in Topology”), which is often cited as the first print occurrence of the term topology. In 1851 the German mathematician Bernhard Riemann considered surfaces related to complex number theory and, hence, utilized combinatorial topology as a tool for analyzing functions. The German geometers August Möbius and Felix Klein published works on “one-sided” surfaces in 1858 and 1882, respectively. Möbius’s example, now known as the Möbius strip, may be constructed by gluing together the ends of a long rectangular strip of paper that has been given a half twist. Surfaces containing subsets homeomorphic to the Möbius strip are called nonorientable surfaces and play an important role in the classification of two-dimensional surfaces. Klein provided an example of a one-sided surface that is closed, that is, without any one-dimensional boundaries. This example, now called the Klein bottle, cannot exist in three-dimensional space without intersecting itself and, thus, was of interest to mathematicians who previously had considered surfaces only in three-dimensional space.
Work by many mathematicians, including the four mentioned above, preceded the 1895 publication of Analysis Situs, in which Poincaré established a basic context for using algebraic ideas in combinatorial topology. Combinatorial topology continued to be developed, especially by the German-born American mathematician Max Dehn and the Danish mathematician Poul Heegaard, who jointly presented one of the first classification theorems for two-dimensional surfaces in 1907. Soon thereafter the importance of associating algebraic structures with topological objects was clearly established by, for example, the Dutch mathematician L.E.J. Brouwer and his fixed point theorem. Although the phrase algebraic topology was first used somewhat later in 1936 by the Russian-born American mathematician Solomon Lefschetz, research in this major area of topology was well under way much earlier in the 20th century.
Simultaneous with the early development of combinatorial topology, 19th-century analysts, such as the French mathematician Augustin Cauchy and the German mathematician Karl Weierstrass, investigated Fourier series, in which sequences of functions converged to other functions in a sense similar to convergence of sequences of points in space. From another point of view, mathematicians such as the German Georg Cantor and the French Émile Borel studied the relationship between Fourier series and set theory. Two initiatives arose from these efforts: establishing a rigorous mathematical setting for major problems of analysis and providing a general setting for mathematical ideas related to convergence of sequences. In 1899 the German mathematician David Hilbert proposed an axiomatic setting for general geometry beyond what the ancient Greeks had considered. In 1905 the French mathematician Maurice Fréchet proposed a consistent scheme of axioms for convergence in an abstract set and also axioms for a metric space, which is a set supplied with a distance function (or “metric”). In 1910 Hilbert suggested axioms for neighbourhoods of points in an abstract set, thereby generalizing properties of small disks centred at points in the plane. Finally, the German mathematician Felix Hausdorff in his Grundzüge der Mengenlehre (1914; “Elements of Set Theory”) proposed the foundational axiomatic relationships among the metric, limit, and neighbourhood approaches for general spaces (see Hausdorff space). Although it was not until 1925 that the Russian mathematician Pavel Alexandrov introduced the modern axioms for a topology on an abstract set, the field of general topology was born in Hausdorff’s work.
During the period up to the 1960s, research in the field of general topology flourished and settled many important questions. The notion of dimension and its meaning for general topological spaces was satisfactorily addressed with the introduction of an inductive theory of dimension. Compactness, a property that generalizes closed and bounded subsets of n-dimensional Euclidean space, was successfully extended to topological spaces through a definition involving “covers” of a space by collections of open sets, and many problems involving compactness were solved during this period. The metrization problem, which sought a topological description of the spaces for which the topology could be induced by a metric, was settled following considerable work on the notion of paracompactness, a property that generalizes compactness.

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Wednesday, 26 June 2013

Can We Resolve Quantum Paradoxes by Stepping Out of Space and Time? [Guest Post]

By George Musser | June 21, 2013 | Scientific America

Next month will be the 100th anniversary of Bohr’s model of the atom, one of the foundations of the theory of quantum mechanics. And look where we are now: we still don’t know what the darned theory really means. One of the most radical interpretations (which is saying something) has got to be the so-called Transactional Interpretation, whereby particles send a type of signal backward in time. This past fall, University of Maryland philosopher Ruth Kastnerpublished a book that tries to make sense of it. I’ve invited her to guide us through it.

In the June issue of Scientific American, physicist and writer Hans Christian von Baeyer describes the current state of “deep confusion about the meaning of quantum theory” and discusses one proposal—a denial that the theory describes anything objectively real—for rendering some of the quantum perplexities “less troubling.” Von Baeyer also lists several other possible interpretations, but leaves out what I think is the most promising approach.

The idea, known as the Transactional Interpretation, was first proposed by University of Washington physicist John Cramer in the 1980s and has its roots in the ideas of renowned physicists John Wheeler and Richard Feynman. This interpretation makes use of a concept known technically as “advanced action,” which is characterized not by the usual positive energy but by negative energy. Though it may seem counterintuitive at first, it turns out to provide a natural way to understand certain aspects of the theory that currently seem arbitrary or ad hoc, such as the rule for calculating the probabilities of measurement outcomes.

In the transactional picture, the entities described by quantum states, which are characterized by positive energy, are only half the story. The other half of the story is the absorption of those emitted states, which is accompanied by a negative-energy (advanced) response. Cramer himself compared his account to the handshake of a financial transaction: the emitted state is the offer and the response state is the confirmation. Absorption is the key to untying the interpretational Gordian Knot presented by quantum theory, which has given rise to such perplexities as the famous Schrödinger’s Cat thought experiment. It is absorption that collapses the quantum superposition and saves the poor cat from the fate of being both dead and alive at the same time.

Cramer’s original version of the interpretation, although promising, did not receive widespread acceptance. Physicists and philosophers had trouble making sense of advanced propagation, which is usually considered synonymous with back-in-time propagation and therefore seemed to raise the possibility of causal-loop paradoxes, such as being able to go into the past and kill one’s own parents. In addition, some critics felt that the notion of absorber was not well-defined. My research is aimed at resolving these types of challenges and providing a clear account of what constitutes an absorber. By incorporating principles from relativistic quantum theory, which were absent from the original transactional picture, I have been able to obtain a clear criterion for the boundary between the microscopic quantum realm and the macroscopic classical realm, which is the point at which collapse is overwhelmingly likely to occur (although the collapse process is fundamentally indeterministic).

My development of the Transactional Interpretation makes use of an important idea of Werner Heisenberg: “Atoms and the elementary particles themselves … form a world of potentialities or possibilities rather than things of the facts.” This world of potentialities is not contained within space and time; it is a higher-dimensional world whose structure is described by the mathematics of quantum theory. The Transactional Interpretation is best understood by considering both the offer and confirmation as Heisenbergian possibilities—that is, they are only potential events. That removes the possibility of causal-loop inconsistencies, since neither the positive-energy offer wave nor the negative-energy confirmation wave carries real energy, and neither is contained in spacetime. It is only in the encounter between the two that real energy may be conveyed within spacetime from an emitter to an absorber—and when this occurs, all the energy is delivered in the normal future direction.

The Transactional Interpretation, in this new possibilist version, provides not only a clear physical account of measurement but also a new understanding of quantum reality in which dynamic possibilities give rise to observable physical events through the transactional process. It also renders harmless the “spooky action at a distance” that troubled Einstein. Quantum correlations do not violate the relativistic speed limit because these correlations exist only at the level of possibility.

The transactional picture is conceptually challenging because the underlying processes are so different from what we are used to in our classical world of experience, and we must allow for the startling idea that there is more to reality than what can be contained within spacetime. As is evident from von Baeyer’s article, quantum theory truly challenges us to think outside the box—and, in this case, I submit that the box is spacetime itself. If this seems farfetched, consider the eloquent point made by physicist and philosopher Ernan McMullin: “Imaginability must not be made the test for ontology. The realist claim is that the scientist is discovering the structures of the world; it is not required in addition that these structures be imaginable in the categories of the macroworld.” Only if we face the strange non-classical features of the physical world head-on can we have a physical, non-observer-dependent account of our reality that solves longstanding puzzles such as the problem of Schrödinger’s Cat.

Images courtesy of Ruth Kastner

About the Author: is a contributing editor at Scientific American. He focuses on space science and fundamental physics, ranging from particles to planets to parallel universes. He is the author of The Complete Idiot's Guide to String Theory. Musser has won numerous awards in his career, including the 2011 American Institute of Physics's Science Writing Award. Follow on Twitter @gmusser.

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Monday, 4 March 2013

The Fourth Dimension

The concept of the Fourth Dimension has been used as an explanation for the manifestation of claimed psyhic, and mystical phenomena. Unfortunately, like Quantum Mechanics its understanding is incomplete without the development of Multi-Dimensional Science..

http://www.p2pfoundation.net/Multi-Dimensional_Science

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

In mathematics, four-dimensional space ("4D") is an abstract concept derived by generalizing the rules of three-dimensional space. It has been studied by mathematicians and philosophers for almost three hundred years, 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 thus not a Euclidean space.

[edit] History

[edit] Vectors

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

$\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.$

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

$\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}; \mathbf{e}_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix},$

so the general vector a is

$\mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.$

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,

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

and calculate or define the angle between two vectors as

$\theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.$

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:

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

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.

[edit] Orthogonality and vocabulary

In the familiar 3-dimensional space that we live in 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 length measured along the w axis can be called spissitude, as coined by Henry More.

[edit] Geometry

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 polychora (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 polychora, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform polychora, analogous to the 13 semi-regular Archimedean solids in three dimensions.

Regular polytopes in four dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A₄	BC₄		F₄	H₄
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"), and a cylinder may be extruded to obtain a cylindrical prism. 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 ^[9]. 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]}.

[edit] Hypersphere

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.

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:

$\mathbf V = \begin{matrix} \frac{1}{2} \end{matrix} \pi^2 R^4$

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.^[10]

[edit] Cognition

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.^[11] 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."^[11] In another study,^[12] 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.^[13] 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).

[edit] Dimensional analogy

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.^[14]
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.

[edit] Cross-sections

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 balloon passed through a sheet of paper, a being on the paper would see a circle gradually grow larger, then smaller again. 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 sphere.^[15]

[edit] Projections

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.

[edit] Shadows

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 direction), 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.)

[edit] Bounding volumes

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.

[edit] Visual scope

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]}

[edit] Limitations

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 $C = 2\pi r$ and the surface area of a sphere: $A = 4\pi r^2$ . One might be tempted to suppose that the surface volume of a hypersphere is $V=6\pi r^3$ , or perhaps $V=8\pi r^3$ , but either of these would be wrong. The correct formula is $V = 2\pi^2 r^3$ .^[8]

[edit] See also

Wikisource has original text related to this article:

Flatland

Euclidean space
Euclidean geometry
4-manifold
Exotic R⁴
Dimension
Four-dimensionalism
Fifth dimension
Sixth dimension
Polychoron
Polytope
List of geometry topics
Block Theory of the Universe
Flatland, a book by Edwin A. Abbott about two- and three-dimensional spaces, to understand the concept of four dimensions
Sphereland, an unofficial sequel to Flatland
Charles Howard Hinton
Dimensions, a set of films about two-, three- and four-dimensional polytopes
List of four-dimensional games

[edit] References

^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 141.
^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., pp. 142–143.
^ Rudolf v.B. Rucker, editor Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, p. vii, Dover Publications Inc., 1980 ISBN 0-486-23916-0
^ Hinton, Charles Howard (1904). Fourth Dimension. ISBN 1-5645-9708. http://www.archive.org/details/fourthdimension00hintarch.
^ Gardner, Martin (1975). Mathematical Carnival. Knopf Publishing. pp. 42, 52–53. ISBN 0 14 02.2041 0.
^ Hermann Minkowski, "Raum und Zeit", 80. Versammlung Deutscher Naturforscher (Köln, 1908). Published in Physikalische Zeitschrift 10 104–111 (1909) and Jahresbericht der Deutschen Mathematiker-Vereinigung 18 75–88 (1909). For an English translation, see Lorentz et al. (1952).
^ C Møller (1952). The Theory of Relativity. Oxford UK: Clarendon Press. p. 93. ISBN 0-19-851256-2.
^ ^a ^b Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 119.
^ J. Scott Carter, Masahico Saito Knotted Surfaces and Their Diagrams
^ Ray d'Inverno (1992), Introducing Einstein's Relativity, Clarendon Press, chp. 22.8 Geometry of 3-spaces of constant curvature, p.319ff, ISBN 0-19-859653-7
^ ^a ^b Ambinder MS, Wang RF, Crowell JA, Francis GK, Brinkmann P. (2009). Human four-dimensional spatial intuition in virtual reality. Psychon Bull Rev. 16(5):818-23. doi:10.3758/PBR.16.5.818 PMID 19815783 online supplementary material
^ Aflalo TN, Graziano MS (2008). Four-Dimensional Spatial Reasoning in Humans. Journal of Experimental Psychology: Human Perception and Performance 34(5):1066-1077. doi:10.1037/0096-1523.34.5.1066 Preprint
^ John McIntosh's four dimensional maze game. Free software
^ Michio Kaku (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension, Part I, chapter 3, The Man Who "Saw" the Fourth Dimension (about tesseracts in years 1870–1910). ISBN 0-19-286189-1.
^ Gamow, George (1988). One Two Three . . . Infinity: Facts and Speculations of Science (3rd ed.). Courier Dover Publications. p. 68. ISBN 0-486-25664-2. http://books.google.com/books?id=EZbcwk6SkhcC., Extract of page 68

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