Dimension

This article is about the geometrical concept of dimension. For a plane of existence, see plane.

A dimension is the number of degrees of freedom of a particular space or object, or the number of independent coordinates necessary to specify a particular point on it. A line has one dimension, because there is only one possible direction (or its opposite) to move along it, and any point on it can be described by one coordinate. So too a circle: one can only move clockwise or counterclockwise, and any point can be uniquely described by specifying its angle relative to some reference point. The interior of a circle, on the other hand, is two-dimensional, as is the surface of a sphere; the interior of a sphere is three-dimensional, as are humans and the material objects that surround them.

In some senses, an object's dimensionality—its number of dimensions—does not have to be an integer. It has been proposed that a degree of freedom be considered half a dimension if movement in one direction is possible but not the opposite—so, for instance, a space in which it is possible to move north, west, and east but not south would have one and a half dimensions—but this idea has not caught on. More widespread is the idea of fractional dimensions to characterize certain objects that exhibit self-similarity at different scales. An object with such a non-integer dimension is called a fractal.

A single point is considered to have zero dimensions, since there are no degrees of freedom within it, and no coordinates are necessary to specify a point within it (since it only comprises one point). Negative dimensions are undefined, although by some conventions the empty set is considered a −1-dimensional space.

Manifolds

There are some simple types of manifold that have analogues in all dimensions. An n-dimensional version of one of these manifold families is specified by prefixing the number n (generally followed by a hyphen) to the general name of the family, although there are often separate names for the two-, three-, and often four-dimensional members. (The general name for the three-dimensional version (or, in the case of the plax, the two-dimensional version), supplemented by the prefix "hyper-", is often used by three-dimensional beings to refer to the general family of objects.) Perhaps the simplest of all is the plax, or hyperplane, which is merely a flat space (that is, a space with no or minimal curvature) embedded within a space of higher dimension. A 1-plax is simply a line; a 2-plax is a plane, and a 3-plax is a realm. A 4-plax is called a flune. Another very simple shape is the cog, or hypersphere, which comprises all points a set distance (called the cog's radius) from a particular point. A 1-cog is a pair of points (a brace); a 2-cog is a circle; a 3-cog a sphere; and a 4-cog a glome.

Still another common category of manifold is the polytope. An polytope is a finite region in n dimensions bounded by finitely many (n−1)-dimensional places. Thus a one-dimensional polytope is a region in a line bounded by two points, which is to say a line segment. A two-dimensional polytope is a polygon, a three-dimensional polytope is a polyhedron, and so on. An n-dimensional polytope comprises a number of (n−1)-dimensional polytopes, its "facets", connected by their (n-2)-dimensional facets. A polytope is "regular" if all its facets are themselves regular (for this purpose, all line segments are considered regular), and if all adjacent pairs of facets meet at equal angles. In two dimensions, there are infinitely many different regular polygons—for all integers n>2, there exists a regular polygon with n sides. (More than one, in fact, for n>4, if one counts stellated polygons.) In three dimensions, there are only five convex regular polyhedra, plus four concave stellated ones. In four dimensions, there are six convex polychora, plus ten stellated ones. Above four dimensions, only three regular polytopes exist in each dimension, all of which are convex, and each of which belongs to a family with corresponding members in all dimensions above two: the simplex, the teerling (or hypercube), and the orthoplex.

Although all of the above manifolds, whether or not they are themselves flat, can be embedded in a flat space of one higher dimensionality, that is not the case for all manifolds. There exist n-manifolds that can only be embedded without self-intersection in spaces of n+2 or more dimensions. Two simple examples are the bleson and the Klein bottle, both of which are two-dimensional but neither of which can exist in flat three-dimensional space without self-intersection. (Three-dimensional representations of Klein bottles are fairly common, but all involve the bottle's surface intersecting itself.) Higher-dimensional examples also exist; the bleson, for instance, has a three-dimensional analogue in the ploce, and in general the n-dimensional equivalent is called a plegma.

Snarls and snares

One familiar property of one-dimensional manifolds embedded in 3-space is that they can be knotted, contorted in such a way that, with their ends joined, they cannot be manipulated into circles by continuous transformations without passing one portion through another or removing them from their embedding. (Technically, a manifold that can be so transformed into a circle is still held to compose a special kind of knot called the "unknot".) This property does not exist in lower dimensions (there is no such thing as a knot in 2-space), and disappears in higher dimensions, as any knot can be easily untied without removing the manifold from its embedding. However, while one-dimensional manifolds cannot be knotted in higher dimensions, something similar can be done with two-dimensional manifolds in four dimensions, as counterintuitive as that may be to organisms residing in three-dimensional space, and with three-dimensional manifolds in five dimensions. These operations are known as torts and straggles, respectively; in general, the n-dimensional analogue of a knot is known as a snarl.

Related to snarls are snares, the n-dimensional version of what for one-dimensional manifolds are called links. A link exists when two or more separate closed one-dimensional manifolds are so arranged that it is impossible without passing one of them through another to move them onto opposite sides of an arbitrary plane—like the links in a chain. Essentially, a link is a knot involving more than one manifold. Like knots, links do not occur in dimensions higher or lower than three. However, while one-dimensional objects cannot be linked in higher dimensions, again there is an equivalent phenomenon with two-dimensional objects in four dimensions (the yoke), with three-dimensional objects in five dimensions (the shackle), and so on for still higher dimensions. (Contrary to what one might assume from generalizing from links, it is not necessarily the case that n-dimensional manifolds can be snared only in n+2-dimensional space. Rather, for instance, two 10-dimensional manifolds can be snared in any number of dimensions between 12 and 21, except, for some reason, for 17, and two 102-dimensional manifolds can be snared in 181-, 182-, or 183-dimensional space.)

Coordinate systems

While an n-dimensional object requires n coordinates to describe it, the choice of coordinate is not unique. For two-dimensional planes, for instance, there are two different coordinate systems in wide use: rectangular, or Cartesian, coordinates, and polar coordinates. Even here, such choices as scale and origin are arbitrary. In three dimensions, Cartesian coordinates, cylindrical coordinates, and polar coordinates all are commonly employed. Four dimensions yields the familiar Cartesian as well as trochilical, phalaggomatic, and glomeric coordinates, and so on. In each dimension, more exotic coordinate systems also exist, though less often used.

The standard names for the axes and coordinates vary. By convention, the rectangular coordinates in two dimensions are usually symbolized by x and y, and for physical objects are referred to as length and width. Three dimensions adds z and depth; four dimensions w and ginth. (Sometimes height is regarded as a dimension, replacing either length or depth.) The angle of polar, cylindrical, or trochilical coordinates is usually symbolized by the Greek letter θ, the additional angle of spherical or phalaggomatic coordinates by φ, and the third angle in glomeric coordinates by ψ (the distance in all of these systems from the origin or the appropriate axis being most usually called the radius, r.)

Symmetries and Transformations

In any dimension, certain transformations exist between spaces, and symmetries of objects and spaces unchanged by these transformations. While other more esoteric transformations also exist, the three most basic are those of translation, rotation, and reflection.

Translation

A translation is a transformation that moves every point in a space to another point a fixed distance away, in a fixed direction. Essentially, translation simply moves objects; it changes their position while maintaining their shape and orientation. Although finite objects cannot have translational symmetry—after they're moved, they necessarily end up in a different place—it is a property that can hold for infinite objects and spaces. It is possible for translational symmetry to hold only for specific distances and/or directions. An infinite cylinder, for instance, would have translational symmetry along its axis, but not in any other direction. An infinite grid would have translational symmetry along its grid lines, but only for distances that are exact multiples of the distance between grid lines.

Rotation

A rotation involves maintaining the distance of each point in an n-space from a particular (m−2)-plax, called the "cnodax", while changing the angle from the cnodax (relative to a fixed reference angle). In two dimensions, the cnodax is a point; for rotation in three dimensions, the cnodax is a line (referred to as the axis of rotation); in flunespace, one rotates about a plane, and in five-space a realm. Multiple rotations can duplicate the effect of any translation, but the reverse is not true. An object or space unchanged by rotation (other than rotation of multiples of 360 degrees, which will obviously leave any object unchanged) is said to have rotational symmetry. As with translational symmetry, rotational symmetry may apply only in particular directions and degrees; a sphere has perfect rotational symmetry in any direction, for instance, but a regular tetrahedron only has rotational symmetry through axes passing through the centers of two opposite faces or edges, and only for rotations of multiples of 120 degrees for the former and 180 degrees for the latter.

The rotations between two different parallel cnodaces differ only by a translation. This means that in planespace, there's fundamentally only one way to rotate; all points are parallel, so no matter what point an object is rotated about, ignoring translations it's the same as rotating it about any other point. Above two dimensions, though, the orientation matters. One can rotate about three independent axes in three dimensions; one can rotate about six independent (perpendicular) planes in four dimensions, and ten perpendicular realms in five. In general, there are [math]\displaystyle{ \binom{n}{2} }[/math] mutually perpendicular n-2-places in n-dimensional space, a particular application of the more general formula that there are [math]\displaystyle{ \binom{n}{n-m} }[/math] mutually perpendicular m-places in n-space.

Reflection

A reflection is a transformation that carries each point in an n-dimensional object or space through an m-plax referred to as the "speculum" to a point an equal distance from the speculum on the other side. If m−n is even, then the reflection is equivalent to a rotation, but if m−n is odd, then the reflection replaces an object with its mirror image. If an object and its mirror image are identical, then the object is said to possess reflectional symmetry, or to be amphichiral. Most often, it is assumed that m = n−1—that is, a reflection involves a speculum only one dimension lower than that of the reflected object or space. In two dimensions, one usually refers to reflection through a line, and in three dimensions through a plane. A reflection through a lower number of dimensions can always be reduced to a series of reflections through (n−1)-places. For instance, reflecting an object through a line in realmspace is equivalent to rotating it 180 degrees around that line, and to reflecting it through two perpendicular planes that contain that line.

As with rotations, the results of reflecting through two different parallel specula differ only by a translation. This means that, ignoring translations, all reflections in one dimension are equivalent, since the reflections can only be through a point, and all points are parallel. In two dimensions, reflections can be made through lines of infinitely many orientations. However, only those through orthogonal axes are independent; others can be emulated by combinations of these. Therefore, since there only can exist two mutually perpendicular axes in two dimensions, there are two possible independent reflections. In three dimensions, there are three, and in flunespace four, and so on... this is a special case of the formula [math]\displaystyle{ \binom{n}{n-m} }[/math] formula when n = m−1, since [math]\displaystyle{ \binom{n}{1} }[/math] is simply equal to n.

Multiple reflections can duplicate the effect of any translation or rotation.

Abstract dimensions

Dimensions need not represent physical space in which objects and life forms can exist, but may be applied in a more abstract sense as well to represent the degrees of freedom in a system. In classical dynamics, physicists sometimes find it useful to treat position and momentum as separate dimensions, and more generally in science similar "phase spaces" of any number of dimensions—even infinite—may be used to model the possible states of a system.

Perhaps the best-known such abstract dimension is time, which in relativity is treated as a dimension on par with that of physical space. This has become common knowledge on Earth to the extent that time is frequently thought of or referred to as "the fourth dimension". The idea of time as a fourth dimension didn't originate with relativity, actually; it was proposed in an article by Jean d'Alembert as early as 1754, and famously utilized by H. G. Wells in his well-known story The Time Machine. And, indeed, the concept of time has been proposed, and even used scientifically in different ways, in cosmoi other than Xi where relativity does not apply. Still, its use in relativity has lent the idea an air of scientific legitimacy on Earth that it may not have previously possessed, and has led to many people being convinced that time as the fourth dimension is a scientific fact.

Insisting that "the fourth dimension" can only refer to time, however, shows a gross misunderstanding of the principles involved. For one thing, the fact that time is treated mathematically as a dimension in relativity doesn't mean that it's a dimension in the same sense as the spatial dimensions—even the fact that some relativistic transformations convert time into space, or vice versa, is open to other physical interpretations. Even physicists don't treat time and space completely interchangeably, but refer to "spacelike" and "timelike" dimensions (and there has been at least one proposed Grand Unified Theory that involved the existence of two timelike dimensions!)—Hermann Minkowski may have said that "space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality," but this seems to have proved something of an exaggeration... scientists do refer to "spacetime", but that doesn't prevent them from also referring to space and time and treating them somewhat differently. But even if one does regard time as an actual dimension, it doesn't follow that it's necessarily the fourth dimension. It's a fourth dimension, if one regards it in addition to the familiar three spatial dimensions. But nothing in principle prevents there being more than three spatial dimensions, so it's quite possible to have a fourth spatial dimension and relegate time to the fifth, or sixth, or some higher number. For that matter, there's no particular reason why time has to be counted last; if one wants to number the dimensions, it's just as valid (albeit unusual) to count time as the first dimension, and length, width, and depth are the second, third, and fourth.

Still, while timelike dimensions and the dimensions of phase spaces may not apparently possess the same sort of immediate physical existence as the familiar spacelike dimensions, it may not be completely out of the question that it is possible somehow to treat them as such... to "transform coordinates", as Woleshensky said in Miles J. Breuer's short story "The Gostak and the Doshes", so that one is moving through time, or through some dimension of phase space, the same way one moves through space (and perhaps so interchanging the dimensions that what was formerly the spacelike dimension then, to the "rotated" individual, plays the part of the timelike or phasespace dimension he has rotated into). Exactly what this would entail is difficult to conceive—though that hasn't stopped some science fiction authors from trying.

Converting between dimensions

Moving an object from a lower dimension into a higher dimension, or portraying such a lower-dimensional object in a higher dimension, poses no problems. The lower-dimensional object can simply be embedded as is in a higher-dimensional space. If desired, it can even be given some finite thickness to increase its dimensionality.

The reverse process is not so simple. There is not, in general, an easy way of rendering a higher-dimensional object in a lower dimension without loss of information. It may, in principle, be possible—mathematically, the cardinality of ℝⁿ is equal to 2^ℵ₀ for all n∈ℕ^*, which implies that it is possible to find a bijective transformation between two spaces of different (positive) dimensionality, mapping spaces in a higher dimensionality faithfully and uniquely in a lower. However, the applicability of such a mathematical operation to the transformation of actual physical shapes is questionable. There are, however, a number of methods in common use to represent higher-dimensional objects in lower-dimensional spaces which are not unique or bijective but still may have their uses.

Probably the two most common methods of "converting" objects from higher dimensions to lower are projection and cross section. In the former, an n-dimensional object is essentially "flattened" into an m-dimensional image by choosing an m-dimensional image manifold and a series of space-filling (n−m)-places each of which touches the image and conflating all points in the object that lie within each plax into the single point where it intersects the image manifold. The chosen places are often parallel, though they need not be; among map projections used to project a sphere (such as the surface of the Earth) onto a plane (such as a map), some, such as equal-area cylindricalprojections, use straight rays radiating cylindrically from an axis; others, such as azimuthal projections (as well as the central cylindrical projection, rarely used because of its extreme distortion away from the equator) use rays radiating spherically from a point; and still others use more seemingly outré arrangements. For a cross section, only a "slice" through the object is portrayed—that is, the intersection of an object with a lower-dimensional plax.

Neither method is injective—that is, the results cannot be uniquely inverted. A two-dimensional circle could be a projection of a three-dimensional sphere, or an ellipsoid, or a cylinder, or a lemon, or a superegg, or a more exotic shape. A square with its diagonals could be a projection of a four-sided pyramid, of a regular tetrahedron, of an orthohedron, of a Steinmetz solid, or of a cube with two (neither adjacent nor opposite) corners lopped off (along planes that each pass through three of the cube's vertices), among many other possibilities. As for cross sections, a circle could be a cross-section of a sphere, a cylinder, an ellipsoid, a cone, half of a ring torus, or infinitely many other shapes. A regular hexagon could be a cross-section of a hexagonal prism, a six-sided pyramid, a cube, a cuboctahedron, a triakis tetrahedron, a disdyakis dodecahedron, a disdyakis triacontrahedron, and so on, and so forth.

These issues can somewhat be ameliorated by a section matrix, which comprises an arrangement of parallel cross sections. To produce a section matrix portraying an n-dimensional shape in m dimensions, the intersections are taken of the shape with an array of m-places, and these cross sections are then laid out in an (m−n)-dimensional array. If n > 2m, so that m−n > m, then it becomes necessary to have an array of arrays—or, if n > 3m, an array of arrays of arrays, and so forth. When n is much larger than m, the section matrix may be impractical, but when n and m are within a few dimensions of each other it provides a relatively clear way of "visualizing" a higher dimension. Of course, because the section matrix uses finitely many cross sections, it isn't a completely unambiguous representation of the higher-dimensional object, though the more cross sections that are used the more accurate the representation it will be. The section matrix is not, however, unique; by using cross-sections taken at different angles, very different-looking section matrices may be produced representing the same object.

Otherdimensional worlds

When the number of dimensions of a world is stated, it is usually only the readily perceptible macroscopic spacelike dimensions that are counted. For instance, if string theory is correct, Tamamna may have as many as ten spacelike dimensions, all but three of which are tightly "curled" and indiscernible on the macroscopic scale. Furthermore, like other worlds, it possesses a timelike dimension in addition to the spacelike ones. However, whatever other dimensions it may in some sense have, Tamamna is considered a three-dimensional world, because it has only three evident spacelike dimensions on the macroscopic scale; the motion of an object in Tamamna has three appreciable independent degrees of freedom.

All other worlds currently described in the Wongery are also three-dimensional. This is not, however, to say that worlds of other dimensionality cannot exist. They can, but the different dimensionality would lead to different laws of physics and very different ways of experiencing the world. In a two-dimensional world, for instance, it isn't possible to have a hole or channel go all the way through something without splitting it in two, which means any beings there with digestive tracts would have to either use the same orifice for ingestion and excretion, be pleotic, or have "bridges" spanning their digestive tracts some of which can open at a time to let their food pass but all of which cannot open at once. If gravity exists on the world in question, and its inhabitants are confined to the outer perimeter of a circle just as those of a three-dimensional planet are confined to the surface of a sphere, then their motion, barring climbing, jumping, or burrowing, would essentially be confined to one dimension; faced with a barrier they couldn't somehow go over or under, their only recourse in order to get to the other side would be to go all the way around the world. (The world need not be flat or simply connected, however, and sufficiently twisted and multiply connected world shapes may provide additional paths between points on the world's surface.)

While higher dimensions hold some surprises, there are some consistent rules that can be laid out. In an n-dimensional world, most inhabitants' visual field will actually be n−1-dimensional—with perhaps a limited sense of the nth dimension analogous to our depth perception, but no real ability to "see" in a full n dimensions unless they possess penetrating vision. Thus the inhabitants of a two-dimensional world would, as already mentioned, "see" everything as a line, though to them, of course, this would appear completely natural and normal. The inhabitants of a four-dimensional world, on the other hand, would "see" in three dimensions, and would be able at a glance to see the interior of a three-dimensional object, or the internal anatomy of a three-dimensional life form, just as we at a glance can see the entirety of a drawing or another two-dimensional surface. In terms of physics, the inverse square law pervasive in three-dimensional worlds would be replaced in n-space by a law concerning the power of (n−1). This is not to say that all physical forces would follow such a "law", but it would be quite common. Thus many forces in a two-dimensional world would be likely to follow a straight inverse law, in a four dimensional world an inverse cube law, in five an inverse tesseract law, and so on.

Traveling between worlds of different dimension is possible, but requires considerable alteration in the traveling object or life form (just as does the travel between worlds with different physical laws in general). Beings traveling to worlds of lower dimensionality often find such worlds stifling and unpleasantly restrictive; those traveling to higher dimensions find them confusing and overwhelming. However, there are exceptions, and some individuals do manage to easily adapt to other dimensions, perhaps enjoying the simplicity of lower dimensions or the freedom of higher. As with intercosmic travel or other travel between worlds of different physical laws, the conversion to fit into the new dimension may be "lossy" (particularly when traveling from higher to lower dimension), and generally the method of interdimensional travel also includes a mechanism to record the details of the traveler's original state so that it can be restored if he returns. If this is not the case, then it may not be possible to return to one's original dimension by the same means, and even if one does find a way back, he may arrive substantially altered in some way from when he set out.

See also

• Dimension on Wikipedia

External links

• Glossary for Hyperspace
• Multi-dimensional Glossary
• Dimension on Wolfram MathWorld