Definitions

# Fourth dimension

In physics and mathematics, a sequence of n numbers can be understood as a location in an n-dimensional space. When n=4, the set of all such locations is called 4-dimensional space, or, colloquially, the fourth dimension.

Such a space differs from the familiar 3-dimensional space that we live in, in that it has an extra dimension, an extra degree of freedom. This extra dimension may be interpreted either as time, or as a literal fourth dimension of space, a fourth spatial dimension.

## The fourth dimension as time

Usually, when a reference is made to four-dimensional coordinates, it is the temporal interpretation which is meant. In this case, the four coordinates are understood to represent 3 dimensions of space plus 1 dimension of time. Such a space is called a Minkowski space or "(3 + 1)-space", and is the space used in Einstein's theories of special relativity and general relativity.

## The fourth dimension as space

Sometimes, the fourth dimension is interpreted in the spatial sense: a space with literally 4 spatial dimensions, 4 mutually orthogonal directions of movement. This is the space used by mathematicians when studying geometric objects such as 4-dimensional polytopes. To avoid confusion with the more common Einsteinian notion of time being the fourth dimension, however, the use of this spatial interpretation should be stated at the outset.

Mathematically, the 4-dimensional spatial equivalent of conventional 3-dimensional geometry is the Euclidean 4-space, a 4-dimensional normed vector space with the Euclidean norm. The "length" of a vector

$mathbf\left\{x\right\} = \left(p, q, r, s\right)$
expressed in the standard basis is given by

$| mathbf\left\{x\right\} | = sqrt\left\{p^\left\{2\right\} + q^\left\{2\right\} + r^\left\{2\right\} + s^\left\{2\right\}\right\}$

which is the natural generalization of the Pythagorean Theorem to 4 dimensions. This allows for the definition of the angle between two vectors (see Euclidean space for more information).

### Orthogonality

In the familiar 3-dimensional space that we live in, there are three pairs of cardinal directions: up/down (altitude), north/south (latitude), and east/west (longitude). These pairs of directions are mutually orthogonal: they are at right angles to each other. Mathematically, they lie on three coordinate axes, usually labelled x, y, and z. The z-buffer in computer graphics refers to this z-axis, representing depth in the 2-dimensional imagery displayed on the computer screen.

A space of four spatial dimensions has an additional pair of cardinal directions which is orthogonal to the other three. This additional pair of directions lies on a fourth coordinate axis perpendicular to the x, y, and z axes, usually labelled w. Attested terms for these extra directions include ana/kata (sometimes called spissitude or spassitude), vinn/vout (used by Rudy Rucker), and upsilon/delta. These extra directions lie outside (and indeed, perpendicular to) the three observable directions in our 3-dimensional world.

### Vectors

The fourth spatial dimension can be understood in terms of geometric vectors. A vector consists of a direction and a length (also known as its magnitude). It may be regarded as a description of how to get from one point to another by moving along a certain direction for a certain distance. A special vector, called the zero vector, is a vector of zero length. It may be regarded as not moving at all.

#### Vector operations

A vector may be scaled by changing its length while preserving its direction. This may be thought of as walking along the direction of a vector for a different distance than the original vector. A vector of negative length is equal to a vector of the equivalent positive length pointing in the opposite direction. This may be thought of as walking backwards while facing the direction of the vector.

The net motion resulting from following two vectors, attached end-to-end, is a third vector called the sum of the two vectors. For example, if one starts at some location A, follows the first vector and reaches B, and then, starting from B, follows the second vector to reach C, then the vector that starts from A and goes directly to C is the sum of the first two vectors.

#### Vector span

Given a set of vectors, they may be freely scaled and added to obtain new vectors. The set of all possible vectors obtained this way is called the span of the set of vectors. The span may be thought of as the set of all possible locations one can reach by traveling along some combination of directions (vectors) in the set.

A set of vectors is said to span a geometric object X if, starting from a single point in that object, every other point in the object can be reached just by scaling and adding vectors from the set.

#### Vector basis

The smallest set of vectors that spans an object X is called a basis of X. Not all sets of vectors are bases, because they may include redundant vectors. A vector is redundant if it can be obtained from the other vectors in the set by some combination of scaling and adding. For example, if there are two parallel vectors in the set, then one of them can be removed and every point in X can still be reached, since whatever can be reached by the removed vector can also be reached by the other vector which is parallel to it. Or, if one of the vectors is the sum of another two, then it may also be safely removed. The zero vector is always redundant, because it does not allow one to go anywhere beyond what one can already reach.

#### Dimensionality

We can obtain a basis for a geometric object X by removing all redundant vectors from any set that spans X. Depending on which vectors one starts with, one may obtain different bases that span X; however, it can be proven that all of these bases will always have the same number of vectors. This number is called the dimension of X. In other words, X is n-dimensional if a minimum of n of vectors are needed to span it.

Intuitively, the dimension of an object may be thought of as the number of independent directions one needs to travel in order to reach every point in it.

For example, a point is a zero-dimensional object. No vectors are necessary to span it since if one starts at the point, one has already reached all of it.

A line is a one-dimensional object. Starting at some point on the line, one needs a vector that points in the direction of the line in order to reach the other points on the line. Only one vector is necessary, since scaling it by different amounts allows one to reach any other point on the line.

A plane is a two-dimensional object. Given some starting point on the plane, at least two vectors, not parallel to each other, are needed to span it. With only one vector, only points that lie on a straight line can be reached. A second vector, not parallel to the first, is needed to move "sideways" to points on the plane outside that line. Only two directions are necessary, since one can move forwards (or backwards) along the first vector by different distances and then move sideways by different distances to cover every point on the plane. One may think of the plane as a "stack" of parallel lines; to get from one point to another in the two-dimensional plane, one first travels along the line in one direction, and then travels "across" the parallel lines in a second direction.

Space, as we perceive it, is three-dimensional. In order to reach some point in space, one not only needs to move forwards or backwards, and sideways; one needs to move upwards or downwards as well. In other words, a third vector is necessary to cover all of space. One may think of space as a "stack" of parallel planes: to travel from one point to another in space, one may move forwards or backwards along one direction, and then move sideways along a second direction, and finally move upwards or downwards in the third, vertical direction.

Four-dimensional space is a space where four independent directions are needed to cover all of it. Such a space may be visualized as a "stack" of many parallel three-dimensional spaces. To understand this concept, think of putting pieces of paper side by side. The sheets do not extend into the third dimension until one puts them on top of one another. In the same way, in order to reach into four-dimensional space, it is necessary to move in a new direction, a direction outside three-dimensional space. To reach each point in four-dimensional space, not only does one need to travel forwards and backwards, left and right, up and down, but also along a new pair of directions, ana and kata.

### Dimensional analogy

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.

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 being able to remove objects from a safe without breaking it open (by moving them across the third dimension), being able to see everything that from the two-dimensional perspective is enclosed behind walls, and remaining completely invisible by standing a few inches away in the third dimension.

Another example of this concept is seen in the Wii Video Game Super Paper Mario, where the protagonist is able to shift from a two-dimensional perspective to a three-dimensional one.

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 demonstrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.

#### 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 3-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 trapezoids 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 the 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.

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 each of the corners connected. Similarly, if the wireframe of a four-dimensional cube were lit from "above" (in the fourth direction), its shadow would be that of a three-dimensional cube within another three-dimensional cube.

#### 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 squares. 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. This helps one understand features of such projections that may otherwise be very puzzling.

#### 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. 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.

#### 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 $C = 2pi r$ 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 $V = 2pi^2 r^3$.

## Geometry

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

Just as in 3 dimensions, one may construct polyhedra from polygons, in 4 dimensions one may construct polychora (4-polytopes) from 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. In 3 dimensions, there are 13 Archimedean solids, whereas in 4 dimensions, there are 58 convex uniform polychora (64 including the regular polychora).

In 3 dimensions, one may extrude a circle to form a cylinder. In 4 dimensions, there are several different cylinder-like objects. One may extrude a sphere to obtain a spherical cylinder (a cylinder with spherical "caps"), or one may extrude a cylinder to obtain a cylindrical prism. One may also take the Cartesian product of two circles 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. 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. Another such surface is the real projective plane.