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The Fourth Dimension
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Fourth dimension
There are three conventional spatial dimensions: length (or depth),
width, and height, often expressed as x, y and z. x and y axes appear on
a plane Cartesian graph and z is found in functions such as a "z-buffer"
in computer graphics, for processing "depth" in imagery. The fourth
dimension is often identified with time, and as such is used to explain
space-time in Einstein's theories of special relativity and general
relativity. When a reference is used to four-dimensional co-ordinates,
it is likely that what is referred to is the three spatial dimensions
plus a time-line. If four (or more) spatial dimensions are referred to,
this should be stated at the outset, to avoid confusion with the more
common notion that time is the Einsteinian fourth dimension.
If time is the "fourth dimension", an additional spatial dimension would
be referred to as the fifth dimension. The implications of another
spatial dimension are now discussed. This would be orthogonal to the
other three spatial dimensions. The cardinal directions in the three
known dimensions may be referred to as up/down (altitude), north/south
(latitude), and east/west (longitude). When speaking of the fourth
spatial dimension, an additional pair of terms is needed. Attested terms
include ana/kata (sometimes called spissitude or spassitude), vinn/vout
(used by Rudy Rucker), and upsilon/delta.
Concepts
The fourth spatial dimension and orthogonality
A right angle is defined as one quarter of a revolution and "orthogonal"
(from the Greek) refers to co-ordinates or functions that are at right
angles to each other. Cartesian geometry arbitrarily chooses orthogonal
directions through space, which means that they add height. The fourth
dimension is therefore the direction in space that is at right angles to
these three observable directions.
Vectors
The fourth spatial dimension can be thought of in terms of vectors,
analogous to arrows, fixed from some single place in space which we call
the origin, that point to other places. These are called geometric
vectors.
A point is a zero-dimensional object. It has no extension in space, and
no properties. If one were to think of this point as a geometric vector,
like an arrow, it would have no length. This vector is called the zero
vector.
A line is a one-dimensional object. If we pick some nonzero vector in
some direction, this vector has some definite length. That vector has a
head at some point in space and a tail at the origin. If we think of
stretching that vector so it is twice as long, three times as long, and
so on and even stretching it backwards so it takes all possible lengths
it can (even zero length, to get the zero vector), we get a single line
with one dimension of length. All the vectors that describe points on
this line are said to be parallel to each other. Even though any line we
can draw must have some small thickness (so that we can see it), this
theoretical line does not.
A plane is a two-dimensional object. It has a finite length and breadth
but no thickness — somewhat like a sheet of paper (but paper too has
some thickness). Thinking of a plane in terms of vectors can be a little
more challenging. If we think of taking one vector and moving it so that
its tail is touching the head of the first and forming a vector with its
tail at the origin and the head at the head of the repositioned second
vector, we have a reasonable way of talking about adding vectors. If we
have two vectors that are not parallel, we can talk about all the points
we can reach by stretching either of the vectors (or not stretching
them), and, adding these vectors together, these points form a plane. We
say that the two vectors span the plane.
Space, as we perceive it, is three-dimensional. We can think of putting
a line together with a "stack" of planes. These planes are "stuck
together" like a sandwich, with the line passing through them like a
skewer. To get to some point in space, we can imagine traveling up the
line and then moving across the plane to the point. We then have three
vectors to think about, one to travel some distance up the line and two
to get to some point in space.
The fourth spatial dimension, then, can be described by "sticking
together" several three-dimensional spaces in a row. To get to some
point in the four-dimensional space, one travels along the
three-dimensional spaces, and also across the fourth dimension. The
total number of vectors involved is four.
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{x} = (p, q, r, s)
expressed in the standard basis is given by
\|\mathbf{x}\| = \sqrt{p^{2}+q^{2}+r^{2}+s^{2}}
which is the natural generalisation of the Pythagorean Theorem to 4
dimensions. This allows for the definition of the angle between two
vectors (see Euclidean space for more information).
Geometry with four spatial dimensions
In four spatial dimensions, Euclidean geometry provides for a greater
variety of shapes to exist than in three dimensions. Just as
three-dimensional polyhedrons are spatial enclosures made out of
connected two-dimensional faces, the four-dimensional polychorons are
enclosures of four-dimensional space made out of three-dimensional
cells. Where in three dimensions there are exactly five regular
polyhedrons, or Platonic solids, that can exist, six regular polychorons
exist in four dimensions. Five of the six can be interpreted as natural
extensions of the Platonic solids, just as the cube, itself a Platonic
solid, is a natural extension of the two-dimensional square.
The pentachoron is constructed out of 5 tetrahedrons for cells and 10
triangular faces, and is the four-dimensional analogue of the
tetrahedron. The tesseract, or hypercube, is made out of 8 cubic cells
and 24 squares, and is the four-dimensional hypercube. The tesseract's
dual, the 16-cell, is the equivalent of the octahedron, as they are both
cross-polytopes.
The 120-cell and 600-cell are duals of each other, and are analogous to
the dodecahedron and icosahedron, respectively. The 24-cell is the
unique regular polychoron in that it has no three-dimensional
equivalent.
There are also a large set of semiregular polychora, called convex
uniform polychoron, most of which can be derived from the 6 regular
forms above.
Just as the sphere, or 2-sphere, is a curved two-dimensional surface
made up of all points equidistant from a given central point in
three-dimensional space, the 3-sphere, a kind of hypersphere, is the
space containing all points equidistant to a given central point in
four-dimensional space. Every three-dimensional cross section of a
3-sphere is a 2-sphere.
Dimensional analogy
To make the leap from three spatial dimensions into four, a device
called dimensional analogy is commonly employed. Dimensional analogy is
studying how (n – 1) dimensions relate to n dimensions, and then
inferring how n dimensions would relate to (n + 1) dimensions.
For example, in Edwin Abbott's book Flatland, he writes about a square
that lives in a two-dimensional world, like the surface of a piece of
paper. A three-dimensional being has seemingly god-like powers from the
perspective of this square: such as being able to remove objects from a
safe without breaking it open (by moving them across the third
dimension), 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. 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.
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 removing
information about the third dimension. In this case, depth is removed
and replaced with indirect information. The retina of the eye is a
two-dimensional array of receptors but it can allow the brain to
perceive the nature of three-dimensional objects using indirect
information (such as shading, foreshortening, binocular vision etc.).
Artists use perspective to give 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 then 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 using indirect information contained in the
images it receives in its retina. Perspective projection from four
dimensions produces similar effects as in the three-dimensional case,
such as foreshortening. This adds four-dimensional depth to these
three-dimensional pictures.
Dimensional analogy also helps in understanding such projections. 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 mathematically: 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 in understanding features of such projections that
may otherwise be very puzzling.
Likewise the concept of shadows can help us better understand the theory
of four dimensions. If you were to shine a light on three dimensional
object, it would cast a two dimensional shadow. Therefore light on a
two-dimensional object would cast a one-dimensional shadow (in a
two-dimensional world), 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. This idea can be used in the other direction; light
on a four-dimensional object would cast a three-dimensional shadow.
As an example of this, imagine that light is shone down through a
wireframe cube onto a flat surface. The shadow that results is that of a
square within a square with each of the corners connected. Similarly, if
a four-dimensional cube were lit "from above", its shadow would be that
of a three-dimensional cube within another three-dimensional cube.
Being three-dimensional we are only able to see the world with our eyes
in two dimensions; a four-dimensional being would see the world in
three. Thus it would be able, for example, to see all six sides of an
opaque box simultaneously. Not only so; it would also be able to see
what was inside the box at the same time, just like in Flatland, where
the sphere sees objects in the two-dimensional world and everything
inside them simultaneously. Analogously, a four-dimensional viewer would
see all points in our 3-dimensional space simultaneously, including the
inner structure of solid objects and things obscured from our
three-dimensional viewpoint.
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 = 2πr and the surface area of a sphere:
A = 4πr2. One might be tempted to suppose that the surface volume of a
hypersphere is V = 6πr3, or perhaps V = 8πr3, but either of these would
be wrong. The correct formula is V = 2π2r3.
The "fourth dimension" in popular culture
* Fourth Dimension is an album by Power Metal band Stratovarius
* The fourth dimension has been a subject of popular fascination since
at least the 1920s. See Into the Fourth Dimension (1926) by Ray
Cummings, the comic Eugene the Jeep or "—And He Built a Crooked House—"
by Robert A. Heinlein
* Donnie Darko uses the Fourth Dimension as a plot for Time Traveling.
The reference is related to water being a fourth dimensional tool for
time traveling.
* Alan Moore's graphic novel From Hell uses the fourth dimension as a
reference to the insanity of Jack the Ripper's character.
* Star Ocean: Till the End of Time uses the Fourth Dimension as
"reality".
* Cube 2: Hypercube (2002), the second in the cult-classic Cube series,
is set with the characters in a booby-trapped tesseract-shaped series of
rooms.
* Slaughterhouse-Five by Kurt Vonnegut features space-aliens that exist
along Jupiter and its moons who refer to the fourth dimension as time
and space.
* The Time Traveller in "The Time Machine" by H.G. Wells identifies time
as a fourth dimension, as does the Doctor from the first episode of
Doctor Who.
* On Jimmy Neutron, the title character has a small cube (which he calls
a hypercube), which serves as a portal to the fourth dimension. He uses
this purely for storage.
* The videogame Blinx: The Time Sweeper refers to itself as "The World's
First 4D Action Game", with the player having control over the game's
flow of time. Many other games with similar time-bending abilities (such
as Prince of Persia: The Sands of Time and Viewtiful Joe) or internal
clock coordination (as in Animal Crossing and Metal Gear Solid 3: Snake
Eater) are often referred to as "4D games."
* In the novel A Wrinkle in Time, the fourth dimension represents time,
as the first three represent length, width, and depth.
* In the TV-series Threshold, an alien race - who are staging an
invasion on earth - came to earth using spaceships who intersect the
fourth dimension.
* Several references to the fourth dimension in the science fiction
trilogy Back to the Future, like when Doc says "Marty, you're not
thinking fourth dimensionally!"
* In William Sleator's "The Boy Who Reversed Himself," the main
characters get lost in the fourth spatial dimension, where they
encounter highly intelligent beings who are represented by three
dimensional cross sections of themselves, which is all that can be
perceived of them by the book's three dimensional characters.
* In Michael Atkinson's 2006 Science Fiction novella "Combing Back
Through Time", a history recording probe "combs" space around itself in
order to regress in the Fourth Dimension.
* Most motion simulator attractions use 4-D as a marketing term,
referring to the moving seats as the fourth dimension.
Further information: Motion Simulator
* In the videogame Mother, the main character can use a special PK
ability called "Fourth Dimension Slip" to instantly get out of any
battle.
* In Frank Herbert's "Dune" the fourth dimension is used when folding
space to allow instantaneous travel through space.
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