Five-dimensional space

In physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space.

Abstract five-dimensional space occurs frequently in mathematics, and is a legitimate construct. Whether or not the real universe in which we live is somehow five-dimensional is a topic that is debated and explored in several branches of physics, including astrophysics and particle physics.

In physics, the fifth dimension is a hypothetical extra dimension beyond the usual three spatial dimensions and one time dimension of Relativity. The Kaluza-Klein theory used the fifth dimension to unify gravity with the electromagnetic force; e.g. Minkowski space and Maxwell's equations in vacuum can be embedded in a 5-dimensional Riemann curvature tensor [1] . Kaluza-Klein theory now is seen as essentially a gauge theory with gauge group the circle group. M-theory suggests that space-time has eleven dimensions, seven of which are "rolled up" to below the subatomic level. Physicists have speculated that the graviton, a particle thought to carry the force of gravity, may "leak" into the fifth or higher dimensions which would explain how gravity is significantly weaker than the other three fundamental forces.

In 1993 the physicist Gerard 't Hooft put forward the holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature of path of a moving infinitesimal (test) particle. Hooft has speculated that the fifth dimension is really the spacetime fabric.

Contents

Five-dimensional geometry

Polytopes

Main article: 5-polytope

In five or more dimensions, only three regular polytopes exist. In five dimensions, they are:

  1. The 5-simplex of the simplex family, with 6 vertices, 15 edges, 20 faces (each an equilateral triangle), 15 cells (each a regular tetrahedron), and 6 hypercells (each a 5-cell).
  2. The 5-cube of the hypercube family, with 32 vertices, 80 edges, 80 faces (each a square), 40 cells (each a cube), and 10 hypercells (each a tesseract).
  3. The 5-orthoplex of the cross polytope family, with 10 vertices, 40 edges, 80 faces (each a triangle), 80 cells (each a tetrahedron), and 32 hypercells (each a 5-cell).

A fourth polytope can be constructed as an alternation of the 5-cube, and is called a 5-demicube, with half the vertices (16), bounded by alternating 5-cell and 16-cell hypercells.

Regular and semiregular polytopes in five dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A5 BC5 D5

5-simplex
5-cube
5-orthoplex
5-demicube

The 5-simplex is self-dual, and the 5-cube and 5-orthoplex are dual to each other. The 5-demicube can be constructed by removing alternate vertices of the 5-cube.

The following are three projected images of the edges of a 5-cube:

Hypersphere

A hypersphere in 5-space (also called a 4-sphere due to its surface being 4-dimensional) consists of the set of all points in 5-space at a fixed distance r from a central point P. The hypervolume enclosed by this hypersurface is:

V=\frac{8\pi ^2r^5}{15}

The "fifth dimension" in popular culture

In popular usage, the "fifth dimension" is often used to refer to unexplored or unknown aspects of the universe, and not necessarily to the mathematical concept of a 5-dimensional space. For example, the opening narration of The Twilight Zone begins: "There is a fifth dimension, beyond that which is known to man." In the fictitious universe of DC Comics, the "fifth dimension" is said to be the place from which Mister Mxyzptlk comes. In 1966, The Byrds released an album titled Fifth Dimension, using the fifth dimension as a metaphor for unexplored and unknown aspects of the universe and oneself. The 5th Dimension is the name of an American vocal music group popular in the late 1960s and early 1970s. In Hindu philosophy, the fifth dimension of love of the Divine is termed by the Gaudiya Vaisnavas as turyatita, the dimension of the soul's Soul. The original Doctor Who episode hints at the 5th dimension being key to the abilities of the TARDIS.

Other uses of the "fifth dimension" are closer to its mathematical meaning. For example, the novel The Boy Who Reversed Himself features 4-dimensional and 5-dimensional spaces, using the mathematical fact that a 3-dimensional object can be turned into its mirror image if additional spatial dimensions were available for it to rotate through. The characters in Madeleine L'Engle's novel A Wrinkle In Time use the fifth dimension as a "dimensional shortcut" to travel through space. A similar concept appears in the Powerpuff Girls episode Bring Back Jojo, where a creature unable to see higher dimensions takes a dimensional shortcut through a fifth dimension to travel through time. The Red Dwarf episode "Parallel Universe" refers to the fifth dimension as the space in which multiple four-dimensional spaces exist.

Not all references to the "fifth dimension" in the mathematical sense involve time travel or space travel; Douglas Adams' book Mostly Harmless advances the idea of the fifth dimension being probability.

In Mikhail Bulgakov's The Master and Margarita, valet Koroviev in an explanation to Margarita attributes the expansion of a small apartment into the size of a large auditorium to the fifth dimension.

In the "original" or "boxed" version of the Dungeons & Dragons roleplaying game rulebooks (discontinued in 2000), consisting of the Basic set, and then the Expert, Companion, Master and Immortals expansion sets, a five-dimensional model was proposed in the Masters set where characters from the ordinary plane had the first, second and third dimensions as their three-dimensional home, but other-dimensional beings called the third, fourth and fifth dimension home. Upon perceiving each other, each thought the other kind to be horribly deformed "demons".

In Family Guy, Mayor Adam West sends Alex Trebek, host of Jeopardy, to the fifth dimension by making him say his name backwards, commenting, "Only saying his name backwards can send him back to the fifth dimension where he belongs."

In a well-received 2010 CBS commercial for How I Met Your Mother, Accidentally on Purpose, Two and a Half Men, The Big Bang Theory, and The Late Show with David Letterman, an announcer states that "Everyone has 3-D, but only CBS has comedy in 5-D." This was a reference to the growing popularity of 4-D film in the later half of 2009, suggesting that certain programs on CBS feature 5-dimensional effects. By the end of the commercial, however, it is revealed that the letter D in 5-D does not stand for "Dimension," but stands for "Delightful, Delicious, Daring, Demented, and Dave."

See also

References

  1. ^ Embedding.pdf eq. 37

External links