Embedded Files
See also: Kaluza-Klein, Supergravity and String theory
Before we can have conversations about higher dimensions, we need to establish what a dimension even is. It is the minimum number of coordinates needed to specify a point within that space. For example: you need two points (x, y) to define a point in two-dimensional space. You need three points (x, y, z) to define a point in a three-dimensional space. So on and so forth... In this image, we can see representations of the first 4 dimensions starting with zero-dimensional point (line segment, square, cube and tesseract).
Euclid
Carl Friedrich Gauss
János Bolyai
Nikolai Lobachevsky
We had Euclidean geometry, prior to the 1800s with the work of Gauss, Bolyai and Lobachevsky, then later by Riemann.
Gauss claims to have discovered some notions of non-Euclidean geometry, however, never had his ideas published. The proposal of a consistent geometry that was not based on Euclid's axioms was significant for a number of reasons. The most obvious is perhaps that our description of space and time, general relativity, relies on precisely the non-Euclidean geometry of Bernhard Riemann.
Bolyai was a Hungarian mathematician and is regarded as one of the fathers of non-Euclidean geometry for his published work in 1832. The discovery of non-Euclidean geometry was freeing for mathematicians from the abstract mathematics that may or may not have anything to do with the physical world.
Lobachevsky is known for having proposed hyperbolic geometry. In hyperbolic geometry, the sum of the angels of a right triangle add up to the less than 180 degrees.
Bernhard Riemann
Bernhard Riemann, a German mathematician, it should be noted, was perhaps the first to consider geometries in higher dimensional space. Riemann felt that the flat 3-dimensional geometry of Euclid was based more on common sense and human experience than on actual mathematical logic. To Euclid, nothing can be 4-dimensional. There is no such thing. A point has 0 dimensions, a line has 1 dimension, a sheet or membrane would have 2 dimensions (x, y) and a solid construction would have 3 dimensions (x, y, z). That's it in the world of Euclid's geometry. Aristotle and Ptolemy also proposed that the 4th spatial dimension was not possible. Riemann was challenged by Carl Friedrich Gauss, whom he had met at the University of Gottingen, to re-propose the foundations of geometry. Sadly, this led to Riemann's nervous breakdown in 1854.
It was Riemann's lecture at the University of Gottingen and his essay: "On the Hypothesis Which Lie at the Foundation of Geometry." The lecture was eventually given and the work was published 12 years later in 1868. Riemann's idea was a metric. This metric was a collection of numbers at every point in space: a tensor. These numbers would describe how bent or curved that space was. Riemann proposed that 10 numbers at each point were necessary to describe the curvature of some 4-dimensional manifold. This worked, no matter how curved the manifold was. It seemed to work for any arbitrary kind of curvature. The greater the number on the tensor, the greater the curvature.
This will give birth to Riemannian geometry and to the mysterious 4th dimension used by Einstein in his theories of relativity. This is the birth of a new idea that will shock the world of physics: physical laws are simplified in higher dimensional space. Furthermore, Einstien's general theory of relativity will prove to be very successful at describing how gravitation is a consequence of the curvature of spacetime.
General relativity was so successful, in fact, that Einstein continued to work, in isolation, to incorporate electromagnetism into his theory of general relativity. This unified field theory would unite his theory of gravity with Maxwell's theory of electromagnetism. This was his quest for a classical unified field theory of fundamental interactions. This is a theory that would be able to explain all of the laws of nature, from atomic to cosmic scales. This theory of everything would have been the crowning achievement of Einstein's life. Einstein termed this sought after idea: the unified field theory. Einstein once summarized his quest for unification with the analogy of wood and marble. Marble was the beautiful world of geometry. Wood was the chaotic world of matter. Wood determined the structure of the marble. This was a problem for Einstein. The geometry of spacetime was determined by the presence of matter. Einstein wanted to create a universe of pure geometry. Sadly, after about 30 years of searching: Einstein was ultimately unsuccessful in his pursuit of unification, as he passed away in 1955.
Johann Zollner
Henry Slade
Slade claimed to be able to make right handed snail shells left handed and remove an object from a sealed bottle.
The first time that the idea of the 4th dimension began to circulate in the mind of the public was in 1877.
Henry Slade, was a "psychic" who claimed to be able to summon spirits from the 4th dimension. He was arrested for fraud. However, many eminent physicists came to his defense, one of them being: Johann Zollner, who among others, proposed that Slade's feats were possible if one could manipulate objects in the 4th dimension. Zollner's defense of Slade was made public all throughout 19th century London and drove much controversy.
In fact, Zollner's proposal was tested in 1877, that objects could be sent through the 4th dimension. Slade's skills as a medium were tested by Zollner in some controlled conditions. Of course today, we know that this kind of feat would require a miracle, or technology that supersedes our current capabilities. Most of the attempts were failed, however, Slade was able to fool Zollner in some...
At any rate, it is fascinating that the next dimension to be discovered was not one of space, as Zollner expected, however, one of time, as in Einstein's theories of relativity. This would not be known for several decades.
Charles Howard Hinton - 1880
Shadow of a tesseract or hypercube - the analogue of a cube to 4-dimensions
"Unraveled" tesseract
Charles Howard Hinton wrote an article in 1880. Hinton was fascinated with coming up with mathematical ways to visualize higher dimensional space, in particular, four-dimensional objects. Hinton coined the term "tesseract" to describe the analogue of a 4-dimensional cube. This is also known as a hypercube.
We cannot visualize the hypercube, however, can unravel it's components. This unraveled hypercube would unfold into a kind of cross geometry, composed of 3-dimensional constituent cubes. Sadly, we cannot grasp how these cubes fold into a hypercube with our 3-dimensional minds.
We can also say that, in the same way that a 3-dimensional cube casts a 2-dimensional shadow: the shadow of a 4-dimensional tesseract would be a 3-dimensional cube within a cube. The shadow of a hypercube would be a cube within a cube.
Edwin A. Abbott
Flatland
Edwin A. Abbott wrote the work Flatland: a Romance of Many Dimensions by a Square in 1884. Abbot was taking a stab at the prejudice in Victorian England.
Mr. Square is the protagonist. All of the other citizens are some sort of polygon.
Woman - are lines
Nobility - polygons
High priests - circles
The more sides one has, the higher your social rank. Also, it is forbidden to discuss the "3rd dimension". This is a swipe at those who refuse to believe in these other worlds.
Kaluza-Klein theory and Supergravity theory, although they were fascinating mathematical tricks, proved to not be realistic models of our physical world. They were nonrenormalizable. This means that they couldn't be used to make meaningful physical predictions and they were plagued with mathematical anomalies like infinities that render the theory useless.
Indeed, both of these theories had problems. However, a newer and even more unique theory of higher dimensions seems a bit more promising, as it has yet to be eother proved or disproved and can answer some serious questions about the inconsistency or antagonism between general relativity and quantum mechanics. That theory is known as string theory, and it exists in 10 (superstring theory) or 11 (M-theory) dimensions:
String theory demands the presence of extra dimensions for it's mathematical consistency. There are two ways that physicists believe that they may be hidden:
They may be compactified on a very small scale, perhaps, on the geometry of a Calabi-yau manifold.
Our universe, could be localized on a 4-dimensional brane, in a higher dimensional bulk or hyperspace.
Compactification in string theory
"We can also picture, that if superstring theory is correct, then at every point of 3-dimensional extended space, there are also 6 compactified dimensions, taking on the Calabi-yau geometry that strings are allowed to vibrate in. These dimensions would exist everywhere as a component of the spatial fabric. However, they are so tiny that we are completely unaware of their existence."
M-theory, however, has 7 compact dimensions, not 6. Thus the compact geometry is not a 6-dimensional Calabi-yau, however, in "this case one considers 7-dimensional manifolds of G2 holonomy. (Six-dimensional Calabi-Yau's have SU(3) holonomy.) The math is very challenging. Also, there are lots of issues if one wants to get realistic physics this way." This quote is from one of my email conversations with John H. Schwarz.
Braneworld scenario
M-theory, is the 11-dimensional counterpart of the 10-dimensional superstring theory. In M-theory, there is not only 1-dimensional strings, however, there can exist membranes of different dimensions. In the braneworld approach, however, our universe, is this 4-dimensional entity, embedded in a higher dimensional space. Our universe could be restricted to existing on something called a brane. A brane, is a concept in string theory, that generalizes the notion of a point-particle to higher dimensions, for example:
Point particles are called "zero-branes".
A string is a "one-brane". This is a 1-dimensional object defined by only its length.
A membrane is a "two-brane". This is because it is defined by it's length and it's width. A membrane can be embedded in 3-dimensional space, however, these surfaces themselves are 2-dimensional.
Our universe could be some sort of "three-brane". These are 3-dimensional objects with length, width and breadth.
Branes can also be generalized to higher dimensions. The variable we use to decide the number of dimensions is p. Thus, we call these p-branes.
In superstring theory, the 6 extra higher dimensions are wrapped into the tiny geometry of the Calabi-yau manifold. These are compactified dimensions that are too small to be observed. It would be impossible to enter these higher dimensions. They are too small. However, these extra dimensions in M-theory, are not necessarily small and compact. These extra dimensions, may be quite large. They may even be observable in the laboratory.
Thus, in the braneworld scenario, our universe is restricted to a 4-dimensional brane, embedded in a higher dimensional bulk, or, hyperspace. Our universe could be a membrane floating in a much larger universe. This opens up the possibility that there can be other branes, or even, other universes. These dimensions could be huge or even infinite in extent!
A pair of D-branes in string theory. The end points of open strings have to lie on D-branes. All of the forces, except for gravity, would be confined to the brane. Gravity can propagate through the bulk as closed strings.
How one could visualize multiple branes in a higher dimensional bulk or hyperspace.
This is a graphic that I produced based on how physicists make the extra dimensions of string theory an interesting feature:
Compactification: extra dimensions are curled up small and invisible to low energy experiments on Planck scale (10^-35 meters, the scale of quantum gravity and of string).
Calabi-yau: 6-dimensional geometry that satisfies these requirements. Named for Euginio Calabi and Shing-Tung Yau. It was shown to provied N=1 supersymmetry by Edward Witten, Andrew Strominger, Gary Horowitz and Philip Candelas in the mid-1980s.
Mirror symmetry: a duality between two seemingly different Calabi-yau manifolds.
Brane world scenarios: Our universe and the Standard Model (not the graviton) is localized on a 4-dimensional brane in a higher dimensional bulk or hyperspace. Gravity is free to propagate through the bulk and this could explain the hierarchy problem or the problem of the relative weakness of gravity to electromagnetism, and the strong and weak nuclear force.
Large extra dimensions: (the ADD model: Nima Arkani-Hamed, Savas Dimopolous and Gia Dvali). Extra dimensions could be larger than the Planck length, some millimeters in size.
Warped extra dimensions: Randall-Sundrum models; Lisa Randall, Raman Sundrum; RS1; Two branes, Gravitybrane and Weakbrane ;RS2: Only Gravitbrane
Google Sites
Report abuse