Space: Dimensions
Space: Dimensions
| Topics: |  Cosmology General Relativity Astrophysics |
Introduction - Hidden Dimensions
We take it pretty much for granted that space has 3 dimensions to it, but the idea that space has a limited number of dimensions may not be an old one at all. In all of Greek mathematics, no references seem to have been directly made to these aspects of space, perhaps considered too obvious to state, even in Euclid's detailed systematization of plane geometry. John the Grammarian ca 575 AD seems to have been among the first to consider space as explicitly 3-dimensional, incorporeal and different from any body placed in it.
"...it is pure dimensionality void of all corporeality; indeed as far as matter is concerned, space and void are identical." —John the Grammarian
By the 1800's, the idea that space might have more than 3-dimensions had been considered and dismissed in a number of speculative essays beginning in the 6th century AD.
Early Essays on Multi-Dimensional Space
In the Commentaries by Simplicius he writes that "The admirable Ptolemy in his notebook 'On Distances' well proved that there are not more than three distances". Stifel (1486-1567) described "going beyond the cube just as if there were more than three dimensions...which is against nature. John Wallis (1616-1703) protests in his Algebra that
"...Length, Breadth and Thickness take up the whole of Space. Nor can Fansie imagine how there should be a Fourth Local Dimension beyond these Three." —John Wallis
Lagrange, in his 1796 book "Theory of Analytical Functions" had, however, refered to classical mechanics as a 4-dimensional geometry. So at least by the start of the 19th century, some manner of extension of the world into a 4-dimensional arena had received serious consideration, but only as a footnote to other more pressing issues in mathematics.
Arthur Cayley (1821 - 1895) in 1843 is among the first mathematicians known to have pioneered the study of spaces with dimensionalities greater than 3 as a serious mathematical subject. This caught the attention of such mathematicians as William Clifford who went on to develop higher-dimensional projective geometry. George Green (1793 - 1841) in 1833, and Augustine Cauchy (1789 - 1857) in 1847 had also made general comments such as "We shall call a set of n variables an analytical point...", which seemed to presage the concept of an N-dimensional manifold, but by the mid-1800's the subject of "hyperspace" was not really being rigorously considered.
N-Dimensional Geometry
This all changed abruptly when Georg Riemann gave his Ph.D. lecture in 1854 which was published in 1866 and translated to english by Clifford in 1873. By 1870, this subject received increasing attention so that by 1911 over 1830 references had accumulated in the literature on the subject of N-dimensional geometry. While N-dimensional geometry was being rapidly investigated, a number of other new, but odd, ideas made their appearance as well. The stage for this new level of discussion about space had been set by Carl Friedrich Gauss (1777 - 1855) and Georg Bernhard Riemann (1826 - 1866) who demonstrated that there was more to geometry than Euclid's postulates. In fact, Riemann discovered through his powerful new mathematical techniques that one could just as easily speak of the geometries of space for any dimensionality one wished to choose. Our 3-dimensional space, or at least its mathematical equivalent, was only one of an infinite number of logically self-consistent possibilities, and Euclidean space only one of a large number of geometric systems for describing the characteristics of these N-dimensional manifolds.
In the history of the exploration of space, Riemann's new non-Euclidean geometry was the first new way of thinking about space in 2000 years. But it could not have occurred without several thousand years of refinement in mathematical thinking beyond what the Ancient Greeks had enjoyed. By the middle of the 1800's, the idea that some Newtonian Absolute Space existed had become an irrelevancy in-so-far-as actual dynamical calculations were concerned. In the face of calculations that had to be performed, and solar system dynamics that had to be straightened out, very few pragmatic physicists worried much about these seemingly philosphical matters. Maxwell had pretty well adopted the relativist point of view that, "...We cannot describe the time of an event except by reference to some other event" and similarly for location. The existence of Absolute Space had evaded all means of experimantal detection, and Ernst Mach himself refers to "...the conceptual monstrocity of absolute space".
Whether space was measureably Euclidean or not had been investigated by Gauss ca 1827 by surveying the angles in a triangle formed by the mountains, the Brocken, the Hoher Hagen, and the Inselberg, with sides of 69, 85 and 107 kilometers. No deviation from a sum of 180 degrees was found which led Gauss to conclude that physical space was Euclidean. Riemann wasn't satisfied with merely enlarging the dimensional arena to physical space, he also challenged the need for imagining space to be well-behaved and 'flat' at the microscopic scale. In his 1854 paper, 'On the Hypotheses that Lie at the Foundations of Geometry', he conjectured that a flat 'Euclidean' geometry was only one possibility to the shape of space at its smallest scales. There seemed to be nothing that physically required this very special shape, and he proposed that space could have all manner of complicated connectivities between points which might render space at its smallest separations extraordinarly complex in its topology. Riemann's new view of space was a far-reaching conception that was far in advance of the rigid, homogeneous and static notion so common to Newtonian mechanics.
Only William Clifford (1845 - 1879) seemed to have appreciated the implications of Riemann's conjectures for unifying space, geometry and physics. In the annals of history, Clifford is most remembered as a popularizer of mathematics and physics, although he made significant contributions to the study of the topology of Riemann surfaces. A series of articles by J. J. Sylvester appearing in "Nature" in 1870 titled "A Plea for Mathematicians" mentioned that Clifford had by that time already speculated on the existence of spaces of more than 3 dimensions. This predates by three years Clifford's translation of Riemann's lecture into English in this same journal.
W. I. Stringham (1847 - 1909) of the University of California described pictures of projections of 4-D regular solids upon our 3-D space. The astronomer Simon Newcomb at Johns Hopkins University showed the possibility of turning a clossed shell inside out without tearing.
Felix Klein (1849 - 1925) showed that knots could not be tied in the 4th dimension; Gusippe Veronese (1854 - 1917) at Padua University proved that a body could be removed from a closed box without breaking its walls. The May 1, 1873 issue of "Nature" is of interest, not only for Clifford's translation of Riemann's famous lecture, but because it contains a curious article "On Space of Four Dimensions" written by G. F. Rodwell, F.C.S. In it he mentions that among others, Clifford
"..had indicated in some of [his] profoundest mathematical demonstrations that [he] possessed, 'an inner assurance of the reality of [4-dimensional] space'." — G. F. Rodwell, F.C.S.
The brief essay then describes how our perception of space is determined by our state of motion and the experiences to which we become familiar over time. He ends by speculating that the many different directions and magnitudes of motion of matter in the universe,
"...lead us to the very threshold of transcendental space, and once on the threshold, we may look wonderlingly beyond."
Dimensions Manifest at the Atomic Level
The possibility that space may have more than 3 dimensions was also raised by philosopher Charles Hinton (1853 - 1907) in 1887 who published a book "What is the Fourth Dimension?" It was a delightful little booklet that described in elegant and intuitively appealing prose how on the basis of various geometric analogies, 4-D objects would appear as solid bodies in our 3-D world. Most of the discussion centered on a demonstration that 4-D space was not such a radical idea at all, and simply a modest extension of what we know about the properties of ordinary points, lines, planes and volumes. Toward the end of the book, however, he offers a rather remarkable proposition. Where else would nature choose to manifest its 4-D character but at the atomic scale? He proposed that atoms may actually be thin, 4-D threads whose cross sections we can only perceive as microscopic, 3-D bodies. He went on to consider how,
"...it would probably be in the ultimate particles of matter that we should discover the fourth dimension, for [their] sizes in the three dimensions are very minute, and the magnitudes in all four dimensions would be comparable". — Charles Hinton
Time as the Fourth Dimension
By the time that Hinton's little book appeared, Victorian schoolmaster Edwin Abbott Abbott (1838 - 1926) had already published his famous book "Flatland" in 1884. Flatland told the story of creatures living in a 2-D world, and communicating with beings from the third dimension. Soon after this book appeared, H.G. Wells wrote the "Time Machine" in which the fourth dimension became identified with time itself, not an independent spatial direction. These speculations still had some of the trappings of legitimate scientific speculation, but they certainly do not exhaust the range of ideas that were also popular at this time.
The notion that spirits hailed from some other dimension can be found in the works by philosopher Henry More (1614 - 1687) who in 1671 proposed that since spirits must occupy space, they could only do so if they existed in the fourth dimension. As Rudy Rucker notes in his 1984 book "The Fourth Dimension," More defined a new term called spissitude to denote extension in the fourth spatial dimension. But it was astronomer Johan Zollner (1834 - 1882) from the University of Leipzig who really popularized this idea. He was an enthusiastic champion of the American Medium Henry Slade's assertion along these same lines. His book "Transcendental Physics of 1878" purported to give experimental evidence of the existence of a 4-dimensional spirit world.
General Relativity
The advent of general relativity in 1915 ultimately opened the door to exploring, at least mathematically, the consequences of adding additional dimensions to spacetime, and exploring their physical consequences. There were three episodes in this exploration that continue to show consideral promise today.
- Kaluza-Klein Cosmology ca 1925.
- Dimensionally-extended supergravity theory ca 1978.
- String Theory ca 1982.
- M-theory ca 1995.
External Links
- "Scientists May Soon Have Evidence for Exotic Predictions of String Theory" - NASA, "Beyond Einstein," Issued by Northeaster University, January 26, 2006.
Preview Image
A visual analog for the space-filling framework of hidden dimensions. (Source: Unknown-Sten Odenwald.)
Citation
Odenwald, Sten, Ph.D. (Contributing Author); Bernard Haisch (Topic Editor). 2009. "Space: Dimensions." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published January 5, 2008].
<http://www.cosmosportal.org/articles/view/137588/>
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