We perceive three spatial dimensions, but theoretical Mathematics is easily capable of dealing with many more. Conventional graphical analysis frequently represents Time as a dimension, encouraging consideration of it as the "fourth dimension". The possible existence of Parallel Worlds displaced from ours along a fourth spatial dimension (in the same way that a series of two-dimensional universes might lie next to one another like the pages of a book) is a popular hypothesis in sf, and such worlds are frequently referred to as "other dimensions". The Cosmology of Einstein's General Theory of Relativity (1916), which proposes a four-dimensional model of the Universe in which the notions of space and time are collapsed into a single "spacetime continuum", offered considerable encouragement to sf notions of a multidimensional Universe or Multiverse. Many modern occultists and pseudoscientists have followed in the tracks of Johann Zöllner (1834-1882), author of Transcendental Physics (1865), who borrowed mathematical notions to "justify" the idea of the "astral plane" beloved by spiritualists and Theosophists (see Theosophy). J W Dunne used the notion to explain prophetic dreams, eventually constructing a theory of the "Serial Universe", and P D Ouspensky (1878-1947) built a more complex model of the Universe in which time "moves" in a spiral and there are six spatial dimensions.
The possible dimensional limitations of human existence and perception were dramatized by Edwin A Abbott in Flatland (1884) as by "A Square", in which Flatland is a world of two-dimensional beings, one of whom is challenged to imagine our three-dimensional world – encouraging readers, by analogy, to attempt to imagine a four-dimensional world. The challenge was taken up by C H Hinton, whose many essays on the subject attempt to "explain" ghosts and to imagine a four-dimensional God from whom nothing in the human world can be hidden. In his story "An Unfinished Communication" (in Stella and An Unfinished Communication, coll 1895) the afterlife involves freedom to move along the time dimension (see Time Travel) to relive and reassess moments of life; he also wrote a Flatland novel, An Episode of Flatland (1907). H G Wells borrowed Hintonian arguments to "explain" the working of the device in The Time Machine (1895). The eponymous figure of E V Odle's The Clockwork Man (1923) could perceive many dimensions when working properly, but while malfunctioning could do no more than flutter back and forth in time, offering the merest hint of the quality of multidimensional life. Algernon Blackwood's "The Pikestaffe Case" (in Tongues of Fire, coll 1924) attempts to evoke the non-Euclidean geometry of a dimensional trap lurking within a mirror.
Early Genre-SF writers who found the notion of dimensions fascinating included Miles J Breuer, most notably in "The Appendix and the Spectacles" (December 1928 Amazing) and "The Captured Cross-Section" (February 1929 Amazing); Donald Wandrei in "Infinity Zero" (October 1936 Astounding); and Nelson S Bond in "The Monster from Nowhere" (July 1939 Fantastic Adventures), which makes play with the fact that a four-dimensional entity's various limbs or digits could appear as separate solid objects as the creature intersects our 3D world. (Wandrei's "The Monster from Nowhere" [23 November 1935 Argosy] was wrongly cited as a dimensions-related story in past editions of this encyclopedia, almost certainly because the above-mentioned Bond story with the identical title is wrongly credited to Wandrei in the first edition of The Best of Science Fiction [anth 1946] edited by Groff Conklin.) In E E "Doc" Smith's Skylark of Valeron (August 1934-February 1935 Astounding; 1949) the heroes briefly enter a four-dimensional reality, and in Clifford D Simak's "Hellhounds of the Cosmos" (June 1932 Astounding), 99 men enter the fourth dimension in a single grotesque body to fight a four-dimensional Monster. E E Smith's Lensman universe features various Aliens whose existence in ultra-frigid environments depends on their "extension into the hyper-dimension", as Smith phrases it in First Lensman (1950). In lower dimensionality, Madeleine L'Engle's A Wrinkle in Time (1962) features a brief, uncomfortable visit to a two-dimensional planetary reality.
As sf became more sophisticated, slick trickery often replaced rigorous thought. Henry Kuttner's and C L Moore's classic "Mimsy Were the Borogoves" (February 1943 Astounding) as by Lewis Padgett features toys from the far future (see Toys in SF) which educate children into four-dimensional habits of thought which remain carefully unexplored: like most stories of the period, this uses dimensional shenanigans casually to tie up its plot with a neat knot. Similarly, Mark Clifton's "Star, Bright" (July 1952 Galaxy) and Alan E Nourse's The Universe Between (stories March, September 1951 Astounding; exp fixup 1965) both take it for granted that comprehending the fourth dimension will grant special powers, here Teleportation. The "adult" treatment in Ian McEwan's "Solid Geometry" (July 1974 The New Review) draws an analogy between paper-folding and human sexual positions, climaxing with the folding of a woman into other-dimensional nothingness. Greg Bear's "Tangents" (January 1986 Omni) is a knowing and sophisticated homage to this whole class of stories.
The mathematical discipline of topology has inspired several dimensional fantasies: Möbius strips feature in Martin Gardner's "No-Sided Professor" (January 1947 Esquire) and "The Island of the Five Colors" (in Future Tense, anth 1952, ed Kendell Foster Crossen), Theodore Sturgeon's "What Dead Men Tell" (November 1949 Astounding), Arthur C Clarke's "The Wall of Darkness" (July 1949 Super Science Stories), A J Deutsch's "A Subway Named Möbius" (December 1950 Astounding) and Homer Nearing Jr's "The Hermeneutical Doughnut" (in The Sinister Researches of C.P. Ransom, coll 1954). The Klein bottle (a one-sided self-intersecting surface with no edges) and the tesseract or hypercube (the four-dimensional analogue of a 3D cube) feature in "The Last Magician" (January 1953 F&SF) by Bruce Elliott, "– And He Built a Crooked House" (February 1941 Astounding) by Robert A Heinlein and "Star, Bright" (July 1952 Galaxy) by Mark Clifton. Occam's Razor (1957) by David Duncan also deploys some topological jargon – expressed through lively discussion of the properties of soap films stretched on wire frames – to shore up its dimensional speculations. George Gamow's popularization of ideas in modern physics, Mr Tompkins in Wonderland (coll 1939), dramatizes certain odd situations very well (although its contents are fictionalized didactic essays rather than stories).
The notion that Spaceships might make use of a fourth-dimensional Hyperspace in order to evade the limiting velocity of light is very common in sf, having been initially popularized by Isaac Asimov among others, but few stories actually attempt to describe it. It is usually imagined as a chaotic environment which utterly confuses the senses, as in Frederik Pohl's "The Mapmakers" (July 1955 Galaxy), Clifford D Simak's "All the Traps of Earth" (March 1960 F&SF), Madeleine L'Engle's above-cited A Wrinkle in Time (which invokes the tesseract, though very loosely, as a Faster Than Light travel facilitator) and Bob Shaw's Night Walk (1967). In Larry Niven's Known Space sequence, hyperspace appears or fails to appear as a sanity-threatening "Blind Spot" which human and other visual systems edit out. The dimensional chaos that might be associated with Black Holes has received closer attention, though these too are most often used as Wormholes permitting very long journeys to be taken more or less instantaneously. Among the more effective representations of experience in dimensionally distorted environments are Norman Kagan's "The Mathenauts" (July 1964 If), David I Masson's "Traveller's Rest" (September 1965 New Worlds) and Christopher Priest's Inverted World (1974). Douglas Adams deals humorously with the notion in one Space Flight sequence of The Hitch Hiker's Guide to the Galaxy (1979), whose imagery is comically surreal.
In the 1980s C H Hinton's ideas were entertainingly revived by Rudy Rucker, who has used dimensional and other Mathematics very extravagantly in a number of his novels and short stories, including the afterlife fantasy White Light (1980), the comedy of fourth-dimensional intrusions The Sex Sphere (1983) and many of the shorter pieces first published in The 57th Franz Kafka (coll 1983) and reprinted, with others, in Transreal! (coll 1991). Rucker is perhaps the only modern author to have answered the challenge of "A Square" with authentic verve and authority, but A K Dewdney's The Planiverse (1984) is an interesting drama-documentary about a two-dimensional world whose topography echoes Hinton's Flatland rather than Abbott's: this addresses such seemingly insuperable 2D engineering difficulties as building houses with working doors, and even a plausible steam-engine. More in the arena of Imaginary Science, Liu Cixin's Santi (May-December 2006 Kehuan Shijie; 2007; trans Ken Liu as The Three-Body Problem 2014) includes the bizarre notion of dimensionally unpacking a single proton into a vast 2D surface upon which the circuitry of a super-Computer is inscribed to produce – after restoration to 3D particle size – an inimical AI-controlled proton.
Vague references to other or higher dimensions – without concern for the mathematical aspect – have long been used in sf as a loose synonym for Parallel Worlds. Thus Superman impish tormenter Mister Mxyzptlk hails from a fantasy "fifth dimension", and Aliens from "higher dimensions" can be found as late in the twentieth century as Walter Jon Williams's Knight Moves (1985).
Relevant theme anthologies include Fantasia Mathematica (anth 1958) and The Mathematical Magpie (anth 1962), both edited by Clifton Fadiman, Science Fiction Adventures in Dimension (anth 1953) edited by Groff Conklin, and Mathenauts: Tales of Mathematical Wonder (anth 1987) edited by Rudy Rucker. [BS/DRL]
see also: Antimatter; Cthulhu Mythos; Invasion; Science and Sorcery.
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