Exploring a Fourth-Dimensional World
Flatland, by Edwin Abbott, is a book in which a 3-D creatureattempts to explain his world to a Flatlander, a creature of onlytwo dimensions. Joshua Warach has transposed this theme in anattempt to explain a four-dimensional world to the reader. Thepaper was written for Math 17, a course in linear algebra andmultivariable calculus, taught by Prof Marianne Brown.
Geometry, the mathematics of figures, lines, boundaries, space, thought and logic, is a beautifully precise and rational science. It concerns the inter-relationships of physical, albeit hypothetical, entities. The knowledge of geometric insight, of the "scientific method," may well be utilized as a constructive asset in our complex society. In contrast, the human mind is a deep and intricate amalgam of instincts and diffuse emotions. Edwin Abbott's novel Flatland delineates the dichotomy between the intrinsic logic of geometry and the basic irrationality and emotionalism of the mind.
Flatland might easily be described as a novel of mathematical fantasy. The book describes the imaginary "reality" of "Flatland": relating the customs, traditions, structure, and life, of its two-dimensional inhabitants. Upon closer inspection, Flatland emerges as more than an amusing tale of science fiction. It confronts the juxtaposition of the logic of mathematics with the intrinsic nature of Man and his beliefs.
One of the inhabitants of "Flatland," as related in Edwin Abbott's narration, is able to surmount his prior beliefs and prejudices with the realization that there indeed exist geometrical dimensions beyond his own. Just as members of "Lineland" (1 dimension) are unable to conceptualize the existence of a 2-dimensional space, so too members of the 2-dimensional "Flatland" are hard pressed to imagine the existence of still higher dimensions, e.g. "Spaceland."
The one individual in "Flatland" who finally does reach mathematical enlightenment in his affirmation of the reality of 3 space is branded a heretic and meets with hostility and incarceration. In the face of mathematical proof and logical argument, prejudices and preconceptions become dominant. As with most novel ideas, that of another dimension cannot be accepted - not even by the alleged savants and guardians of the Zeitgeist. Mr. Abbott has touched upon a sensitive issue - one of paramount philosophical significance which continues to confront modern Man. The incorporation of new and unconventional wisdom into our Weltanschauung is a matter which society must ultimately cope with in a rational manner, divorced from emotion and jaundiced predisposition.
Apart from actual visual observation, at times a matter of some difficulty, the method of analogy may be utilized in the conceptualization of dimension. This method is delineated in Abott's Flatland. It utilizes the inherent reliability of geometrical and arithmetic progressions. For example; a point has no sides, a line as 2 "sides," a square has 4 sides, a cube has 6 boundaries, and the 4-dimensional analog should, therefore, have 8 "boundaries," being the fifth member in the arithmetic sequence: 0, 2, 4, 6, 8. One can construct such sequences, though their physical interpretation may be less clear.
In an attempt to examine the 4-dimensional cube, commonly referred to as a "tesseract" or "hypercube," we may construct a number of progressions:
hypercube point line square cube (tesseract) 2 4 6 8 boundaries 0 points lines squares cubes vertices 0 0 4 8 16 faces 0 0 1 6 24 edges 0 1 4 12 32 volume O a a2 A3 a4
(length) (volume) (area)
Though we may gain some insight into the nature of the hypercube, "visualization of the tesseract is another story," we are told by Kasner and Newman in Mathematics and Imagination.
The tesseract has often been drawn in the following manner, however: with the apology that "Our difficulties in drawing this figure are in no way diminished by the fact that a three-dimensional figure can only be drawn in perspective on a two-dimensional surface - such as this page - while the four-dimensional object on a twodimensional page is only a perspective of a perspective." (Kasner and Newman)
Let us now consider the geometry of yet another 4-dimensional figure: the "hypersphere." Just as the sphere might be thought of as being the composite of an infinite number of circles, so might the hypersphere be imagined as an infinite number of spheres. And it might look like this:
A third 4-dimensional figure might be called the "hyperpyramid," with its characteristics as follows:
triangular triangle pyramid hyperpyramid (?) boundaries 3 lines 4 triangles 5 pyramids vertices 3 4 5 edges 3 6 12 faces 1 4 ?
Our hypothetical hyperpyramid might "appear" as follows:
We have ascertained that it is possible to attempt the construction of a 4th-dimensional geometry, on the basis of a paradigm involving the first, second, and third dimensions. It should be noted that our 4th-dimensional constructions are based on a Euclidian model; that those rules governing 3-space will remain viable in 4-space.
The implications of a 4th dimension are already of demonstrated significance in such areas of human knowledge as physics. Just as an inhabitant of the 3rd dimension can "look into" the "insides" of the 2nd dimension, so might the 4th dimension represent a new perspective on the third dimension.
Our physical conception of the universe is rapidly changing as we realize the mathematics of relativity and of the 4th dimension. The concepts of "worm holes," bent-space, and "time-warp" will form an integral part of the physics of the future. The science fiction of the present portends the reality of the future.
We must proceed judiciously, lest our initial apprehensions and reluctance to delve into the realm of 4-space and higher dimensions stifles developing mathematics and scientific curiosity.
"However successful the theory of a four-dimensional world may be, it is difficult to ignore a voice inside us which whispers: 'At the back of your mind, you know that a fourth dimension is all nonsense.' I fancy that voice must often have had a busy time in the past history of physics."
In conclusion, Abbott's Flatland is more than parody. Aside from its amusing story-line, it presents us with a scheme which can be used in our extrapolation of four-dimensional geometric constructions. Abbott's analogy, however simplistic, is a useful basis for imagining the fourth dimension. But again, we must try to combat our initial skepticism regarding higher dimensions. Any attempt to deny the possibility of the 4th dimension represents a repression of knowledge. Mathematical dimension is but a model of the physical universe. The more sophisticated this model, the better our appreciation and understanding of the universe will be.
