Sphereland
Encyclopedia
Sphereland: A Fantasy About Curved Spaces and an Expanding Universe is a 1965 novel by Dionys Burger
, and is a sequel to Flatland
, a novel by "A Square" (a pen name of Edwin Abbott Abbott
). The novel expands upon the social and mathematical foundations on which Flatland is based. It is markedly different from the first novel in that it has a more prosaic ending and treatment of society.
s (who are appointed as priests/leaders of Flatland due to their many sides, or an appearance thereof) do not take A Square's revelation about a third dimension
to be accurate, and A Square is ostracized by his community. Then after some time, society becomes more open to the ideas of Spaceland and, overall, to change and advancement. However, when a prominent surveyor finds a Triangle
with more than 180 degrees
, he is fired from his job and generally considered a crackpot, since such a construction is not possible in Euclidean geometry
. He eventually makes friends with the grandson of A Square, A Hexagon, because he is a mathematician and scientist. Together, they come upon a theory to explain the unusual measurements: they actually live on a very large sphere
, and the Triangles have more than 180 degrees due to being inscribed on a non-planar surface.
With help from the sphere from the first novel, they are able to prove this theory. However, the established scientific community is not able to comprehend the idea proposed by the two, and thus they do not attempt to enlighten Flatland. Furthermore, as the residents of Flatland advance, they begin to travel in space; they see distant worlds like their own, and the surveyor tries to find the distance between their world and these distant worlds, using trigonometry
and radar
. From his calculations, he and the hexagon determine that the universe is expanding; again they try to reveal this theory to the outside world, but again it is not accepted. Therefore, like his grandfather in the previous novel, the hexagon writes a book that is not to be opened until the theory of the expanding universe is discovered and accepted by others. Then they live an inferior existence without any more contact with the sphere.
are intended to encourage consideration of how higher dimensions would appear to us - specifically, the concept that the universe is the three-dimensional surface of a four-dimensional hypersphere
.
Dionys Burger
Dionys Burger was a Dutch mathematician and author of the novel Sphereland....
, and is a sequel to Flatland
Flatland
Flatland: A Romance of Many Dimensions is an 1884 satirical novella by the English schoolmaster Edwin Abbott Abbott. Writing pseudonymously as "A Square", Abbott used the fictional two-dimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture...
, a novel by "A Square" (a pen name of Edwin Abbott Abbott
Edwin Abbott Abbott
Edwin Abbott Abbott , English schoolmaster and theologian, is best known as the author of the satirical novella Flatland .-Biography:...
). The novel expands upon the social and mathematical foundations on which Flatland is based. It is markedly different from the first novel in that it has a more prosaic ending and treatment of society.
Plot summary
The CircleCircle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s (who are appointed as priests/leaders of Flatland due to their many sides, or an appearance thereof) do not take A Square's revelation about a third dimension
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
to be accurate, and A Square is ostracized by his community. Then after some time, society becomes more open to the ideas of Spaceland and, overall, to change and advancement. However, when a prominent surveyor finds a Triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
with more than 180 degrees
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
, he is fired from his job and generally considered a crackpot, since such a construction is not possible in Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. He eventually makes friends with the grandson of A Square, A Hexagon, because he is a mathematician and scientist. Together, they come upon a theory to explain the unusual measurements: they actually live on a very large sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, and the Triangles have more than 180 degrees due to being inscribed on a non-planar surface.
With help from the sphere from the first novel, they are able to prove this theory. However, the established scientific community is not able to comprehend the idea proposed by the two, and thus they do not attempt to enlighten Flatland. Furthermore, as the residents of Flatland advance, they begin to travel in space; they see distant worlds like their own, and the surveyor tries to find the distance between their world and these distant worlds, using trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
and radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
. From his calculations, he and the hexagon determine that the universe is expanding; again they try to reveal this theory to the outside world, but again it is not accepted. Therefore, like his grandfather in the previous novel, the hexagon writes a book that is not to be opened until the theory of the expanding universe is discovered and accepted by others. Then they live an inferior existence without any more contact with the sphere.
Purpose
As with Flatland itself, the Flatlanders' encounter with the third dimensionThree-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
are intended to encourage consideration of how higher dimensions would appear to us - specifically, the concept that the universe is the three-dimensional surface of a four-dimensional hypersphere
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
.
Editions in print
- Sphereland appears in one edition bound back-to-back with Edwin A. Abbott's Flatland, each novel thus having one of the two covers of the book; (1994), HarperCollins, ISBN 0-06-273276-5