''Sphereland: A Fantasy About Curved Spaces and an Expanding Universe'' is a 1965 translation of , a 1957 novel by

Dionys Burger Dionys Burger (10 July 1892, Amsterdam - 19 April 1987) was a Dutch secondary school physics teacher and author of the novel ''Sphereland ''Sphereland: A Fantasy About Curved Spaces and an Expanding Universe'' is a 1965 translation of , a 1957 no ...

, and is a sequel to ''

Flatland ''Flatland: A Romance of Many Dimensions'' is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dime ...

'', a novel by "A Square" (a pen name of

Edwin Abbott Abbott Edwin Abbott Abbott (20 December 1838 – 12 October 1926) was an English schoolmaster, theologian, and Anglican priest, best known as the author of the novella ''Flatland'' (1884). Biography Edwin Abbott Abbott was the eldest son of ...

). 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

Circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

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 (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...

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 A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

. He eventually makes friends with the grandson of A Square, A

Hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

, 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 A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, 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 () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...

and

radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...

. 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.

Adaptations and parodies

In 2012 the creators of '' Flatland: The Movie'' released an educational sequel film, ''Flatland 2: Sphereland'' based in part on Burger's ''Sphereland'' that dealt with the topics of curved spaces, symmetry and congruence, an expanding universe, and the multiverse. Actors

Kristen Bell Kristen Anne Bell (born July 18, 1980) is an American actress. Beginning her acting career by starring in stage productions while attending the Tisch School of the Arts at New York University, she made her Broadway stage debut as Becky That ...

Danny Pudi Daniel Mark Pudi (born March 10, 1979) is an American actor. His roles include Abed Nadir on the NBC sitcom ''Community'' (2009–2015), for which he received three nominations for the Critics' Choice Television Award for Best Supporting Actor in ...

Michael York Michael York OBE (born Michael Hugh Johnson; 27 March 1942) is an English film, television and stage actor. After performing on-stage with the Royal National Theatre, he had a breakthrough in films by playing Tybalt in Franco Zeffirelli's ''Ro ...

Tony Hale Anthony Russell Hale (born September 30, 1970) is an American actor and comedian. He is known for his role in the Fox comedy series ''Arrested Development'' as Buster Bluth. Hale played Gary Walsh on the HBO comedy ''Veep'' from 2012 until its ...

Danica McKellar Danica Mae McKellar (born January 3, 1975) is an American actress, mathematics writer, and education advocate. She played Winnie Cooper in the television series ''The Wonder Years'' from 1988 to 1993, and since 2010 has voiced Miss Martian in th ...

, and

Kate Mulgrew Katherine Kiernan Maria Mulgrew (born April 29, 1955) is an American actress and author. She is best known for her roles as Captain Kathryn Janeway on '' Star Trek: Voyager'' and Red on '' Orange Is the New Black''. She first came to attention ...

provide voices for the 36-minute animated movie.

References

{{Flatland 1965 novels 1965 science fiction novels Dutch novels adapted into films Fictional dimensions Novels about mathematics Sequel novels

Plot summary

Adaptations and parodies

See also

References