''Flatterland'' is a 2001 book written by mathematician and science popularizer Ian Stewart about

non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

. It was written as 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 ...

'', an 1884 novel that discussed different dimensions.

Plot summary

Almost 100 years after A. (which we find out stands for Albert) Square's adventures that were related in ''Flatland'', his great-great-granddaughter, Victoria Line (Vikki), finds a copy of his book in her basement. This prompts her to invite a sphere from Spaceland to visit her, but instead she is visited by the "Space Hopper" (a character looking somewhat like the "

Space Hopper A space hopper (also known as a moon hopper, skippyball, kangaroo ball, bouncer, hippity hop, hoppity hop, sit and bounce, or hop ball) is a rubber ball (similar to an exercise ball) with handles which allow one to sit on it without falling off. ...

" children's toy with a gigantic grin, horns and a spherical body). The Space Hopper, more than being able to move between Flatland and Spaceland, can travel to any space in the ''Mathiverse'', a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of all imaginable worlds. After showing Vikki higher dimensions, he begins showing her more modern theories, such as fractional dimensions and dimensions with

isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...

Topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

are also discussed, as well as the Projective "Plain" (complete with intersecting "lions") and the

quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...

level. Hopper and Victoria also visit the Domain of the Hawk King to discuss

time travel Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a w ...

and the

theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...

How to Escape from a Black Hole

This is a

Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...

explaining how to escape from a black hole as mentioned in the book. # You are outside the

black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...

. # You have fallen into the black hole. A future version of yourself (4) appears and gives you a portable

white hole In general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy-matter, light and information can escape from it. In this sense, it is the reverse of a black ho ...

. You use the portable white hole to escape the black hole. # Another future version of yourself (6) appears and gives you a

time machine Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a w ...

. # You go back into the black hole and give the past version of yourself (2) the portable white hole. # You use the time machine to go far enough into the future (i.e. millions of years) that the black hole has evaporated due to

Hawking radiation Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...

. # You then travel back in time and give the past version of yourself (3) the time machine. # You are now outside of the black hole. The dashed red line indicates the path of the portable white hole (clockwise). The dashed blue line indicates the path of the time machine (counterclockwise). Movement from the bottom towards the top generally indicates movement forward in time (not to scale) and vice versa.

Real-world references

Ian Stewart often includes puns and topical references in his popular writing, and ''Flatterland'' is no exception. *The heroine's name,

Victoria Line The Victoria line is a London Underground line that runs between in south London and in the north-east, via the West End. It is printed in light blue on the Tube map and is one of the only two lines on the network to run completely undergr ...

, and her mother's,

Jubilee Line The Jubilee line is a London Underground line that runs between in east London and in the suburban north-west, via the Docklands, South Bank and West End. Opened in 1979, it is the newest line on the Underground network, although some secti ...

, are both lines on the

London Underground The London Underground (also known simply as the Underground or by its nickname the Tube) is a rapid transit system serving Greater London and some parts of the adjacent ceremonial counties of England, counties of Buckinghamshire, Essex and He ...

. *Her great-great-grandfather's name is the fictional

Albert Square Walford is a fictional borough of east London in the BBC soap opera ''EastEnders''. It is the primary setting for the soap. ''EastEnders'' is filmed at Borehamwood in Hertfordshire, towards the north-west of London. Much of the location work i ...

in London, from the BBC soap opera ''

EastEnders ''EastEnders'' is a Television in the United Kingdom, British soap opera created by Julia Smith (producer), Julia Smith and Tony Holland which has been broadcast on BBC One since February 1985. Set in the fictional borough of Walford in the Ea ...

''. Her father's name,

Grosvenor Square Grosvenor Square is a large garden square in the Mayfair district of London. It is the centrepiece of the Mayfair property of the Duke of Westminster, and takes its name from the duke's surname "Grosvenor". It was developed for fashionable re ...

, is a square situated in the Mayfair district of London. *Hawk King is a simple pun on the surname of the famous astrophysicist Stephen Hawking, whose research includes the Theory of Relativity and

. *Vikki, while travelling in the Topological Dimension, also meets a one-sided cow named Moobius (derived from the

Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...

) who sells her milk in

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...

s (the strip and the bottle both being one-sided topological figures). *The Doughmouse, the Harsh Mare, and the Mud Hutter are the counterparts in Topologica (the rubber-sheet continent) of Carroll's Dormouse, March Hare, and Mad Hatter, respectively. The Doughmouse works with a dough tea set, dough being flexible.

Editions

* Original hardback from Macmillan * Hardback edition by Perseus * Paperback edition by Perseus * Mass-market edition by Pan

References

{{Flatland Books by Ian Stewart (mathematician) 2001 British novels Novels about mathematics Fictional dimensions 2001 science fiction novels British science fiction novels Metaphysical fiction novels Quantum fiction novels Perseus Books Group books