Plot
The story
describes a two-dimensional world occupied by geometric figures,
whereof women are simple line-segments, while men are polygons
with various numbers of sides. The narrator is a square named
A Square, a member of the caste of gentlemen and professionals,
who guides the readers through some of the implications of life
in two dimensions. The first half of the story goes through the
practicalities of existing in a two-dimensional universe as well
as a history leading up to the year 1999 on the eve of the 3rd
Millennium.
On New Year's
Eve, the Square dreams about a visit to a one-dimensional world
(Lineland) inhabited by "lustrous points", in which
he attempts to convince the realm's monarch of a second dimension;
but is unable to do so. In the end, the monarch of Lineland tries
to kill A Square rather than tolerate his nonsense any further.
Following
this vision, he is himself visited by a three-dimensional sphere
named A Sphere, which he cannot comprehend until he sees Spaceland
(a tridimensional world) for himself. This Sphere visits Flatland
at the turn of each millennium to introduce a new apostle to the
idea of a third dimension in the hopes of eventually educating
the population of Flatland. From the safety of Spaceland, they
are able to observe the leaders of Flatland secretly acknowledging
the existence of the sphere and prescribing the silencing of anyone
found preaching the truth of Spaceland and the third dimension.
After this proclamation is made, many witnesses are massacred
or imprisoned (according to caste), including A Square's brother,
B.
After the
Square's mind is opened to new dimensions, he tries to convince
the Sphere of the theoretical possibility of the existence of
a fourth (and fifth, and sixth ...) spatial dimension; but the
Sphere returns his student to Flatland in disgrace.
The Square
then has a dream in which the Sphere visits him again, this time
to introduce him to Pointland, whereof the point (sole inhabitant,
monarch, and universe in one) perceives any communication as a
thought originating in his own mind (cf. Solipsism):
"You
see," said my Teacher, "how little your words have done.
So far as the Monarch understands them at all, he accepts them
as his own ¨C for he cannot conceive of any other except himself
¨C and plumes himself upon the variety of Its Thought as an instance
of creative Power. Let us leave this God of Pointland to the ignorant
fruition of his omnipresence and omniscience: nothing that you
or I can do can rescue him from his self-satisfaction."[4]
¡ªthe Sphere
The Square
recognizes the identity of the ignorance of the monarchs of Pointland
and Lineland with his own (and the Sphere's) previous ignorance
of the existence of higher dimensions. Once returned to Flatland,
the Square cannot convince anyone of Spaceland's existence, especially
after official decrees are announced that anyone preaching the
existence of three dimensions will be imprisoned (or executed,
depending on caste). Eventually the Square himself is imprisoned
for just this reason, with only occasional contact with his brother
who is imprisoned in the same facility. He does not manage to
convince his brother, even after all they have both seen. Seven
years after being imprisoned, A Square writes out the book Flatland
in the form of a memoir, hoping to keep it as posterity for a
future generation that can see beyond their two-dimensional existence.
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