Flatland was
originally published under the pseudonym of A. Square. Since Edwin A. Abbott’s middle and last names were both “Abbott” (His
mother and father were first cousins, each having the last name of Abbott),
it is possible that his friends might have nicknamed him “A Squared”.
There is no definite evidence to support this, but it is an interesting
idea and a fun play on words. At any rate, A. Square, a square-shaped character
in the book, is obviously a pun in and of itself (Abbott/Stewart, p. 2).
Flatland is
a story about two-dimensional creatures—lines,
triangles, squares, circles, and other polygons—that live on a plane.
The protagonist and narrator of the story, A. Square, visits a one-dimensional
land known as Lineland and is visited by a Sphere from Spaceland. After
the Sphere takes A. Square on a tour of Spaceland and then returns him
to Flatland,
Square decides to share the “Gospel of Three Dimensions” with
other Flatlanders. As a result, Square is imprisoned for life for his belief
in three dimensions.
Written in 1884,
Flatland is a biting satire of English Victorian Society with its rigid
hierarchies that limit opportunities of the common man and
relegate all women to subservient, inferior roles. Abbott, the most famous
schoolmaster for the City of London School was especially interested in
the education of women, which was remarkably limited in Victorian England.
Clearly,
Abbott hoped to challenge Victorian views through his satirical portrayal
of Flatlanders.
But Flatland is also a novella about mathematics, particularly geometry.
It cleverly encourages readers to consider the idea of a fourth dimension
by using the analogy of a two-dimensional being who is introduced to a three-dimensional
world. Victorians were intrigued by the idea of a fourth dimension and conversations
on the subject were frequent in the late 1800s.
As stated by a staff writer for
the “Math Forum @ Drexel” site,
One way to understand what the
fourth dimension "looks like" is
to carefully examine what the 3rd dimension looks like to "creatures" living
in a 2-dimensional world. If we can understand this, then we can understand
some of what the fourth dimension looks like
to us creatures living in
the 3-D world by using appropriate analogies. (Fourth Dimension.)
A. Square is to space of three dimensions as a Victorian human is to space
of four dimensions. Abbott was influenced by the writing of Charles
Howard Hinton, who laid the groundwork for this analogy used in Flatland.
For more biographical
information on Edwin A. Abbott, please see Additional
Resources.
Flatland is divided into two parts.
Part I: This World
The first part of the book is
more heavily social satire.
Part II: Other Worlds
The second section of the book is
more heavily scientific
Before embarking
on the journey with A. Square, Abbott advises (in his preface to the 2nd
edition) that we "decline to say on the one
hand, 'This can never be,' and on the other hand, 'It must needs be precisely
thus,
and we know all about it.'" (Abbott, p. x)
In other words, keep an
open mind and don't be a know-it-all!
Excellent advice for all endeavors!
With that admonition, let's begin!
Teachers: For a Study Guide Questions sheet complete
with answers, please email Sandy
Stuart. Sorry students, but this offer is not available for you,
not even for chocolate.
Part One: This World
1—Of the
Nature of Flatland
Who is the narrator of Flatland?
Since Flatland
is a plane, all shapes must appear as what?
How many dimensions
does Flatland have?
Using shapes
drawn and cut from paper, demonstrate how residents of Flatland appear
to each other.
2—Of the
Climate and Houses of Flatland
The houses
were what shape? Why?
What helps Flatlanders
determine direction?
3—Concerning
the inhabitants of Flatland
How long were
most inhabitants?
What was the
shape and ranking of women?
What are the
various shapes and ranks of the men?
Abbott depicts
social class as visible by physical form in Lineland. In what way is
social class physically
visible in Spaceland
(or our world)?
How did male
children differ from their fathers? What was the significance of this?
Additional Information:
Abbott’s description of a circle as a polygon of so many sides
that it can’t be distinguished from a circle is an example of
Abbott using exemplary math. A regular n-gon with n very large is approximately
a circle, but the radius r of that circle depends on the length of
side
d of the polygon.
r = (d/2) cosec(/n).
In order to approximate a given circle by a series of regular n-gons,
for increasing n, the sides must shrink as n increases. For a circle
of radius r, the side of the polygon must get closer and closer to
2r/n.
(Abbott/Stewart, p. 44)
Darwinian influence was shaking up Victorian England in the late
1800s. Many in Victorian society accepted a misreading of Darwin, allowing
for
the worst kind of "Social Darwinism”—it’s OK
to ignore the poor because it will lead to an improved human species.
(However,
Abbott’s school, the City of London School, was actually quite
progressive and egalitarian.)
(Abbott/Stewart, p.45)
4—Concerning
the Women
Why do women
have a separate entrance?
What must women
do when walking in public places?
Describe the
characteristics of Flatland women. What does this satirical writing say
about the Victorian view
of women?
5—Of Our Method of Recognizing
One Another and 6—Of Recognition by Sight
What are the three methods Flatlanders
use to recognize one another? What are the advantages and disadvantages of
each?
7—Concerning
Irregular Figures
This chapter makes
a case for regularity or symmetry. Although stressing symmetry, Abbott
is mathematically careful when he states that “If
our sides were unequal our angles might be unequal.” He
is obviously aware that there are exceptions.
Give an example
of a polygon with unequal sides, but with equal angles.
Give
an example of a polygon with equal sides, but with unequal angles.
Rigid Victorian England had little
tolerance for irregularity or lack of conformity. Abbott’s satire is particularly biting here, as he explains
that Flatlanders propose “painlessly and mercifully” consuming
irregular offspring. This is very much like Swift’s A
Modest Proposal for Preventing the Children of the Poor People
in Ireland from being a Burden
to their Parents or Country and for Making Them Beneficial to
the Public (1729):
I have been
assured by a very knowing American of my acquaintance in London, that a
young, healthy child,
well-nursed, is at a year old a most delicious,
nourishing and wholesome food, whether stewed, roasted, baked, or boiled,
and I make no doubt that it will equally serve in a fricassee, or a ragout.
(Swift as quoted by Abbott/Stewart, p. 78)
Discuss possible
effects of statements such as Abbott’s or
Swift’s. What are the pros and cons of using irony or satire
to make a point? Is this approach likely to be effective in changing
attitudes?
Will members of Victorian society “get it” and reconsider
their own behaviors and attitudes? Or will the extremeness of the
satire cause
them
to be outraged and feel morally superior? Give your opinion and explain
why you have that opinion.
How do you think
Victorians would deal with people with special needs such as physical
or mental challenges?
What would their view be of special
education? What might Victorians have done with Einstein (who was a very
poor student
in school)?
8—Of the
Ancient Practice of Painting and
9—Of the Universal Colour Bill
The Victorians worried about the destructive influence of popular culture
on the classics—“dumbing down” the culture. In Abbott’s
story, they worried about the reduction of the Art of Sight Recognition
because it was less needed with the addition of color. Give an example
of this worry
about “dumbing down” the culture today.
Explain how the
coloring of women and priests could cause women to be mistaken
for priests? Demonstrate this with paper circles and lines.
In the passage
on mistaking women for priests on pages 28-30 in Flatland, how does
this discussion imply more intelligence in women than earlier passages
in Flatland?
10—Of the
Suppression of the Chromatic Sedition
During the Victorian era,
the English social system was struggling toward greater equality—first for the common man and then for women, noble or common.
(Oddly enough, it was the conservative views of Queen Victoria—a woman,
obviously—who discouraged universal rights.)
Discuss how the
Color Sedition demoralized the Circles and brought about the suppression
of color.
Color came to
be allowed only for illustrating some mathematical properties. How is
color used today to illustrate math?
11—Concerning our Priests
and
12—Of the Doctrine of Our Priests
Explain the Flatland
version of “gene therapy”?
In Flatland and
in Victorian England, social pedigree outweighs everything else. What
is the discussion about “invisible” irregularities
in women saying about Victorian society?
Although A. Square
makes an appeal for educating women, what reason does he give? Why
do you think Abbott uses this kind of reasoning?
Part II: Other Worlds
A tour of alternative
dimensions begins in Part Two.
13—How I had a Vision of Lineland
In Lineland,
what shape are the King and Men? What shape are the women?
Why do you
think Abbott takes A. Square to Lineland, a simpler world
of only one dimension?
What is he preparing
the Victorian reader to consider?
14—How
I vainly tried to explain the nature of Flatland
How does the
King of Lineland determine length (what Lineland calls Space)?
A. Square
tries to explain Flatland by Cartesian coordinates. What does
this mean and how does Square do this? What is the result—how does the
King react?
15—Concerning
a Stranger from Spaceland and
16—How the Stranger Vainly Endeavored to Reveal to Me in Words
the Mysteries of Spaceland
When
A. Square shows his grandson (the bright little Hexagon) how to determine the
area
of a
square by squaring
a side (a square with three inch sides,
for instance, has an algebraic meaning—3 squared, and also
has a geometric meaning—the area of a square), what analogy
does the bright
little Hexagon
suggest? What is A. Square’s reaction?
What is the only
way the visiting Sphere can exhibit his shape to A. Square?
Demonstrate
how Sphere exhibits his shape in Flatland, using a sphere (such as
a ball) and a flat surface.
A being from
Spaceland can view the “insides” of
a being from Flatland. (The interior of a polygon is not visible
to a Flatlander because the edges get in the way, but to a Spacelander,
the “insides” are
visible.) By analogy, it follows that a creature
from the fourth dimension could do what?
Just as A. Square
turns on Lord Sphere because he doesn’t understand
and is afraid, give one or more examples of how people
in our world tend to do this with anything they don’t understand or that
seems too unusual.
17—How the Sphere, Having
in Vain Tried Words, Resorted to Deeds and
18—How I came to Spaceland, and What I Saw There
Although Abbott
is a clergyman and is serious about his profession, he
doesn’t
mind poking fun at it as well. What does he call the message he has come
to proclaim to A. Square?
After A. Square is pushed out
of Flatland into Spaceland, the Sphere attempts to visit Flatland’s
High Council and explain the third dimension. Why do you think the High
Council silenced
the Sphere? Do you think governing
bodies
ever do this today (silence what they don’t agree with or understand)?
Give a possible example of this happening.
What happened to A. Square’s
brother, the Chief Clerk? Why did it
happen?
19—How, Though the Sphere
Showed Me other Mysteries of Spaceland, I Still Desired More; and What Came
of It
At first, A. Square has trouble
perceiving a solid as it looks irregular to him. This is due to being unaccustomed
to what three elements
that allow two-dimensions
to perceive three (as our eyes must do)?
Accepting the three-dimensional
world, A. Square now wants Sphere to show him the next world of further
dimensions in which one may peer into
the insides
of the Sphere. Explain how A. Square continues the analogy as proof
of a fourth dimension, using terminal points. And, how does he do this,
using bounding
points?
20—How the Sphere encouraged
me in a vision
A. Square dreams of Pointland with
no dimensions where a single, lone
point is, yet he is self-content. He states, “To be self-contented is to
be vile and ignorant, to aspire is better than to be blindly and impotently happy.” What
do you think this statement means? In what ways do you agree or disagree
with this statement?
21—How I Tried to Teach
the Theory of Three Dimensions to my Grandson, and With What Success and
22—How I then Tried to Diffuse the Theory of Three Dimensions by
Other Means, and of the Result
A Square discovers he cannot
demonstrate the third dimension to his grandson because an initial state
that lies
entirely within the Flatland plane can never
leave that plane. As stated by Stewart, “A. Square can no more
move things “Upward,
not Northward” than we can make a rabbit disappear from our universe
by giving it the right push.” (Abbott/Stewart, p.190) What is the
grandson’s
reaction to Square’s explanations?
What happens after A. Square
forgets himself at the Local Speculative Society meeting and tells
about his experiences with the third dimension?
r/n.