Flatland
Flatland
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The book Flatland was written in 1880 by Edwin Abbott. A copy is
available at www.geom.uiuc.edu/∼banchoff/Flatland.
Flatland is inhabited by 2-dimensional geometric figures. The more
sides somebody has the more status they have. Flatland is run by the
circles, who can be thought of as figures with infinitely many sides.
In the movie a sphere comes to flatland with the hope of educating
the population by introducing one flatlander to the third dimension.
He then takes Arthur Square into this new dimension. While there
Arthur encounters 0-dimensional and 1-dimensional worlds. He has a
hard time understanding their limited view of the universe while he
has a hard time understanding the third dimension.
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After visiting the third dimension Arthur ponders dimensions higher
than the third.
The circles are trying to keep knowledge of the third dimension
hidden.
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Some Things to Think About
How does Arthur Square appear when he goes into a 1-dimensional
universe. Is this reasonable?
What kind of shape(s) do you get when you cut a three-dimensional
shape by a plane? This shows up in the movie.
The sphere opens Arthur Square up to the idea of a higher dimension
by taking him to the third dimension. How does the sphere react to
Arthur’s broadening of his horizons?
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Clicker Question
When flatlanders rotate 1/2
turn, their eye always is right
side up. Do you think this would
happen to actual creatures in a
2-dimensional universe?
A Yes
B No
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Clicker Question
The sphere is trying to open up Arthur’s mind to the idea of the third
dimension. But, he thinks considering a fourth dimension is a
ridiculous idea. Are you surprised by his attitude?
A Yes
B No
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Clicker Question
There is a followup movie called Sphereland. Was watching Flatland
interesting enough for us to watch Sphereland later in the semester?
A Yes
B No
The issue of the sphere’s attitude toward the fourth dimension is
explored further in Sphereland.
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Fractals
The pictures drawn in the interior of flatlanders and elsewhere are
mathematically based images called fractals. We may discuss these
later in the semester.
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Next Time
Next week’s topic is graph theory. There is more than one notion of a
graph in mathematics. The one we will discuss isn’t the one that
comes up in algebra courses. It is a more modern idea that has many
applications to computer science and applied problems.
We’ll discuss aspects of graphs, how it arose historically, some of its
applications, and how it can be used to help understand and classify
different types of surfaces.
Flatland
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