Flatland 2: Sphereland

Flatland 2: Sphereland
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Sphereland the Book
The book Sphereland, by Dionys
Burger, was published in 1965.
It is a followup to Edwin
Abbott’s book Flatland. One
fundamental idea is how fourth
dimensional hypersphere would
interact in the third dimension.
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Flatland 2: Sphereland the Movie (2012)
Here is the plot summary from IMDB.com: A young scientist from
2D space must solve mathematical mysteries to save a doomed space
mission and discover the true shape of her universe. In a world of only
two dimensions, length and width, the geometrical inhabitants of
Flatland dismiss the idea of the third dimension. But when a scientist
(Danny Pudi) discovers a mathematical anomaly, he turns to the only
person who might listen - Hex (Kristen Bell), an outcast
mathematician shunned for her ideas of the third dimension. An
unexpected reunion with Spherius (Michael York), a 3D being,
plunges the two Flatlanders into a journey across dimensions as they
race to discover the true shape of their universe.
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Some Things to Think About
In Flatland Spherius thinks the idea of the fourth dimension is
ridiculous. See what he thinks about it in the movie.
The following clicker questions involve issues that show up in the
movie.
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Clicker Question
Could a creature in Lineland change left and right?
A Yes
B No
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A Not by staying in Lineland. However, we’ll see how he changes left
and right in the movie.
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Clicker Question
Q How many degrees is the sum of the angle measurements in a
triangle?
A In plane geometry the sum is 180◦ . We’ll see that this fact is a key
point in the plot of the movie.
Let’s now watch the movie.
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Spherical Geometry
Puncto discovered that the sum of the angles of a triangle was
coming out larger than 180◦ . This is because Flatland turned out to
be the surface of a sphere.
In spherical geometry, lines are great circles; circles which cut a
sphere into two identical hemispheres.
A triangle in spherical geometry is formed by three great circles; the
sum of the angles can vary, but is greater than 180◦ .
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The larger the triangle relative to the size of the sphere, the greater
the sum of the angles. The sum can be up to 270◦ .
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Moving in Lineland
The king of Lineland cannot change left and right by remaining in
Lineland. Puncto pulls him out and puts him back after accidentally
rotating him. That is how he gets his orientation changed.
The same thing happens in Flatland. Hex wants Spherius to flip the
orientation of the ship, which he can do by rotating the ship in the
third dimension.
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The Sierpinski Triangle
This is an example of a fractal. We saw some fractals in Flatland
(and in Sphereland). We’ll take about fractals later in the semester.
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Some Fractals from Flatland
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Next Week
We’ll discuss how the idea of proportionality and scaling allowed
people in antiguity to find indirect measurements, such as the height
of a pyramid. We’ll also discuss how this idea shows up in filming of
movies.
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