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Non-Euclidean Geometry
Rick Roesler
I can think of three ways to talk about non-Euclidean geometry. I’m pretty sure they
are all equivalent, but I can’t prove it.
1. The Parallel Postulate
Euclidean geometry is called ‚Euclidean‛ because the Greek mathematician Euclid
developed a number of postulates about geometry.
One of these was the parallel postulate:
Given a line, L, and a point, P, not on L: there is exactly one line through P that’s
parallel to L.
For a really long time (2000 years?) people tried to prove that the parallel postulate
could be derived from the other postulates; in fact, it’s independent. And because it’s
independent, you can invent a geometry that has a different postulate - these are the
non-Euclidean geometries.
Euclidean geometry is the geometry of a ‘flat’ space - like this piece of paper or
computer screen (a plane) -- or Newtonian space-time. There are two archetypal non-
Euclidean geometries: spherical geometry and hyperbolic geometry. I’ll mostly talk
about spherical geometry because it’s easier to picture, and I found some good graphics
on the web. In spherical geometry, there are no parallel lines – not even one! So what’s
a ‚line‛on the surface of sphere? In Euclidean geometry (a plane), when we have a way
to measure distances between points, one way to define a line (segment) is that it’s the
curve between two points that has the minimum length. In a plane, the curve isn’t
curved, it’s a straight line, but it satisfies the ‘minimum distance’ principle.
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In math, these curves of minimum distance are called geodesics. Get a ball (whose
surface is a sphere) and a piece of string; mark two points on the ball and stretch the
string tightly along the surface of the ball between the two points. You’ll find that the
string is part of a great circle; a great circle is the intersection of the sphere with a plane
passing through the center of the sphere. Because we’ve stretched the string as tightly
as possible, the curve that the string defines must have the shortest length between the
two points; if it didn’t, we could stretch it even tighter. So a great circle path is a
geodesic on a sphere.
Figure 1. Three examples of great circles on a sphere. Each great circle is in a plane
that includes the center of the sphere. The center of the sphere is also the center of
each great circle.
On the earth, the equator is a great circle, as are the lines of constant longitude. (But not
the lines of constant latitude. Why?) Now you can convince yourself that on a sphere
there are no ‚parallel lines‛. Why?
The surface of a sphere satisfies all the other Euclidean axioms, but not the parallel
postulate. So it’s non-Euclidean. By the way, you now understand why a flight from
Dallas to Tokyo goes through Alaska. Why? (And this is a great example of an
‘everyday use’ of non-Euclidean geometry. But gravity is better and more exciting, and
I’ll get to that later!)
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2. A Triangle’s Angles Don’t Have to Sum to 180˚
In a plane (Euclidean geometry), if you draw a triangle and measure the three included
angles, you’ll find that the sum always add up to exactly 180˚. Now draw a triangle on
a globe (spherical, non-Euclidean, geometry). The first side goes from the North Pole to
the equator via the prime meridian (0˚ longitude, near London). For the second side,
start at the North Pole and go to the equator via New Orleans (90˚W longitude). The
third side goes along the equator, connecting the first two sides. Now look at the
angles - what’s the measure of each one? What’s the total?
You should have found that the total is greater than 180˚; in fact, for this example, it
should have been 270˚. This will always be true in a spherical geometry, but how far it
deviates from 180˚ depends on the area of the triangle. (In hyperbolic geometry, by the
way, the sum of the angles is always less than 180˚.) So the second definition of non-
Euclidean geometry is something like: ‚if you draw a triangle, the sum of the three
included angles will not equal 180˚.‛
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Figure 2. Example of a spherical triangle. In this illustration the angle at the North Pole is
50˚ rather than the 90˚ angle we constructed in the text; here the sum of the angles is 230˚.
When we construct smaller triangles on the sphere (see the inset), we end up with angles
that sum to almost exactly 180˚.
Remember, I said that the deviation from 180˚ depended on the size of the triangle. In
the picture above, the inset shows that if you draw a triangle in your back yard and
measure the angles, you’ll get something very, very close to 180˚; your back yard is
locally flat and approximates a Euclidean plane - even though your back yard is really
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curved because it’s part of the Earth’s surface . That’s why for a very, very long time,
people thought the earth was flat. Because it is – locally.
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The spherical triangle that we constructed with three 90˚ angles has an area 1/8 that of the earth’s surface
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(A ~ 5 x 10 m ). If you drew an isosceles right triangle in your back yard with each side being 1m, the
earth
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triangle’s area would be 0.5 m which is 10 times smaller – a million billion times smaller -- than the surface of
the earth. And that’s why the earth seems flat to us.
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