Tuesday, December 27, 2011

35: One Messed-Up Triangle

    You have probably heard the term "Non-Euclidean Geometry"
mentioned in science fiction novels or modern physics discussions
about oddly curved spaces.  The name seems to connote some kind of
weird form of geometry that doesn't match with our usual assumptions
about the world.   And indeed, non-euclidean geometry does make it
possible to construct bizarre figures that seem to defy our basic
assumptions.  But what exactly does "non-euclidean
geometry" mean? 
    To start with, let's review what Euclidean geometry is.
Basically, this is the type of geometry taught in most high schools,
based on the works of the greek mathematician Euclid over two thousand
years ago.  Central to this system of geometry are five postulates, or
basic assumptions about how the world works, which are used as the
basis for proving more complex theorems.  Four of the postulates are
very simple:  any two points determine a line, any line segment can be
extended indefinitely, any line segment can be the radius of a circle,
and all right angles are congruent.  The fifth postulate, or "parallel
postulate", is a little more complicated to state:  Given a line and a
point not on the line, exactly one line can be drawn through that
point that is parallel to the first line.  Quite a mouthful compared
to the other postulates, don't you think?  Because it is so much more
complex to state, many mathematicians over the years tried to find
ways to get rid of it entirely, and prove it based on the other
    If you draw a few pictures on a piece of paper, you'll soon
realize that the fifth postulate has to be true.  Well, drawing
pictures on paper might not be the best way to figure this out, since
you'll soon see that it's hard to even draw something the supposedly
violates this postulate.  For example, suppose
that instead of the parallel postulate, *no* line can be drawn through
an external point that is parallel to a given line.  That would mean
that if you draw two lines forming a right angle, and then a third
line that also forms a right angle with the second, then the third and
first lines must intersect somewhere, forming a triangle with two
right angles!  Otherwise, the third line would be parallel to the
first, violating our modified postulate.  Due to absurdities like
this, for many years it was assumed that the fifth postulate must be
true, and the only open question was whether it should be a
postulate or a theorem.
    But in the nineteenth century, mathematicians were coming to a
growing realization.  While mathematics is often very useful for
describing the real world, ultimately, it is a system for deducing the
consequences of your basic assumptions, or postulates.  So why not try
modifying the fifth postulate, deducing the consequences, and seeing
where that takes you?  Mathematicians Janos Bolyoi and Nicolai
Lobachevsky independently pursued this idea in the 1820s and 1830s,
developing whole geometries based on modifications of the parallel
postulate.  At first, it looked like they were just playing some silly
symbolic game, and it was several decades before their work was widely
accepted.  But gradually their colleagues realized that these new
geometries were very usefully modelling properties of different types
of surfaces. 
    For example, let's look again at our 'absurd' example of a
triangle with two right angles.  How could such a thing be possible?
On a flat plane, it really is absurd.  But look at a globe of the
Earth.  Draw lines, which are actually great circle segments on the
surface of a sphere, from the equator to the North Pole at the 0 and
90 degree meridians.  Each of these lines forms a 90 degree angle with
the equator, yet they intersect at the pole-- and we really do have a
triangle with two right angles!  And the angle at the pole is also 90
degrees, so there are actually three right angles in this triangle. 
In other words, our modified geometry may not make sense when viewed
on a flat plane, but is an accurate description of the properties of
shapes on the surface of a sphere.  And you can now amaze your friends
by drawing triangles with three right angles, though they may get mad
at you for defacing their globes. 
    The development of non-euclidean geometries became vital when
Albert Einstein began working on his general theory of relativity in
the early 20th century.  In Einstein's models, the three-dimensional
space we live in is curved in the fourth dimension in regions where
matter is present-- so these odd forms of geometry are what actually
describe the real world, rather than Euclid's so-called "obvious"
models!  Of course, whether in a small region of the surface of a
sphere, or in a small region of relativistic space, Euclid's
conclusions are an excellent approximation of reality for most
practical purposes.  And it's still true that Euclid made an
immeasurable contribution by showing how interesting and useful
conclusions could be deduced from a simple set of basic postulates.
But the real world is a lot more complicated than Euclid thought.
    And this has been your math mutation for today.

  • Euclidean Geometry on Wikipedia
  • Non-Euclidean Geometry on Wikipedia
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