Tuesday, December 27, 2011

71: What Goes Around Comes Around

    Let's take a look at that most mundane of mathematical
tools, the number line.  If you draw a number line, the numbers to the
right count upwards: 1, 2, 3, and so on, towards infinity.  The
numbers to the left count downwards: -1, -2, -3, and continue towards
negative infinity.  Certainly the two numbers that are farthest apart
are positive infinity, to the right, and negative infinity, to the
left, right?  It looks that way from our standard number line.  But,
if you view the number line in a slightly different way, both positive
and negative infinity appear to be the same point.
    How does this work?  Well, let's create an alternate view of the
number line, a kind of number circle.  We can define this circle in a
very special way.  Draw a circle of radius 1, tangent to the zero
point of our number line, or in other words touching only the zero
point and resting above the line otherwise.  The zero point on the
number circle is the same as the zero point on the number line.  But
for any other number (n) on the number line, to find its corresponding
point on the circle, draw a line segment from point (n) on the line
to the point at the very top of the circle, two units above the zero
point.  Such a line will intersect the circle exactly once:  that
intersection point is the representation of (n) on our number circle.
    If you think about it for a minute, you will see that numbers
close to 0 will rapidly proceed up the circle, and the first 90-degree
arc actually only contains the numbers 0 through 2.  But then as we
move further up the number line, we proceed ever more slowly up the
right side of the circle, getting closer and closer to the top, but
never quite getting there.  So the infinity point is the point at the
top of the circle.  But now let's travel the other way-- look at the
negative numbers.  Symmetrically, you can see that by -2, we are
halfway up the circle, but then go up more and more slowly-- so
negative infinity *also* corresponds to that very same top point!  If
you can envision a ball spinning around that number circle, and view
its 'shadow' of corresponding points on the line, it will seem that it
travels out to infinity, then somehow comes back from negative
infinity.  So the two infinities are actually the same point! 
    This number-circle representation is actually the two-dimensional
analog of a 3-D mapping known as the "stereographic projection", where
you place a sphere tangent to a plane, and map each point on the plane
to the point on the sphere where it's intersected by a line from the
sphere's north pole.  Such projections can be used to create flat maps
of spherical surfaces like the Earth-- in this case, it results in a
circular map centered around the tangent point.  Of course, since the
north pole becomes the infinity point, such a map can never show 100%
of the planet, but creates a good map for a finite region.  This kind
of projection was first documented by Ptolemy in Roman Egypt, during
the 1st century A.D.  Of course, like in our number circle, the
mapping does not preserve distances, though relations like
between-ness and angles are usefully represented.  Such a projection
is known as a conformal projection.  And, as you would expect, all
lines that head out to infinite distances in any direction on the
plane correspond to lines towards the north pole of the sphere. 
     And this has been your math mutation for today.

Stereographic Projection at Wikipedia

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