Analytic Geometry is a branch of mathematics that treats the
relation of algebraic functions and their respective graphs, or
pictures that can be drawn from these functions. Students are
first introduced to analytic geometry in Algebra II courses, and
delve further into its study in both pre-calculus and calculus
courses. Essentially, this branch of mathematics combines
geometry and algebra to show what certain mathematical
relationships, called functions, look like in the real world.
There are four well known shapes--also known as curves or
functions in math jargon-- that students spend a good time
studying in analytic geometry. These are the circle, parabola,
ellipse, and hyperbola. These curves are called conic sections
because they can be derived in a most curious way from a figure
called a double-napped cone. A double-napped cone is formed by
placing two cones on top of each other, balancing at the tip.
Essentially what you end up with is something that looks like an
hourglass made out of two ice cream cones--without the ice
cream, of course, or the sand. By appropriately slicing the
double-napped cone with a plane, we end up with each of the
different conic sections.
Of the four conic sections, indubitably the circle is the one
with which most people are familiar. One does not have to look
to far to find applications of the circle in everyday life,
although many of these are taken for granted. The circle, by
mathematical definition, is the set of all points which are an
equal distance from another fixed point. If you got that the
fixed distance is the radius and the fixed point is the center
of the circle, take a bow. Well done!
Now what makes the circle so special and why? Well, it is the
precise definition of this conic section that has caused this
shape to have such a monumental effect on the progress of
mankind. For how would man have progressed without the wheel?
Precisely because the points of the circle are all the same
distance from the center, is the smooth roll of the wheel
possible. After all, how efficient would your car or dune buggy
be, for that matter, if the wheels were in the shape of
hexagons? Ah, I bet you never thought of that. Indeed it is hard
to imagine that hungry car salesman making a good pitch for that
new car model with those strange hexagonal tires!
Another point that comes to mind regarding circles is why
manhole covers are round. Think about this in context of what
you now know about circles, and it becomes apparent why this
unique shape works for this application. Why would a cover in
the shape of a pentagon (five-sided figure) not work here? If
the answer is not immediate, think of what might happen to the
poor utility worker standing at the bottom of the hole, while
his careless coworker fiddled with the cover above.
Conic sections. Yes. Now we see how the precise mathematical
definition of a shape as omnipresent as the circle plays a
critical role in determining the useful applications of this
curiously round shape. Next time you see a circle, think of how
lucky you are that your tires are round and not hexagonal. Stay
tuned...
About the author:
Joe is a prolific writer of self-help and educational material.
Under the penname, JC Page, Joe authored the classic of
mathematical ABC's Arithmetic Magic. Joe is also author of the
charmingly pithy and popular ebook, Make a Good Impression Every
Time: The Secret to Instant Popularity; the seminal collection
of verse, Poems for the Mathematically Insecure. For more
information, visit his website at
www.mathbyjoe.com.
