Dana
TeCroney
Archimedes
Archimedes
is one of the most interesting characters in mathematics and science history.
Stories of his life include events such as running naked through the streets of
Syracuse yelling ÒEurekaÓ (Greek for ÔI have found itÕ) after solving a problem
on buoyancy.
Below
I have included some websites that talk more about Archimedes life as well as
his contributions to math and science.
http://www.crystalinks.com/archimedes.html
http://www-history.mcs.st-and.ac.uk/Biographies/Archimedes.html
http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html
http://www.school-for-champions.com/biographies/archimedes.htm
One
accomplishment Archimedes is famous for was his estimation of Pi. Below is a discussion of how Archimedes
did this, sort ofÉ
It was known during his time that the
ratio of the circumference to the diameter was constant (
,
or 
). The Greeks were also good with
constructions. One easily
constructible figure is an equilateral triangle. Consider an equilateral triangle with its circumcircle
drawn.
What
I want to do is get a lower bound for my estimation of the circumference. Clearly, the circumference is greater
than the perimeter of the triangle.
There
is a little known theorem in geometry that says: The
radius of a circle inscribed in an equilateral triangle is equal to one-half of
the radius of the circumscribed circle and equal to one-third the altitude of
the triangle.
So what does this mean about the circumcircle and altitude? Well, one-half the radius equals
one-third the altitude of the equilateral triangle, or, the altitude is
three-halves the radius of the circle, or ¾ the diameter. This implies:
3d/4 = BE
= BO + OE
= d/2 + OE
Therefore,
d/4 = OE
LetÕs put some labels on our diagram:
Using a little, trig., 
,
which implies 
.
Furthermore, 
,
or 
.
Since the circumcircle is constructed using perpendicular
bisectors, AE = CE, so 
.
Therefore, the perimeter of the equilateral triangle is 
.
This could have all been done using the Pythagorean theorem, which
Archimedes may have done, so why the trigonometry?
It turns out the length of each side of the inscribed triangle (or
polygon) is 
where n is the number of sides.
If this is the length of one side, the perimeter is 
This represents a quick way to find one of the bounds because all
you need to know is how many sides there are.
Now, to find the upper bound, I want to find a similar equilateral
triangle that our circle inscribed in it.
To create this triangle, mark the points of intersection of the
perpendicular bisectors and the circumcircle, namely, DÕ, EÕ, and FÕ. Then draw in the lines perpendicular to
the bisectors, through DÕ. EÕ. and FÕ.
Notice
OEÕ = d/2.
From this we can conclude:
Since AÕEÕ = CÕEÕ, 
.
So,
the perimeter of the big triangle is 
,
which also an upper bound for my estimation of the circumference.
More
generally, the side length of larger triangle (or polygon) is 
. Furthermore, the perimeter is 
To
put this all together,
In
general,
As
n increases, the estimation for Pi gets better, so why didnÕt Archimedes come
up with this?
Approaching
this from a purely geometric standpoint, you could use the original points A,
B, C, and the other three points on the circle, DÕ, EÕ, and FÕ to create a
hexagon. By drawing the
perpendicular bisector to each of the segments of the hexagon, you could go
through a very similar process as before to get a better estimation.
As
you can see, by doing this process once, our upper and lower bounds would be
much closer to the circumference of the circle.
Legend
has it that Archimedes began with a hexagon and did this four times to make his
estimation using a 96-gon!
Here
are some websites that describe the method Archimedes may have used:
http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html
http://www.math.utah.edu/~alfeld/Archimedes/Archimedes.html
http://mathworld.wolfram.com/ArchimedesAlgorithm.html
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html