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 Eureka (Greek for I have found it) 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 of

 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

Lets 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, D, E, and F.  Then draw in the lines perpendicular to the bisectors, through D. E. and F. 

 

Notice OE = d/2.

From this we can conclude:

Since AE = CE, .

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 didnt 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, D, E, and F 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