Loadstar 19/019.d81/t.calc.pi

Text File | 2022-08-26 | 7KB | 411 lines

         Math Reflections


  One of the fascinating facets of

mathematics is the huge number of

everyday occurrences that provide

proof for the fundamentals underlying

the structure of mathematics.

  In this instance, we're going to use

the random falling of hail stones to

derive an approximation of the value

of pi.

  First, let's get a grasp of the lay

of the land. Good old Sven Svensen has

planted a huge square plot with wheat.

The area of Sven's acreage is

4,000,000 square meters, which happens

to be 4 square kilometers.

  Now let's imagine a circle inscribed

within this square. There's no good

reason for Sven to imagine such a

circle, but we need the circle for

plot (no pun intended) development.

  Let's stretch our imagination even

further and divide Sven's field into

quadrants, each having an area of

1 square kilometer.

  We can now use the definitions of

Euclidean geometry to derive some

useful information about our imaginary

subdivisions.

  We can determine that the length of

a side of our imaginary quadrant is

1 kilometer. (Length of a side of a

square is equal to the square root of

the area of the square. The area is

1 sq. km, so the side is 1 km.)

  We also know by definition that the

radius of an inscribed circle is equal

to the length of a side of a quadrant

of the square circumscribing the

circle. Isn't this fun?

  So far we haven't found out anything

about pi and, since we know that wheat

has very little to do with pie, we

seem to be out in left field instead

of firmly rooted in Sven's wheat

field. In fact, Sven's field is about

to be devastated by a hail storm.

  But every cloud has a silver lining

and the silver lining here is that we

can use Sven's misfortune to confirm

mathematicians' approximations of the

value of pi.

  How? Obviously not by the very

empirical approach of counting the

hail stones. For one thing, that could

be painful. For another, some of the

hail stones would melt before we got

them tabulated (remember, Sven has a

big field here).

  Instead, we'll use probability

theorems to construct a method by

which we can determine certain useful

information.

  Instead of focussing on Sven's

entire field, let's direct our

attention at one quadrant and its

quarter of a circle.

  Presumably, hail does not fall at

the direction of some Greater

Intelligence, so we can assume that it

is relatively uniform in its

distribution over the land on which it

falls.

  We can understand that every hail

stone that lands in the quarter circle

also lands in the key quadrant. But

the converse is not true, some of the

hail that falls in the quadrant will

fall outside of the inscribed quarter

circle.

  And, because Sven is such a model of

an excellent farmer, we can assume

that he's planted each part of his

field equally as densely as any other

part.

  What we intend to measure is the

number of wheat stalks damaged by the

hail in the quadrant of our interest.

Not only are we interested in the

total number of wheat stalks damaged,

we're also interested in how many of

those stalks fell within the inscribed

circle.

  Probability theory tells us that if

we can measure the damage accurately,

we can derive some good stuff ... like

an approximation of the value of pi.

  Let's leave Sven, mourning over the

destruction of his wheat field, to

contemplate the wonders of probability

theory as they relate to the single

quadrant of Sven's field that we are

interested in.

  The science/art of probability is

not so arcane as to sound like Greek.

In fact, we can intuit the first law

that we're interested in. Let's derive

a rule of probability for determining

how many of the wheat stalks that are

damaged in this quadrant are inside

the quarter circle as well.

  If we simplify the problem by

dividing the quadrant into two equal

halves, we can restate the problem as

"How many of the wheat stalks that are

damaged in this quadrant will be in

the North half of this quadrant?"

  A good guess (and a correct one)

would be half of them, because we're

dealing with half of the area.

  We can generalize this guess to say

that the probability of a wheat stalk

in a given portion of a field being

damaged is represented by the ratio of

the area of the portion to the area of

the whole field.

  For the sake of simplification,

let's call the area circumscribed by

the quadrant Q (for quadrant) and

let's call the area of the quarter

circle that exists inside the

quadrant C (for circle).

  In formula-ese, we can now express

our discovered generality in the

following terms:

                  Area of C
  Prob (w in C) = ---------
                  Area of Q

  As more and more wheat stalks are

randomly damaged by the hail, this

formula can be restated as:

                  N(C)
  Prob (w in C) = ----
                  N(Q)

where N represents the number of

damaged wheat stalks observed in the

area.

  This substitution can also be

intuitively derived. Think of the

quadrant divided in half again. It's

entirely believable that if 100 wheat

stalks are damaged in the whole

quadrant then 50 of them will be found

in the north half. The above formula

just generalizes this rule.

  So much for theory, let's put it to

practice. Remember that the side of

the quadrant is equal to the radius of

the inscribed circle and that exactly

one-quarter of the circle lies within

the quadrant.

  Therefore, substituting the real

values of area into the first equation

gives us the following:

                        2
  Area of C   (1/4) PI R
  --------- = ----------- = (1/4) PI
                   2
  Area of Q       R


  Substituting this result into the

second equation results in this

formula:

     >   N(C)
  PI = 4 ----
         N(Q)


   Does the light begin to dawn?

  We left Sven waiting for the

insurance adjuster to come and inspect

the damage to his wheat.

  If each and every damaged wheat

stalk were counted and the resulting

values inserted into our formula for

pi, were should get a reasonable

approximation of 3.14159... (within .1

of accuracy).

  We don't intend to go out to Sven's

and count wheat stalks. That's too

much like work. Instead, we've devised

a simulation that pretty much depicts

the random destruction of the wheat by

the hail.

  The way it works, each dot

represents a damaged wheat stalk. The

program randomly places these dots,

thus simulating the dropping of hail

on the wheat. At all times, the

program is counting the number of dots

placed and their relationship to the

quadrant and the square.

  Run it from here and see two things:

the total destruction of Sven's wheat

and an interesting approximation of

the value of pi.  To see the current

approximation of pi, press the space

bar.  Then to return to the

destruction, press it again.  To exit

the program, press the 'Q' key.


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