Computer Science is the study of how computers work and how we can use them to solve problems. By studying computer science, we can use technology to make our lives easier and more efficient.

There are three objectives in this project. The first is to derive an upperbound recursive equation for Pi using regular polygons circumscribed about a circle to approximate its circumference. The second goal is to show the equivalence of François Viete's and my Last year's lowerbound expression for Pi. And the last objective is to derive an Algebraic Polynomial of which one root is Pi itself.

Join us in this project to discover the upperbound recursive equation for Pi and the Pi Associates! We will use regular polygons circumscribed about a circle to approximate its circumference and show the equivalence of François Viete's and my Last year's lowerbound expression for Pi. We will also derive an Algebraic Polynomial of which one root is Pi itself.

Title: Pi of Pieces Unlimited: A Continuation

Objectives/Goals

There are three objectives in this project. The first is to derive an upperbound
recursive equation for Pi
using regular polygons circumscribed about a circle to approximate its
circumference. The second goal is
to show the equivalence of François Viete's and my Last year's lowerbound
expression for Pi. And the last
objective is to derive an Algebraic Polynomial of which one root is Pi itself. I
have also found other roots
of this polynomial which I call The Pi Associates.

Methods/Materials

I used regular circumscribed polygons about circle of radius 1 for deriving an
upperbound expression for
Pi. I used the perimeter of each polygon to approximate the circumference of the
circle and from there of
Pi. Starting from a square an 8-sided regular polygon is constructed, doubling
the number of sides. This
procedure can be repeated endlessly doubling the sides of the polygon with every
step. The polygon with
a larger number of sides closely approximates the circle. The side of the
2n-sided polygon can be
determined from the side of the n-sided polygon. This produces a recursive
relationship for the side of the
2n-sided polygon in terms of the side of the n-sided polygon. (For the Algebraic
Polynomial and the
equivalence between Viete's and my expression for Pi I used my results from last
year.)

Results

I was able to derive a recursive equation for Pi using regular polygons
circumscribed about a circle of
radius 1. I have shown that Viete's expression for 2/Pi is equivalent to my last
year's expression for Pi.
Using my last year's expression for Pi from the lower bound I can derive an
Algebraic Polynomial, of
which one root is Pi itself and the others I call the "Pi Associates".

Conclusions/Discussion

I was able to derive a recursive equation for Pi using regular polygons
circumscribed about a circle,
although I did go through quite a bit of trial and error, finding faster and
better ways of deriving it. I also
was able to show that François Viete's expression for Pi is equivalent to my
last year's lower bound
expression for Pi. And Lastly, I have introduced the Pi Associates, numbers
whose properties I have yet to
discover. What kinds of numbers are the Pi Associates? Are they also
transcendental numbers? I would
like to investigate further on these questions.

Summary Statement

I must prove my last year's expression for Pi is equivalent to Viete's
expression for 2/Pi, Pi can be
obtained from an Algebraic Polynomial, and to derive an expression for Pi using
regular polygons
circumscribed about a circle of radius 1.

Help Received

My father has been by me through many sleepless nights, helping me with the
tedious cutting and pasting
that is involved with the making of a board, and making sure I am equipped with
supplies I need. My
Biology teacher has been supportive of me, and put deadlines for different
elements in the project.

Pi of Pieces Unlimited: A Continuation

Design a computer program to derive an upperbound recursive equation for Pi

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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Pi of Pieces Unlimited: A Continuation

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