IlliPascal

A mathematical exploration of Pascal's triangle and tetrahedron

Proposal

IlliPascal is a mathematical exploration of Pascal's triangle and tetrahedron. I have discovered a few things about the triangle over the years, and I want to prove all of them. As soon as I understand the triangle, I will apply what I have learned to the tetrahedron, so I can better understand it. The tetrahedron will most likely contain very similar theorems as the triangle.

Abstract

The project will be based on the summation techniques used to generate levels of Pascal's triangle and tetrahedron. Variations of these techniques are a large part of the project as well. Proofs will accompany, and two basic programs (in C) will be created to give a somewhat graphical perspective to the whole thing. They will produce values found at coordinates according to a rule. It will involve x and y for the triangle program and x, y, and z for the tetrahedron program. Another program, purely graphical, a derivative of the skel, will be created to serve as an interactive tool to view the triangle and tetrahedron. Guests will be able to view these shapes and rotate them appropriately. The project will be heavy on mathematical content; although much programming will be incorporated, this project is still mostly mathematical.

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Documentation

1) Various proofs of my discoveries of Pascal's triangle
- summation technique
- hockey stick theorem (not named by me)
- rectangle (diamond) rule
- seemingly disorganized segment rule
2)Application of such discoveries to Pascal's tetrahedron.
- summation technique
- double hockey stick theorem (kind of named by me)

Narrative

March 12th

- proved summation technique as it applies to Pascal's triangle (two numbers add to find a new number) and tetrahedron (three numbers add to find a new number)

- proved hockey-stick theorem; conjectured the tetrahedron "hockey-stick" theorem to be simply the sum of two hockey sticks

March 19th

- With help of Professor Francis, conjectured if the hockey stick can be bent any more than once; he coins the words "lightning bolt" to describe a more complex process than the simple hockey stick

March 24th

- Expanded the hockey stick theorem to form three other theorems: 1) table rule, 2) rectangle rule, 3) seemingly disorganized segment rule (lightning bolt)

April 1st

- Found a simple way to graph the triangle and tetrahedron. It involves only integers, and it may be a prelude to my eventual computer program that graphs them.

April 7th

- The computer program is here! David Scherba and I write code to generate levels of the triangle! Yippee!

April 16th

- Another one! This time, Chris Appuhn helps me crack out a program that generates levels of the tetrahedron!

Bibliography

Unknown Author, Tripod Website,"http://ptri1.tripod.com/" (this is where I found the phrase "hockey stick")

I would like to credit David Scherba, Professor Francis, and Chris Appuhn for their help!