PRIME MODULO AND PASCAL’S TRIANGLE AS SEEN WITH FRACAL GEOMETRY

PRIME MODULO AND PASCAL’S TRIANGLE AS SEEN WITH FRACAL GEOMETRY

By Matthew Potter

Thesis Adviser: Dr. Andre Khalil

A Lay Abstract of the Thesis Presented

in Partial Fulfillment of the Requirements for the

Degree of the Master of Arts

(in Mathematics)

August, 2010

Looking at Pascal's Triangle there are many patterns that arise and phenomena that happen. Considering the elements of the triangle under a given modulus there are clear triangular patterns that can be mathematically described. The patterns in this structure can be described well with the aid of fractal geometry. Fractal geometry characterizes and describes many fascinating and intricate patterns in nature by comparing sections of an object to other sections.

Taking certain values or sets of values from Pascal's triangle, and under certain conditions (such as taking the modulus) the entire triangle can be recreated or values can be predicted directly from values that are in a given relative position. There are patterns of vertical and horizontal symmetry as well as entire sets that contain the pattern that depict the infinitely large and complete Pascal's triangle. With the concept of fractals in mind, such patterns and symmetries are used to find fractal dimension under given conditions like modulo 2, modulo p, for p prime and to look ahead to what a modulo p to a power n might look like. This would show patterns of divisibility by any such numbers and also patterns in the residues they leave behind. This is used to compare the triangle under various moduli and to describe many interesting properties that any related modulo might share.