# Project Descriptions

## Research in Geometry

Project Mentors: Eric Bahuaud and Dylan Helliwell

In Euclidean geometry, the distance between two points is defined to be the length of the straight line connecting those two points.  Other notions of distance are possible, and with these alternatives come new ways of thinking about geometry.  A well-known example is taxicab geometry, where the distance between two points in the plane is the sum of the horizontal distance and the vertical distance.

We will explore the ways in which standard results in Euclidean geometry change when considering such alternatives.  Questions include:  What are good notions of angle? How can the Pythagorean Theorem be reinterpreted?  How do triangle centers change?  Proceeding from these questions we will delve into deeper aspects of geometry.

## Research in Combinatorics

Project Mentors: Tom Edgar and Tyler Ball

If x and y are two whole numbers, we say that y dominates x if every digit in y is at least as large as its corresponding digit in x, meaning the ones digit of y is at least as large as the ones digit in x; the tens digit of y is at least as large as the tens digit of x; and so on for the hundreds digits, thousands digits, etc. If y dominates x, we write x<<y.

For example, 13 << 25 and 12 <<72. But 13 is not dominated by 72. Nor is 72 dominated by 13.

But there is nothing special about the way that we choose to represent numbers in base 10 with a ones digit, tens digit, hundreds digit, thousands digit, and so on. We can also talk about dominance for integers that are represented in binary: for example, 1001001 << 1101011 because each bit in 1101011 is at least as large as the corresponding bit in 1001001.

Beyond representations in base two, base three, or base ten, we can also explore representations of integers as sums of binomial coefficients, Fibonacci numbers, or factorial numbers. In each of these contexts, there is a way to define "digits" and so we can also define domination for these various representations of numbers. The purpose of this project is to explore the properties of these different digital dominance orders.