## PROJECTIVE GEOMETRY AND ITS APPLICATIONS IN PERSPECTIVE ART

### LOGAN BENSON

Presented on 4/5/2013

Projective geometry has applications in many topics; my primary topic of interest involves its applications in perspective art. For centuries, artists have used the mathematics of perspective to create realistic paintings, distort physical laws on the page to depict endless staircases and melting clocks, and to make breathtaking sidewalk chalk art. I will show how projective geometry can make such artwork possible. Using aspects of projective geometry, we will render three-dimensional, perspective skyscrapers from simple schematics. When faced with questions about “legitimate” viewing perspectives, I will turn to anamorphic art and apply concepts from projective geometry to see what we can discover about chalk art that uses perspective elements to fool the eye. I will then discuss the Fundamental Theorem of Projective Geometry to see how conics fit into the perspective picture.

Logan Benson is a senior at Loras College, majoring in Applied Mathematics and Creative Writing. Son of an art teacher, he has enjoyed every mathematical topic he has encountered but none so much as the study of perspective in art

## GEOMETRIC OPTICS AND SPHERICAL ABERRATION

### MORGAN MAYER

Presented on 4/3/2013

We will start by explaining the study of geometric optics and spherical aberration. Then we will discuss different lens types that can be used to eliminate spherical aberration. We will use the different formulas calculated to determine which type of lens will eliminate spherical aberration the best. We’ll look at different lens combinations as well as some of the techniques used to get the focal points of the lenses. Finally we will look at potential future work.

Morgan Mayer is a senior at Loras College majoring in Mathematics. She intends to pursue a degree in Optometry upon graduation.

## Numerical Solutions of Ordinary Differential Equations

### Rabin Ranabhat

Presented on 10/31/2012

The presentation will provide an introduction to computational methods that approximate the solution of ordinary differential equations (ODEs). Some of the methods that will be discussed in the presentation are Euler’s method, Taylor series method, Runge Kutta method of order 2. The relationship between step size and error will be discussed. Further, the presentation will also discuss the stability of different methods like Euler’s method, Backward Euler’s method and Trapezoidal method.

Rabin Ranabhat is a senior at Loras College, majoring in Mathematics and Engineering.

## How Many Sudokus Are There?

### Cassie Thill

Presented on 10/24/2012

We will start by explaining what a Sudoku puzzle is, what the rules are in solving them, and why we will use mini-Sudoku. We will discuss what it means for two puzzles to be “essentially different” along with other terminology. We will use a discussion of symmetries to begin looking at how many “essentially different” puzzles there are for a specific solution. We’ll look at some theories we have developed to narrow our search for the number of “essentially different” puzzles and prove these theories. Finally we will look at potential future work.

Cassie Thill is a senior Mathematics Major at Loras College.

## Reduction Numbers of Monomial Ideals

### Katie Burke

Presented on 10/24/2012

For a typeset version of the description go here.

Let I be an ideal of the ring R. We may then define the integral closure of I in R as $$\overline{I}=\left\{ f \in R \left | f^n+a_1 f^{n-1}+a_2 f^{n-2}+\ldots +a_{n-1} f + a_n \right . = 0, a_i \in I^i \right \}$$. An ideal J is called a reduction of I if $$J \subseteq I$$ and $$JI^r=I^{r+1}$$  for some r. Monomial ideals possess the unique property that their monomial elements can be represented by an integer point lattice known as the exponent set. We will therefore examine how to determine the exponent set of a monomial ideal and subsequently utilize its exponent set to succinctly define its integral closure. We may then use the property that J is a reduction of I if and only if $$\overline{J}=\overline{I}$$ to devise correlations between a monomial ideal's reduction number and the geometry of its exponent set.

Katie Burke graduated from Loras with Math and Computer Science degrees in December 2012.  She plans on attending graduate school in Mathematics starting in the Fall of 2013.

## Exploring Patterns in Fibonacci Numbers

### Nicole Jess

Presented on 3/30/2012

For a typeset version of this description go here.

A wide variety of interesting patterns may be observed within the well-known sequence of Fibonacci numbers, which is generated by the relationship $$F_n=F_{n-1} + F_{n-2}$$ beginning with $$F_1=F_2=1$$. The goal of this research project was to make conjectures regarding the patterns observed and prove or disprove these conjectures. We draw upon a variety of methods for analysis, including the recursive definition as well as techniques from linear algebra applied to the Fibonacci Q-matrix. We will use an assortment of techniques to prove the conjectures including proof by induction.

Nicole Jess graduated with degrees in Mathematics and Psychology in May 2012. She is currently in graduate school for Psychology at Michigan State.

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