## Faculty Advisor

Dr. Tedi Draghici

## Author Biographical Statement

Philip is a third-year student studying applied mathematics. His main research interests include measure theory, partial differential equations, and the use of numerical methods to solve mathematical problems. Philip mainly enjoys problems which often pose a deceptively simple question, but require much thought to thoroughly solve. After graduating he intends to pursue a PhD in applied mathematics. Apart from math, Philip also enjoys playing classical piano and composing.

## Abstract

In mathematical measure theory, the “Ham-Sandwich” theorem states that any n objects in an n-dimensional Euclidean space can be simultaneously divided in half with a single cut by an (n-1)-dimensional hyperplane. While it guarantees its existence, the theorem does not provide a way of finding this halving hyperplane, as it is only an existence result. In this paper, we look at the problem in dimension 2, more in the style of Euclid and the antique Greeks, that is from a constructible point of view, with straight edge and compass. For two arbitrary regions in the plane, there is certainly no hope for constructing the halving line by straight edge and compass. At the opposite end, for two circles the problem is very easy, as the halving line is simply the line which passes through the centers of both circles. It turns out that for the case of two equilateral triangles the problem is already interesting and challenging from a constructible point of view, and this is the main goal of our paper. We will investigate the “deltoid” of a triangle, that is, three hyperbolas which form an “envelope” of all halving lines of a given triangle, and how to construct halving lines with certain given properties.

## DOI

10.25148/URJ.020117

## Recommended Citation

Son, Philip A.
(2024)
"Constructible Sandwich Cut,"
*FIU Undergraduate Research Journal*: Vol. 2:
Iss.
1, Article 15.

DOI: 10.25148/URJ.020117

Available at:
https://digitalcommons.fiu.edu/undergraduate-journal/vol2/iss1/15