A Mathematical Approach to Splitting Two Fried Eggs
Open Access
- Author:
- Bromberg, David Marvin
- Area of Honors:
- Mathematics
- Degree:
- Bachelor of Science
- Document Type:
- Thesis
- Thesis Supervisors:
- Wenrui Hao, Thesis Supervisor
Ludmil Tomov Zikatanov, Thesis Honors Advisor - Keywords:
- Computational Methods
Numerical Analysis
Python
Mathematics
Computer Science
Optimization
Optimal Cut
Geometry - Abstract:
- Consider an elliptical frying pan cooking two fried eggs. Assume the egg white covers the entire surface of the pan. Assume both yolks are intact and elliptical. Assume the eggs must be shared between two people. The goal of this thesis is to devise a mechanism through which one can find an optimal cut of the eggs, that is, where the yolks remain intact, each person gets one yolk, and the difference in the areas of the two pieces of egg is minimized. The main techniques to be used are computational methods in numerical analysis, and geometry. Mathematically, consider a closed function f(x, y) = 0, and two non-intersecting inner closed functions g1(x, y) = 0 and g2(x, y) = 0. The goal of this paper is to find two points z0 = (x0, y0) and z1 = (x1, y1) which lie on the function f(x, y) = 0 such that (1) (Ω1 − Ω2)^2 is minimized, where Ω1 and Ω2 are the areas of the two domains separated by the straight line connecting z0 and z1, (2) The line connecting z0 and z1 does not intersect g1(x, y) = 0 nor g2(x, y) = 0, and (3) For any pair of points (x, y) and (x', y') , the constraint (y − mx − b)( y' − mx' − b) < 0 is satisfied, where m and b are the slope of the line and the y-intercept respectively. The thesis is especially focused on functions g, g1 and g2 being ellipses of the form f(x, y) ∶= [(x-h)^2]/u^2 + [(y-k)^2]/v^2 ; h, k, u, v ∈ R. The two points z0 and z1 will be found using computational methods in numerical analysis and geometry and will be demonstrated with Python as the primary programming language.