#### Event Title

On the cross section of minimal bands orthogonal to the sides of a polygon

#### Session Number

Project ID: MATH 2

#### Advisor(s)

Michael Keyton; Former Illinois Mathematics and Science Academy faculty

#### Discipline

Mathematics

#### Start Date

22-4-2020 10:05 AM

#### End Date

22-4-2020 10:20 AM

#### Abstract

If Q is a k-gon, the well of Q is the set of all points X for which a perpendicular through each side of Q passes through X. Equivalently, the well of Q is the cross section of the k bands of minimal width which are each orthogonal to a side of Q and fully contain their associated side.

When k=3, the well of Q is trivially non-empty. We establish conditions for existence of wells in cases where k=4 and consider cases where k>4. Conditions for convergence of the sequence of polygons obtained by recursively constructing wells are also established, as well as properties of the centers obtained in the case of convergence. We conclude with a consideration of related constructions and topics for extension.

On the cross section of minimal bands orthogonal to the sides of a polygon

If Q is a k-gon, the well of Q is the set of all points X for which a perpendicular through each side of Q passes through X. Equivalently, the well of Q is the cross section of the k bands of minimal width which are each orthogonal to a side of Q and fully contain their associated side.

When k=3, the well of Q is trivially non-empty. We establish conditions for existence of wells in cases where k=4 and consider cases where k>4. Conditions for convergence of the sequence of polygons obtained by recursively constructing wells are also established, as well as properties of the centers obtained in the case of convergence. We conclude with a consideration of related constructions and topics for extension.