<p> The purpose of this thesis is to investigate possibly interesting classes of polygons within quadrilaterals and hexagons. We utilize zero sets of polynomials using the vertices of these polygons to find characteristics of two new quadrilateral classes and four hexagon classes. We will refer to the polynomials as “forms”. These forms are invariant under translation and rotation and scaled by a factor under dilation. We define three functions: X- a reflection across the <i> x</i>-axis, <i>ф</i>- a relabeling of vertices across the <i> AC</i> diagonal (or in a hexagon, across a long diagonal AD), and <i> ρ</i>- a relabeling of the vertices by rotating them clockwise. </p><p> We find forms that characterize our classes of polynomials based on how they interact with these functions. For these particular classes, there is one form (up to constant multiples) of order 1 that interacts with the functions in the manner that characterizes the class of polygon. A form of order one is scaled by <i>r</i><sup>2</sup> if the polygon is scaled by <i> r</i>. For each class then, we found several forms that are all equivalent, because they all interact with the functions in the same way.</p><p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:10793876 |
| Date | 08 September 2018 |
| Creators | Darch, Melissa |
| Publisher | Southern Illinois University at Edwardsville |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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