Automated Detection of Interesting Properties in Regular Polygons

Abstract

We demonstrate a systematic, automated way of discovery of a large number of new geometry theorems on regular polygons. The applied theory includes a formula by Watkins and Zeitlin on minimal polynomials of $$\cos \frac{2\pi }{n}$$ cos 2 π n , and a method by Recio and Vélez to discover a property in a plane geometry construction. This method exploits Wu’s idea on algebraizing the geometric setup and utilizes the theory of Gröbner bases. Also a bijective function is given that maps the investigated cases to the first natural numbers. Finally, several examples are shown that are all previously unknown results in planar Euclidean geometry.

Supporting Proving and Discovering Geometric Inequalities in GeoGebra by using Tarski

“…0 designate the coordinates of the remaining points of the regular pentagon. See [15] for more details on this technique.…”

“…In this communication we do not go into the further detail on the hidden technical difficulties, but refer to two recent papers: [15] explains the mathematical background on obtaining ratios in regular polygons, while [22] focuses on the RQE related issues, and points to a large set of benchmarks based on our tool. Instead, we focus on the practical use: how our work can be fruitful for the student and the teacher in a classroom.…”

Symbolic Comparison of Geometric Quantities in GeoGebra

“…In this study we do not go into the further detail on technical difficulties, but refer to two recent papers: [10] explains the mathematical background on obtaining ratios in regular polygons, while [19] summarizes the RQE related issues, and points to a large set of benchmarks based on our tool. Instead, our current paper is rather focusing on the practical use: how our work can be fruitful for the student, the teacher and the researcher.…”