A Lesson in Philosophy of High School Geometry-Strange Circle and Discrete Space.

Euclidean geometry is a form of structural geometry grounded in a set of axioms from which all theorems are derived. This branch of mathematics encompasses two fundamental measurement types: angles and distances. It is widely recognized that the Euclidean axioms for measuring distances taught to students during primary, middle, and high school mathematics curricula establish the foundational concepts and structures of geometry they learn. The purpose of this article is to propose an educational resource that is designed for exceptional or gifted students in secondary school. This resource introduces alternative methods for measuring distances that differ from the Euclidean method. The objective of this educational material is to encourage students to think beyond the conventional Euclidean framework and explore alternative worlds of geometric structures. By broadening their horizons, students can recognize that geometric structures are not absolute but relative to the axioms and logical definitions that are agreed upon beforehand. [ABSTRACT FROM AUTHOR]