Symmetry in Geometry

This lecture explores modern symmetry in geometry, covering transformations, isometries, orientations, and practical applications. It delves into isometries in the plane, including identity, point reflection, axial reflection, and compositions of reflections. The lecture also discusses rotations, translations, and special cases like glide reflections. It concludes with a study of symmetry groups for regular polygons and circles, emphasizing the concept of symmetry of revolution. The content highlights the fundamental theorem on the composition of three axial reflections and the normalization of arbitrary axial reflections. The lecture touches on the synthesis of plane isometries and the properties of various geometric figures under different transformations, leading to the study of projective geometry and its invariant properties.