Simetrijske grupe končnih vzorcev - Mencinger, Matej (2021)
Symmetry Groups of Finite Patterns. Finite patterns are defined first intuitively and then also mathematically. Symmetries of finite patterns are defined based on planar isometries.   Four basic types of planar isometries (mirror reflection, rotation, translation and glide reflection) are considered. In the classification theorem for planar isometries it is proven that every isometry coincides with one of the four basic planar isometries. Vectors and matrices are limited to two- and three-dimensional (vector) space. It is proven that every planar isometry with a fixed point is associated with an orthogonal matrix. The chapter on groups includes the information on (sub)groups, order of the group, isomorphism of groups and group generators. The main result is the classification of the symmetry groups of finite patterns into cyclic and dihedral, which is nowadays known as Leonardo’s theorem. Finally, Leonardo’s time and work is described. The textbook is written for students of the architecture and includes several examples, figures and solved exercises.
Keywords: planar isometry; finite pattern; symmetry; finite group; Leonardo’s theorem