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1. Euclidean geometry : The five axioms of Euclidean geometry ; Ruler and compass constructions ; The golden ratio ; Fibonacci numbers
2. Plane transformations : Plane symmetries ; Plane symmetries, vectors, and matrices (optional) ; Groups of symmetries of planar objects ; Frieze patterns ; Wallpaper designs and tilings of the plane ; Tilings and art
3. Similarities, fractals, and cellular automata : Similarities and some other planar transformations ; Complex numbers (optional) ; Fractals: definition and some examples ; Julia sets ; Cellular automata
4. Hyperbolic geometry : Non-Euclidean geometries: background and some history ; Inversion ; Hyperbolic geometry ; Some basic constructions in the Poincaré model ; Tilings of the hyperbolic plane
5. Perspective and some three-dimensional objects : Perspective ; Mathematics of perspective drawing: a brief overview (optional) ; Regular and other polyhedra ; Sphere, cylinder, cone, and conic sections ; Geometry, tilings, Fractals, and cellular automata in three dimensions
6. Topology : Homotopy of spaces: an informal introduction ; Two-manifolds and the Euler characteristic ; Non-orientable two-manifolds and three-manifolds.
2. Plane transformations : Plane symmetries ; Plane symmetries, vectors, and matrices (optional) ; Groups of symmetries of planar objects ; Frieze patterns ; Wallpaper designs and tilings of the plane ; Tilings and art
3. Similarities, fractals, and cellular automata : Similarities and some other planar transformations ; Complex numbers (optional) ; Fractals: definition and some examples ; Julia sets ; Cellular automata
4. Hyperbolic geometry : Non-Euclidean geometries: background and some history ; Inversion ; Hyperbolic geometry ; Some basic constructions in the Poincaré model ; Tilings of the hyperbolic plane
5. Perspective and some three-dimensional objects : Perspective ; Mathematics of perspective drawing: a brief overview (optional) ; Regular and other polyhedra ; Sphere, cylinder, cone, and conic sections ; Geometry, tilings, Fractals, and cellular automata in three dimensions
6. Topology : Homotopy of spaces: an informal introduction ; Two-manifolds and the Euler characteristic ; Non-orientable two-manifolds and three-manifolds.