Theodore Popelensky

Finite geometries — the Platonic solids. A reminder — right actions, orbits, stabilisers, fundamental domains (corrected definition) and the example of $\mathbb{Z} + \mathbb{Z}$ on the plane; the formula $|G| = |\operatorname{Orb}(x)| \cdot |\operatorname{Stab}(x)|$ and the 48 symmetries of the cube; regular polyhedra — Sosinsky's definition and the definition via flags, a stellated counterexample; the five Platonic solids; regular polytopes in $\mathbb{R}^4$ (six) and in $\mathbb{R}^n$ for $n \ge 5$ (the cube, the simplex, the hyperoctahedron) with the coordinates of their vertices; why there are no more than five — face angles and the Schläfli symbol; uniqueness of the gluing and the rigidity theorem; Euler's formula; the finite subgroups of $SO(3)$ — $\mathbb{Z}_n$, $D_n$ and the rotation groups of the tetrahedron, cube and icosahedron; the duality cube–octahedron, icosahedron–dodecahedron; which figures on the sphere realise $\mathbb{Z}_n$, $D_n$ and $\mathbb{Z}_2$.

What geometries there are and what they have in common — Klein's Erlangen programme; the school 3–4–5 triangle and the ellipsoid; the cube and the brick as finite geometries; metric spaces; groups, quantifiers in the axioms; the group of bijections and the order of composition; left and right actions of a group on a set; the metrics $d_1$, $d_2$, $d_\infty$ on $\mathbb{R}^n$; isometries, subgroups; the order of an element; orbits, stabilisers, transitive actions; the list of geometries in the course — the Euclidean plane, the regular triangle and $S_3$, the cube and its 48 symmetries, the Poincaré model of the Lobachevsky plane; fundamental domains; subgeometries — the square in the cube.