Polyhedra, Tilings and Graphs: Discrete Shapes from the Tetrahedron to the Steiner System

The simplest ordinary convex polyhedron, with four triangular faces, six edges and four vertices. It is also the three-dimensional case of a Euclidean simplex, which is why the path starts here.

Eight vertices, twelve equal edges and six square faces. The cube is the three-dimensional member of the hypercube family, which already points to the next stop.

Twelve regular pentagonal faces meeting three to a vertex, with twenty vertices and thirty edges. Plato set it apart from the other four Platonic solids by associating it with the cosmos as a whole, and it is dual to the icosahedron.

The n-dimensional generalization of the square and the cube, with 2 to the n vertices. The four-dimensional case is the tesseract, a word the Oxford English Dictionary traces to Charles Howard Hinton in 1888.

The smallest convex set containing a set of points, pictured as a rubber band pulled taut around them. The term itself appears as early as Garrett Birkhoff in 1935.

A division of the plane by nearest seed point, formalized by Georgy Voronoy in 1908 and dual to the Delaunay triangulation of the same points.

A covering of the plane that never repeats periodically. Roger Penrose introduced the first version in a 1974 paper, and the tilings later matched real quasicrystals.

Every pair of distinct vertices joined by exactly one edge. Drawings of complete graphs on the points of a regular polygon already appear in the thirteenth century in the work of Ramon Llull.

Vertices split into as equal groups as possible, with every edge present between groups and none within. Pal Turan introduced it in 1941 to prove the result now called Turan's theorem, a foundation of extremal graph theory.

Ten vertices and fifteen edges, constructed by Julius Petersen in 1898 as the smallest bridgeless cubic graph that cannot be edge-colored with three colors. It is a standard counterexample in graph theory.

A graph that encodes the abstract structure of a group from a chosen set of generators. Arthur Cayley first considered them for finite groups in 1878, and they are a central tool in geometric group theory.

An n by n grid in which each symbol appears once in every row and column. Choi Seok-jeong published order-nine examples in 1700, before Euler began the general theory, and a completed Sudoku grid is one.

A collection of blocks in which every t-element subset lies in exactly one block. Jakob Steiner reintroduced triple systems in 1853, and Peter Keevash proved in 2014 that nontrivial systems exist for t of 6 or more, by a non-constructive proof.