Polyhedra Quiz

Polyhedra: 3D Shapes with Flat Faces, Straight Edges, and Sharp Corners

• A polyhedron is a three-dimensional shape with flat polygonal faces, straight edges, and sharp corners or vertices.

• Convex polyhedra are the convex hull of finitely many points, not all on the same plane.

• A polyhedron can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices), faces (two-dimensional polygons), and it sometimes can be said to have a particular three-dimensional interior volume.

• The surface of a polyhedron may be orientable or non-orientable, and the Euler characteristic can determine this distinction.

• Polyhedra may be classified and named according to the number of faces, using a naming system based on Classical Greek.

• For every convex polyhedron, there exists a dual polyhedron having the same number of faces.

• For every vertex, one can define a vertex figure, which describes the local structure of the polyhedron around the vertex.

• The surface area of a polyhedron is the sum of areas of its faces, and the volume of a polyhedral solid is given by a formula that uses dot products.

• Dehn invariant is a vector in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges.

• A convex polyhedron is a polyhedron that, as a solid, forms a convex set.

• Many polyhedra are highly symmetrical, and each such symmetry may change the location of a given vertex, face, or edge.

• Important classes of convex polyhedra include the Platonic solids, the Archimedean solids, and the regular-faced Johnson solids.Polyhedra: a summary of geometric shapes

• A polyhedron is a three-dimensional shape made up of flat faces, edges, and vertices.

• The symmetry group of a polyhedron is the collection of symmetries that leave the polyhedron unchanged.

• Symmetry orbits are formed by elements that can be superimposed on each other by symmetries.

• Regular polyhedra are the most highly symmetrical and consist of five convex and four star polyhedra.

• Uniform polyhedra are vertex-transitive and every face is a regular polygon.

• Isohedra are polyhedra with symmetries acting transitively on their faces.

• There are polyhedra with regular faces but lower overall symmetry than regular and uniform polyhedra.

• Apeirohedra is a class of objects with infinitely many faces.

• Complex polyhedra have an underlying space that is a complex Hilbert space rather than real Euclidean space.

• Spherical polyhedra are formed when the surface of a sphere is divided by finitely many great arcs.

• Convex polyhedra can be defined in three-dimensional hyperbolic space.

• The skeleton of every convex polyhedron is a 3-connected planar graph.

• The term "polyhedron" has also been adopted to describe various related but distinct kinds of structures.

• A polytope is a bounded polyhedron.A Brief History of Polyhedra

• A simplicial figure is a topological space that is decomposed into shapes that are topologically equivalent to convex polytopes.

• An abstract polytope is a partially ordered set of elements whose partial ordering obeys certain rules of incidence and ranking.

• Polyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided pyramids of ancient Egypt also dating from the Stone Age.

• The Greeks were interested primarily in the convex regular polyhedra, which came to be known as the Platonic solids.

• During the Renaissance, star forms were discovered and artists such as Wenzel Jamnitzer depicted novel star-like forms of increasing complexity.

• The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation.

• In 1750, Leonhard Euler for the first time considered the edges of a polyhedron, allowing him to discover his polyhedron formula relating the number of vertices, edges, and faces.

• Max Brückner summarized work on polyhedra to date, including many findings of his own, in his book "Vielecke und Vielflache: Theorie und Geschichte" (Polygons and polyhedra: Theory and History).

• Coxeter analyzed The Fifty-Nine Icosahedra, introducing modern ideas from graph theory and combinatorics into the study of polyhedra.

• In the second part of the twentieth century, Grünbaum published important works in two areas: convex polytopes and the accepted definition of a polyhedron.

• Irregular polyhedra appear in nature as crystals.

• For natural occurrences of regular polyhedra, see Regular polyhedron § Regular polyhedra in nature.