What is a polyhedron?

18gl A polyhedron is a solid bounded by flat polygons, which are called faces.


Examples of polyhedra:

Examples of polyhedra

The sides of the faces are called edges, and their corners are called vertices. Depending on the number of faces one can specify a tetrahedron (4 faces), pentahedron (5 faces), etc. A polyhedron is considered to be convex if it is located on one side of the plane of each of its faces. A polyhedron is called regular if its faces are regular polygons (i.e., where all sides and angles are equal) and all polyhedral angles at the vertices are equal. There are five types of regular polyhedra: tetrahedron, cube (regular hexahedron), octahedron, dodecahedron, icosahedron.

In a three-dimensional space a polyhedron (the concept of a polyhedron) is a set of finite number of flat polygons where
1) Every side of a polygon is a side of the other one (but only one), called adjacent to the first one (along this side);
2) Every polygon, which is a part of a polyhedron, can be reached by using another polygon adjacent to it, and this one in turn can be used to reach another adjacent polygon, etc.

These polygons are called faces, their sides are called edges, and their vertices are the vertices of the polyhedron.


polyhedrons vertices polyhedron verticesvertices
polyhedrons edges  polyhedron edgesedges
polyhedrons faces  polyhedron facesfaces

A polyhedron is said to be convex if it is located on one side of the plane of each of its faces.

This definition implies that all faces of a convex polyhedron are flat convex polygons. The surface of a convex polyhedron consists of faces that lie on different planes. At the same time, the sides of polygons are the edges of polyhedron, the vertices of the polyhedron are the vertices of its faces, and the plane angles of the polyhedron are the angles of its polygons-faces.

A convex polyhedron is called a prismatoid if its vertices lie in two parallel planes. Prisms, pyramids and truncated pyramids are particular cases of a prismatoid. All lateral faces of a prismatoid are triangles or quadrangles, while quadrangular faces are trapezoids or parallelograms.


Polyhedra in architecture. Alexandria Lighthouse

The Lighthouse of Alexandria, one of the Seven Wonders, was built in the 3d century BC in an...

Polyhedra made of ribbon

The article in “Science and Life” tells about an interesting way of making polyhedra....

On Cosmonautics Day, let's bring the stars

There are coincidences, which we do not even suspect. April 12 - Cosmonautics Day!...

Prince Rupert's Cube

In issue 25 of the Magic Edges, we turned our attention to the fact that by cutting a cube with a...

Mathematical properties of the Platonic solids

One can specify the following mathematical characteristics in each of the five Platonic solids: 1....

How to assemble polyhedra without glue?

So far, we have actively used glue to assemble polyhedrons from the Magic Edges sets. Moreover, we...

Prisms that saved the world

  The plot of the fantastic blockbuster "The Fifth Element", is built on the legend that there are...