Polyhedrons: basic definitions and classification

To start with we define the angles inside the polyhedrons.

• Dihedral angle: It is the proportion of space limited by two semiplanes that are called faces.

• Angle of the polyhedron: It is the proportion of space limited by three or more planes that meet at a point called vertex. An angle of the polyhedron must measure less than $360^\circ$.

Once we have introduced these two angles we can define what a polyhedrons is.

A polyhedrons is the region of the space delimited by polygon, or similarly, a geometric body which faces enclose a finite volume.

The notable elements of a polyhedron are the following:

• Faces: Each of the polygons that limit the polyhedron.
• Edges: The sides of the faces of the polyhedron. Two faces have an edge in common.
• Vertexes: The vertexes of each of the faces of the polyhedron. Three faces coincide with the same vertex.
• Dihedral angles: Angles formed by every two faces that have an edge in common.
• Polyhedric angles: The angles formed by three or more faces of the polyhedron with a common vertex.
• Diagonals: Segments that join two vertexes not belonging to the same face.

To finish, in all the polyhedrons the so called Relation of Euler is satisfied: $$c + v = a + 2$$

$c$ being the number of faces of the polyhedron, $v$ the number of vertexes of the polyhedron and $a$ the number of edges.

Classification and families of polyhedrons

The polyhedrons can be classified under many groups, either by the family or from the characteristics that differentiate them. More specificly:

According to their characteristics, they differ:

• In a convex polyhedron a straight line could only cut its surface at two points.

• In a concave polyhedron a straight line can cut its surface at more than two points, therefore it possesses some dihedral angle greater than $180^\circ$.

• In a polyhedron of regular faces all the faces of the polyhedron are regular polygons.

• In a polyhedron of uniform faces all the faces are equal.

• Polyhedron of uniform edges is when any edges have the same pair of faces meeting.

• Uniform vertexes polyhedron is when on all the vertexes of the polyhedron there are the same number of faces and on the same order.

These groups are not exclusive, that is, a polyhedron can be included in more than one group.

In addition to the previous classifications, we can also classify the polyhedrons by means of its families:

• Regular polyhedrons: They are called platonic figures.

• Irregular polyhedrons

  • The archimedian figures are convex polyhedrons of regular faces and uniform vertexes but of non uniform faces.
  • The prisms and the antiprisms are the only uniform and convex polyhedrons that we have not introduced. All the prisms are constructed with two parallel faces called bases that identify the prism and a series of parallelograms, enough to close off the figure.
  • Johnson's figures are the convex polyhedrons, with regular faces, but only one uniform.
  • The bipyramids and trapezoides are polyhedrons with uniform faces but with neither regular faces, nor uniform vertexes or edges.
  • The Catalan's solid is a non regular polyhedron where not all of its faces are uniform.
  • We call Deltahedra the figures that are only formed by equilateral triangles, note that they do not constitute an exclusive group of figures.