What is a polyhedron?

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

• Faces: These are the flat polygonal surfaces that bound the polyhedron. Examples include triangles, squares, pentagons, etc.

• Edges: These are the line segments where two faces meet.

• Vertices: These are the points where three or more edges meet.

• Types of Polyhedra: There are many types of polyhedra, classified by their properties. Some important types include:

  • Regular Polyhedra (also known as Platonic Solids): These are polyhedra whose faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
  • Archimedean Solids: These are convex polyhedra made of two or more different types of regular polygons meeting in identical vertices. They are less restrictive than regular polyhedra.
  • Prisms: These have two congruent polygonal faces (bases) that are parallel, and the other faces are parallelograms.
  • Pyramids: These have a polygonal base and triangular faces that meet at a single vertex (apex).

• Euler's Formula: For any convex polyhedron, the number of vertices (V), edges (E), and faces (F) are related by Euler's formula: V - E + F = 2.

• Duality: Many polyhedra exhibit duality, where the vertices of one polyhedron correspond to the faces of another, and vice versa. For example, the cube and octahedron are duals.