Polyhedra Basics

Polyhedra

• A polyhedron is a three-dimensional solid with flat faces, examples include cubes, pyramids, and prisms.
• Spheres and cylinders are not polyhedra because they have curved faces.

Types of Polyhedra

• A regular polyhedron (also known as a Platonic solid) is a polyhedron with all faces being regular polygons that are congruent to one another.
• There are exactly five regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
• These five polyhedra are very symmetrical and are also known as Platonic solids.

Convex Polyhedra

• A convex polyhedron is a polyhedron in which all vertices "go out" and there are no "dimples" or "dents".
• Formally, a convex polyhedron is one where a line segment connecting any two points on the polyhedron is entirely within the polyhedron.

Concave Polyhedra

• A concave polyhedron is a polyhedron with one or more vertices, edges, or faces that "go into" the polyhedron.
• Formally, a concave polyhedron is one where a line segment connecting two points on the polyhedron also contains points outside the polyhedron.

Polyhedron Formula

• Euler's characteristic states that the numbers of faces (F), vertices (V), and edges (E) of any convex polyhedron satisfy the equation: F + V = E + 2.

Polyhedron Models

• A polyhedron model is a representation of a polyhedron, which can be physical or created using computer graphics.
• Physical models can be constructed by cutting out a "net" of the polyhedron and then folding and gluing it together.
• Computer-generated models can be rotated, expanded, and colored to make them easier to understand.

Uses of Polyhedron Models

• Polyhedron models are used in computer-generated imagery (CGI) and in architecture, where buildings are typically polyhedral.