Monday, May 4, 2015

Unizor - Geometry3D - Elements - Polyhedrons

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Polyhedrons are objects in three dimensional space, all faces of which are polygons connected to each other at common edges and edges are connected at common vertices.

The simplest example of a polyhedron is a tetrahedron, all 4 faces of which are triangles. It has 4 vertices and 6 edges.
We can imagine pentahedrons with 5 faces or hexahedrons with 6 faces. So, the number of faces is a characteristic of a polyhedron that is important for its classification.
Obviously, all prisms and pyramids are particular types of polyhedrons.

We will mostly be dealing with convex polyhedrons - those whose surface divides the entire three dimensional space into "inside" and "outside" of a polyhedron, in such a way that all points of any segment connecting two points on its surface are located on its surface or inside of it.

A remarkable property of convex polyhedrons is the correspondence between the number of its vertices, edges and faces expressed in the famous Euler's formula:
V − E + F = 2

A particular kind of polyhedrons are regular ones. All their faces are regular polygons congruent to each other with all congruent angles between the faces. Examples of such regular polyhedrons are regular tetrahedron (all faces are equilateral triangles) and cube - regular hexahedron with square faces. There are others as well, known since antiquity.

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