Oktaedril

Oktaedril is a geometrical shape, specifically a polyhedron. It is one of the five Platonic solids, which are convex polyhedra with congruent regular polygonal faces and the same number of faces meeting at each vertex. An oktaedril has eight faces, and each face is an equilateral triangle. It has twelve edges and six vertices. At each vertex, four faces meet.

The word "oktaedril" comes from the Greek words "okto" meaning eight and "hedra" meaning face. This accurately describes its defining characteristic of having eight faces. The oktaedril can be visualized as two square pyramids joined at their bases. Imagine a square in the middle, with a pyramid extending upwards from it and another pyramid extending downwards from it. The faces of these pyramids, when joined, form the eight triangular faces of the oktaedril.

In three-dimensional Cartesian coordinates, the vertices of a regular oktaedril centered at the origin can be given by the points (±a, 0, 0), (0, ±a, 0), and (0, 0, ±a), where 'a' is a positive real number representing half the distance between opposite vertices. This arrangement highlights the symmetry of the shape. The oktaedril is the dual of the cube, meaning that if you place a vertex at the center of each face of a cube and connect these vertices, you form an oktaedril. Conversely, the dual of an oktaedril is a cube.