You can cut an octahedron in half to reveal a regular hexagon (drag with your mouse to move it):

If you'd like to make a half-octahedron showing this hexagon face, there's a net here.

If you look end on at one face of the octahedron you'll find that the outline of the octahedron is actually a regular hexagon too. Try it with the figure below - look directly end on at the face with red edges, and the hexagonal outline is formed by the blue edges.

Cutting the octahedron in half another way reveals a square:

(Java applet from LiveGraphics3D)

page date: 18Nov04.      I enjoy correspondence stimulated by this site. You can contact me here.