Like a cube, an octahedron has three different types of axes of
rotation. Each axis intersects to two diametrically opposite
points on the octahedron. The left photo shows a point with the
same symmetries as a square (a vertex). The middle photo shows a
point with the same symmetries as an equilateral triangle (a
face). The right photos shows a point with the same symmetries as
a rectangle (an edge). We sell a pattern for an Octahedral Cluster
bead.
Other beaded beads with these axes of rotation: Star Cluster