Octahedral twisty puzzles

A twisty puzzle can be built so that either the sides or the corners twist. This distinction is particularly apparent in Octahedra.

Here, the sides twist. Professor Ernő Rubik had the idea for twisting an octahedron this way.

The same idea, in size 3x3.

In this one, the corners twist.

The corners twist here.

When an octahedron is only 2x2, there is no point in colouring the sides. But you can make a puzzle by colouring at the corners.

This is a larger octahedron with coloured corners. A physical implementation of this was invented by Josef Trajber.