Someone should design a device like a compass gimbal with an extra ring for teaching euler's angles, patent it (Gnu hardware license- world demand is probably 100 pieces), and persuade Hampton research or MitEGen to manufacture it.
The device (picture at (http://sb20.lbl.gov/berry/Euler2.gif), but I'm no artist) consists of 3 concentric rings: The outer ring is mounted by external studs in an F-shaped support so it can rotate about a vertical diameter (alpha, new Z). One of the bearings has a brass disk with degree marks etched and an indicator on the ring reads alpha angle. The central ring is connected to the outer ring by by horizontal bearings, allowing it to rotate about a horizontal diameter. Likewise a brass disk indicates the beta angle. The innermost ring is connected to the central ring by vertical bearings, allowing it to rotate about a vertical axis (when beta and alpha are zero). An indicator read gamma. Inside the inner ring is suspended an asymmetric object like a pointing hand or arrow painted red on one side and green on the other. Also wire axes indicating x,y,z direction in the original crystal. The F-shaped mounting bracket would have x,y,z direction in the target crystal indicated (and they would be the same when alpha, beta, and gamma are zero, i.e. the rings are coplanar). Playing with this would take some of the abstractness out of Euler angles. It would also let the student resolve for herself the apparent contradiction that all orientations can be reached by the inner object, despite the fact that two of the rotations are (initially) coaxial. Ed
