Here is the pentagramma mirificum projected onto a plane; the first two sliders change the pentagramma, and the other two sliders change the location of the tanglent plane the pentagramma is projected onto.



The compound of five octahedra is very interesting, espically when you change the pentagramma to see how the different octahedra move as a result. And what about how the motions of any one octahedron relates to the motions of the other octahedra? Below are a number of animations to help you investigate the interconnected motion of the octahedra.

This is the five octahedra and the corresponding five 90-90-90 triangles of the pentagramma mirificum, with just one octahedron emphasized.            


The pentagtramma with only one octahedron:



Here are two animations of the five octahedra and the 90-90-90 triangles, where just two octahedra are emphasized. The two different animations show the two distinct relations any pair of the five octahedra can have; i.e. two distinct pairs, because both pair interacts differently as the pentagramma changes.


The same as above, but only having the pair of octahedra shown (instead of all five):


And, the same without the pentagramma: