Antiparallelogram-based linkage traces out a three-lobed curve (more linkage curves linked from G+)
That packing thing is interesting. One ML professor in the draft of his book claimed that the sphere in the center of the cube touching four other spheres became a "porcupine" in high dimensions: it sticks through the wholes and outside the cube. That made me scratch my head, because a convex round object should be convex and round in any projection. However, I have figured out that the sphere becomes so big that it starts sticking outside (actually this is clearly backed up with simple math). But it still has to be round, so it won't look like a porcupine (which is not convex).
Apparently, this sticking phenomenon is possible due to the space emptiness. The four other spheres occupy a tiny fraction of the cube space, so, apparently, they do not prevent the central sphere from going (as dim increases). And eventually it can even stick out of the cube! This seems to be an extremely interesting packing paradox.