What is it? (Wikipedia.org)
A Hoberman sphere is based on the principle of using a scissor mechanism for every edge of a regular polyhedron, allowing the length of the edges to change. The scissor mechanisms are connected together at the verticies of the polyhedron by hubs. Because a regular polyhedron must have all edges the same length to maintain its geometry, all of the scissors assemblies must move together and maintain the same length. This just happens naturally when the hubs allow for the freedom of motion necessary for the scissor mechanisms. This is accomplished by having two hubs at each vertex, one connects the outer arms of the scissor mechanisms at that vertex and the other connects the inner arms. As the scissor mechanisms expand and contract the length of each edge the pair of hubs at each vertex move together or apart respectively causing all the other scissor mechanisms connected to them to contract and expand at the same rate. Because all of the scissor mechanisms and hub pairs are inter-connected they will all change together allowing for the uniform scaling of the regular polyhedron. The trick is that the scissor mechanisms must maintain the included angle of the edge throughout their range of motion. This is accomplished by having the arms curved or bent instead of being strait. The angle measured at the pivot point between the ends of the arms defines the angle that will be maintained by the scissor mechanism. (I'll put a link showing this and describing how to calculate this angle later – all of these are based upon the icosododecahedron and therefor use the same angle)
I have designed several assembly kits for making these icosododeca Hoberman spheres. I've documented one here and I'll put up others later.
Press Fit Hoberman using Pins/Pegs