I designed this for a friend I met at an exchange programme to Qingdao because of his interest in math.
This model is mainly a tube with sealed ends. The following series of folding instructions will make this apparent:
Starting with a 60° triangle grid, collapse this crease pattern:
The grey area is a portion of the paper that overlaps, sealing up the gap in the "tube" formed by wrapping the paper around. The collapsed product should look like:
This model has 30 triangles faces. The ends of the "tube" are open but will be sealed in due course. We proceed to hide the extra 10 faces (an Icosahedron has a mere 20). Open the model and fold this crease pattern, which adds a twist to the "tube".
The twist holds the tube together and stabilizes the entire model. The inner flaps of the twist also lock one of the ends of the tube. That end of the model forms a "cap":
Unfold the model one final time and collapse the full crease pattern:
The main step here is to lock shut the flaps from the lower half of the crease pattern. The locking of one of these flaps looks like:
Lock all five flaps like so:
The locking for this end is more sophisticated because the flaps are not conveniently suppressed by inner flaps from a twist, as the other end has. Complete the icosahedron by forming the twist and locking the remaining end again.