How to assemble a 30-unit Kusudama

To assemble a 30-unit kusudama (typically an icosahedron), you connect the individual, pre-folded modular units by inserting a tab from one unit into a pocket of another unit. The structure is built around points where five units meet, forming a pentagonal shape, and triangular pyramids. 

Preparation

• Fold 30 identical units: Most 30-unit kusudama models use the Sonobe unit or a variation of it. Ensure each unit has a "tab" (or flap) and a "pocket" (or slit) for interlocking.

• Crease well: Firm creases will help the finished model hold its shape without glue. Glue or tape can be used for extra stability, especially for a first attempt. 

Assembly Steps

1. Form the first triangular pyramid: Take three units. Each unit has two tabs and two pockets. Insert one tab into a pocket of the next unit, and continue until all three units form a triangle or a pyramid shape. The units should interlock securely.

2. Build a pentagonal base: Continue adding units to the initial triangle, working in a circle. The goal is to create a central point where five units converge, forming a five-sided "star" or pentagonal face.

3. Continue adding units: Keep building by adding new units to the existing structure. Focus on forming a sequence of connected triangular pyramids, always aiming for a point where five units meet.

4. Work in sections: Some folders find it helpful to work on one "side" or a few connected pyramids at a time, then link them together.

5. Maintain the shape: The kusudama will start to curve and form a spherical shape as more units are added. Gently guide the pieces to maintain an even, spherical form.

6. The final units: The last few pieces can be tricky to insert. Take your time and gently bend the model as needed to tuck the final tabs into their corresponding pockets. The structure should become very stable once the last unit is in place.

7. Finishing touches: Once assembled, you can adjust the pieces slightly to ensure the model is tightly formed. You can also attach a string or tassel if desired. 

Assembling a 30-unit kusudama relies on the mathematical principles of an icosahedron (20 triangular faces, 12 vertices). Keeping this geometry in mind can help you understand how the pieces fit together and where the next unit needs to go. 

For any mathematical shape, we can use a unit as edge, vertex or face. See polyhedron property in this chart.

Vertex, Edge, Face in each polyhedron (from Tomoko Fuse's Unit Origami Fantasy)