Pictures on this site show several models of polyhedra that I’ve constructed from paper card stock. The model I’m building in this article is a compound (combination) of a dodecahedron and a icosahedron. These two polyhedra are ‘duals’ of each other – a dodecahedron has 12 faces with 5 sides each and 3 faces meet at each of 20 vertexes, while an icosahedron has 20 faces with 3 sides each and 5 faces meet at each of 12 vertexes. In the model, it appears that each vertex of the dodecahedron penetrates the center of a face on the icosahedron, and vice versa.
STEP 1: Templates for parts
I use a software program (Great Stella written by Robert Webb) to design my models, but many templates are available for free as PDF documents on the web.
STEP 2: Making parts
With the templates in-hand I use a large pin to make small holes at each corner. I’m making the dodecahedron in white and the icosahedron in dark blue. Put the template on some card stock and use the pin to transfer the corners to the card stock. Now use a stylus or point of a school compass with a ruler to score all of the edges and fold lines between the pinholes on the card stock. Cut out the part leaving tabs (as shown above). Carefully fold all tabs and fold lines along the scored lines you made.
STEP 3: Construction
This model requires 20 of the dodecahedral parts and 12 of the icosahedral parts. Each of the pentagonal icosahedral parts is surrounded by five of the dodecahedral parts. I use either a small film of water-soluble school glue (like Elmer’s) or spread a small dab of household cement on one tab, then position the pieces and pinch the tabs until the glue just sets. Sometimes a flat-sided toothpick is necessary to get glue into tight spots. Also tweezers with a rubber band used to hold them closed make good clamps to hold tabs while they set. I like to mix up the tasks and work on making more parts while I wait for the glue to dry. Try to get every glued edge straight and aligned, small errors build into big problems by the time you get to gluing the final part.
PART 4: Finishing
The last piece is usually difficult as you have to work through a small hole and you can’t use clamps anymore. It’s useful to make small tools from bent wire which you can use through the hole to push against tabs while they set. The model in the picture uses metallic card stock and is finished as soon as the last part is set. For normal card stock I spray the completed model with a light coat of matte clear polyurethane. This gives the model a slight protection from water and allows it to be dusted and handled without worrying about fingerprints or smudges. The glued tabs act like ribs and make the model fairly rigid.
Icosahedron + Dodecahedron