Making Math Visible

Cardboard Dome



Time Required: 1.5 hours

Materials:
Notes:

1. The paper dome workshop is a great introduction to this activity and should be done first as it provides all the necessary background.
2. The template files are scaled for a dome four feet in diameter.  They can be scaled for other sizes, but both files must be scaled the same amount.
3. The cardboard struts can be cut with a band saw or laser-cutter.  The laser-cutter file includes a dotted line for folding.  If using a band saw, mark the fold line by hand.
4. The biggest challenge will be to keep track of the two different lengths of struts, as they are similar. You may want to make the difference clearer by color coding with a marker or sticker.
5. Medium-size black binder clips work well to hold the parts while the glue is drying but may be difficult for younger students to open. 
6. Zometool is an ideal manipulative for illustrating the structure of domes.  It has exactly the lengths and angles required for this particular structure.  If you don't have Zometool, you can improvise with other strut-and-node systems, such as straws and pipe cleaners, or toothpicks and marshmallows.
7. Students can work in groups of two to five.

Part A. Minds-On



1. Hand out blue and yellow Zometool struts and a supply of connector balls.  Giving as little instruction as possible, refer back to the paper dome and ask students to make a structure that is isomorphic to it using the Zometool materials.  They will discover that the yellow struts are slightly shorter than the blue struts and will conclude that the yellows should meet five at a vertex, with a blue pentagon surrounding. 

2. Discuss and review the structure and geometric properties of the dome.  In the image above, the yellow struts correspond to the edges of the underlying icosahedron.  The blue struts correspond to the colored lines that were drawn to create half-size triangles.



3. Point out that one can remove a "starfish" of five yellow struts to create a doorway if desired, as shown above.


Part B. Hands-On

1. Before handing out the cardboard pieces, instruct students to be careful not to bend them except on the desired fold lines.  Cardboard is quite strong for its weight if it hasn't been creased.  Do not hand out glue yet, as the initial assembly should just be with clamps.

2. Students can work in either three or six groups as there are six modules to be assembled.  Divide the class up and hand out the cardboard pieces accordingly.



3. Ask students to fold the tabs on the dotted line as they did with the paper: up on the right and down on the left.  Use a ruler, or fold them over the edge of a desk.



4. Remind students of how the tabs go together in a cyclic structure.  You can illustrate this by drawing the above image on the board.  Every joint looks similar to this, but may have either five or six struts meeting.



5. Hand out clothespins or binder clips.  Just as in the paper dome, groups will make a module in which five short struts meet at the center, surrounded by five longer struts making a pentagon around it. 



6. Before joining the modules together, ask students to visualize the entire dome by holding one module horizontally as the top center piece, and placing the other five around it, each with one strut on the floor.  They will see how the five remaining long struts fill in the base to make a circle.

7. Ask students to work together, with some holding the modules in position while others apply the clips.  If the parts don't fit perfectly, double-check the lengths before gluing.

8. After the dome is erected with clips, decide whether or not to make an opening.  If an opening is desired, remove a "starfish" of five short struts.



9. Hand out small amounts of glue and the glue brushes.  Instruct students to work six at a time, applying glue from different sides.  The best method is to remove one clip, apply a small amount of glue to the entire surface of the tab, then clip it back in place.  If you use black binder clips, you can have the convention of folding the clips forward to mark the tabs that have already been glued.  (See image above.)



10. After the glue is dry (up to half an hour) remove the clips.  Enjoy!


Part C. Conclusion

Ask students to compare the cardboard dome with the paper dome and their marked icosahedron to observe the analogous structures.  (This provides another example of an isomorphism, as introduced in the 12-Card Star workshop.)


Possible Extensions

See the Paper Dome Workshop Extensions.