We have colored the example above with green and yellow stain, but students may prefer to try their own creative coloring ideas

- 60 pieces laser-cut from 1/8-inch thickness (3 mm) plywood using this template. (Red=cut. Blue=etch.)
- 150 uncolored 4-inch cable ties (example)

- Diagonal wire clippers (example)

- Optional stain and foam brushes if coloring the wood (e.g., ColorCraft Brusho and foam brush)

1. This is the third of four related geometric sculpture activities. For an introduction to the materials and concepts, students should first do theAutumnandWintersculptures.

2. Only 120 ties are needed for the design, but have extras available because some mistakes will be made, which need to be clipped and discarded.

3. Our example, illustrated above, is stained with Leaf Green on one side and Yellow on the other side. Other colors, a solid color, or no coloring are all options.

1. Laser-cut the parts, including the etched lines. Smoke marks can be minimized by applying laser-safe tape to the bottom surface before cutting or the surfaces can be lightly sanded afterward. A quick light pass on both sides with an orbital sander (150 grit) removes smoke marks, gives the parts a smooth surface, and prepares them for staining.

2. Two of the edges need to be beveled slightly at 45 degrees, as shown in the above image. The other edge does not need to be beveled. Only a small amount of material needs to be removed. It is sufficient to sand down to half the thickness of the wood, as illustrated above.

3. Material should be removed from the front side, i.e., the side with the etching. This is done by placing that face down on disk sander or belt sander with the table set to 45 degrees. Keep the edge parallel to the plane of the sandpaper, so the depth of the sanding is even at all points along the edge.

1. Remind students about safety when using cable ties: Cable ties should only be used for the construction and can be dangerous if placed around any part of the body. (See the previous geometric sculpture activities for additional cable tie instructions.)

2. Organize students in groups of two or three. Hand out three pieces to each group. Ask students to explore the pieces and how three might fit together to make a module.

4. Explain that we will initially make a module of three pieces with three-fold rotational symmetry using the beveled edges. Ask students to figure out what it might be. When they have a solution, they can hold their three pieces together in position and compare it to the rest of the class. Point out groups with the correct arrangement. Seeing how the beveled surfaces mate should make clear the purpose of the beveling. (It is possible to position the parts in a similar way but "inside out," however then the beveling of the edges is not used.) Help everyone see how the 90 degree angles come together like the corner of a cube and the three planes are orthogonal.

6. Check each module to ensure the parts are
properly joined and the ties are tight. You can snip off
the tails with the wire cutter as a mark of which modules you
have checked.

7. For the assembly, pick a table in a central location
that everyone can see. You can tell the class that the
other ends of the parts also meet in groups of three to form
triangles with 3-fold rotational symmetry. (This triangle
is illustrated in the image above.) Students will now be
connecting the modules using the un-beveled end of the part and
the pair of holes near that end. For this connection, the
un-beveled end meets the __front__ (etched, green) side of
the part.

8. At this stage, the process becomes something of a puzzle. Groups should take turns bringing their modules to a central location and keep adding modules to a single growing structure. One student can hold a module in position where it is to be attached while other students connect the cable ties. Encourage each group to have a turn then make room for others. Allow them to answer each other's questions about where to connect everything. Students will see the patterns and figure out how to extend them to complete the sculpture. Any group not working on the growing construction at the center of the room can take three more parts and three more ties and make another module, continuing until nineteen or twenty modules are complete.

9. A key observation you might want to elicit or point out is that the cube corners form an inner layer and the new triangles form an outer layer. Each cube corner lies directly inside of a new triangle.

10. Students will also be surprised to observe the formation of regular pentagon tunnels that pass entirely through the sculpture, like the one shown above. Ask them do determine how many such tunnels there are.

11. Getting the final module in position requires some gentle maneuvering. It is best to get the final few modules into position before connecting any of them with cable ties, so there is some flexibility to maneuver. Alternatively, the final module can be assembled in place from three separate pieces.

12. When complete, check all the connections are correct, all the cable ties are tight, and snip off the ends of the ties.

13. While working, you can ask students who aren't engaged in the construction to figure out how many cable ties are needed all together. From the fact that there are 60 pieces and each touches four others they should determine there are 120 cable ties. That's 60ยท4 = 240 connection locations, but divided by two because each tie joins two connection locations.

8. At this stage, the process becomes something of a puzzle. Groups should take turns bringing their modules to a central location and keep adding modules to a single growing structure. One student can hold a module in position where it is to be attached while other students connect the cable ties. Encourage each group to have a turn then make room for others. Allow them to answer each other's questions about where to connect everything. Students will see the patterns and figure out how to extend them to complete the sculpture. Any group not working on the growing construction at the center of the room can take three more parts and three more ties and make another module, continuing until nineteen or twenty modules are complete.

9. A key observation you might want to elicit or point out is that the cube corners form an inner layer and the new triangles form an outer layer. Each cube corner lies directly inside of a new triangle.

10. Students will also be surprised to observe the formation of regular pentagon tunnels that pass entirely through the sculpture, like the one shown above. Ask them do determine how many such tunnels there are.

11. Getting the final module in position requires some gentle maneuvering. It is best to get the final few modules into position before connecting any of them with cable ties, so there is some flexibility to maneuver. Alternatively, the final module can be assembled in place from three separate pieces.

12. When complete, check all the connections are correct, all the cable ties are tight, and snip off the ends of the ties.

13. While working, you can ask students who aren't engaged in the construction to figure out how many cable ties are needed all together. From the fact that there are 60 pieces and each touches four others they should determine there are 120 cable ties. That's 60ยท4 = 240 connection locations, but divided by two because each tie joins two connection locations.

1. Ask students to relate the structure to a Platonic solid. They should see it has the same symmetry as an icosahedron or dodecahedron. The twenty triangular openings are arranged on 3-fold axes. The twelve pentagonal openings are arranged in six opposite pairs on the 5-fold axes as in a dodecahedron, but there is an interesting difference. In a regular dodecahedron, opposite pentagons do not have their edges parallel---they point opposite ways, i.e., one is rotated 36 degrees (or, equivalently, 180 degrees) relative to the other. But here, opposite pentagons do have their edges parallel, as the photo above makes clear. How is that possible? (Answer: Imagine starting with a regular dodecahedron and twisting each face clockwise about its center 18 degrees. By making the same change everywhere, you preserve symmetry. The 36 degrees of difference is made up for half on each end.)

2. The 2-fold axes are less obvious, but (as always in anything with this symmetry) they can be found by looking halfway between two nearby centers of 5-fold symmetry, or halfway between two nearby centers of 3-fold symmetry.

3. As with the other sculptures in this series, there is no mirror symmetry. So the sculpture is

4. Ask students to look for co-planar pieces---more than one part lying in the same plane. They should observe that each piece has exactly one mate in its own plane, so the sixty parts lie in just thirty planes. Positioning your eye to be in the plane of a piece and sighting along that plane makes it easy to see the co-planarity.

B. Try all four sculpture activities. Can you adapt the others to cardboard?

C. Try the Symmetry Search game described here.