We have colored the example above with several Fall colors, 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.

- 150 four-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 first of four related geometric sculpture activities.

2. To get the most from this activity, students should be familiar with the Platonic solids and their axes of rotational symmetry. We recommend doing that workshop first.

3. Many young students find cable ties challenging. We recommend this activity for grade 5 through adult.

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

5. The first time you build this, don't try to make a color pattern (as in the image above), just focus on the geometry.

1. Laser-cut the parts. 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 and gives the parts a smooth tactile surface; it is optional, but recommended if you are staining the wood.

3. The two longer edges, which meet at a sharp vertex, should ideally be sanded with the table tilted 54 degrees down from horizontal. However, many sanders only allow the table to tilt to a maximum of 45 degrees, and that will work fine. We want to remove material from the back surface, so ensure that that side is down. You only need to remove a slight amount of material---half the thickness of the wood is plenty.

The image above shows what the edges look like
after beveling. Note that it is important that the bevel
be equal all along the length of the edge.

4. For the two short edges, the wood is turned over and the table should be set to 18 degrees. The front face is down so we remove material from the front face.

5. Optionally, the parts can be stained. We used five different Fall colors for the example shown above and followed a specific coloring pattern described in the explorations below. But the first time you build this design, you might want to work with the unstained parts or use a solid color and focus on the geometry. To stain the parts, brush on a water-based stain with a foam brush and wipe off the excess with a paper towel. Let dry.

1. Give students a quick safety briefing about how to use cable ties: Cable ties should only be used for construction and can be dangerous if placed around any part of the body. If students are unfamiliar with them, you may need to explain how the tip must go in to "the flat side" of the box. It comes out "the bump side." And it must loop like a cylinder, not twisted like a Möbius strip.

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

3. In order to familiarize students with the parts, ask them to describe the shape and discuss observations such as the following: All the parts are identical. The boundary consists of various curves and four straight segments, each with a small hole nearby. Two of the segments are longer and meet at a vertex; the other two are shorter and don't meet. Each edge has been beveled slightly. Point out that the straight segments are the connecting edges of each part, and the hole is where a cable tie will be used to connect it to another part. Beveled surfaces will meet with one another.

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. The class now needs to work together to assemble the
modules. One way to begin is to pick five groups for the
first stage and ask one person from each group to bring their
module to the front of the room. Ask them to position them
to make a 5-fold star, as shown above. They will now be
joining the short segments. Suggest that they use symmetry
as a guide for how the parts join, along with the principle that
their beveled surfaces should mate together. They should
see that there are ten places they can now add a cable tie to
connect these modules. (Five are near the center of the
image above and five are on the sides.) While the first
five people continue to hold the parts, ask for a second person
from each group to come up and make two of the cable tie
connections. Hand out ten ties for this.

8. After this, the students should see the pattern and be able
to complete the structure, making similar 5-fold stars
everywhere. Tell them that they should make only 5-way
stars, never 4-way or 6-way. A general tip is that one student
can hold a piece in position where it is to be attached while
another student connects the cable ties. Depending on the
group, you can work in a very structured way, organizing them to
add five modules at a time symmetrically, or you can let them
explore and decide for themselves where and in what order to add
their modules. It is safest to keep adding modules to a
single growing structure. (It would not work to try making
two separate "halves" and hope they fit together!)
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. Those who are waiting can take three more parts
and make another module until all twenty modules have been
built.

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

10. 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 60·4 = 240 connection locations, but divided by two because each tie joins two connection locations.

10. 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 60·4 = 240 connection locations, but divided by two because each tie joins two connection locations.

11. Take group photos and hang it up
on display. This image shows a version stained all in
one color.

12. And here's a version that is stained brown on the
outside and yellow on the inside, assembled at a workshop at
Eastern Carolina University.

1. Ask students to relate the structure to a Platonic solid. They should notice there are twelve pentagonal openings and twenty 3-fold vertices, as in a dodecahedron. There are also thirty curvy openings, corresponding to the edges of the dodecahedron, i.e., each lies between two vertices.

2. Discuss the rotational symmetry axes of the sculpture. Students should be able to find the 3-fold axes (lines that connect opposite 3-fold vertices), the 5-fold axes (through the center of each pentagonal opening), and the 2-fold axes (in the curvy space between any two adjacent vertices). Demonstrate the rotations on these axes which bring the sculpture back to its position. Point out that the axes are arranged in the same way as a dodecahedron or icosahedron. There are ten 3-fold axes, six 5-fold axes, and fifteen 2-fold axes.

3. Discuss whether there is any mirror symmetry in the sculpture. Students should see that (unlike a regular dodecahedron) there is no mirror symmetry, so the sculpture could have been made in a left-hand or right-hand form. It 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. If students are familiar with the rhombic triacontahedron, you can explain that the thirty planes of the sculpture's pieces are the face planes of an imagined (invisible) rhombic triacontahedron in the center of the orb.