Disc-O-Ball

Truncated Icosahedron CD Sculpture

The "truncated icosahedron" is a well-known shape that students will be familiar with because it is the formal name for the shape of a soccer ball.  In this workshop, students discover the deeper mathematical ideas behind the truncated icosahedron by drawing and building the 3D shape themselves.  Students learn about "truncation" by starting with the simpler, familiar example of cutting off the corners of a cube to make a "truncated cube".  Then they move on to the icosahedron and the truncated icosahedron. They are surprised when their drawing turns out to be a drawing of a soccer ball.

This type of activity develops logical and spatial reasoning and helps create a fluency with geometric ideas and vocabulary that is beneficial throughout students' entire mathematical careers.  Translating the soccer ball's underlying structure into a new material helps students appreciate and become comfortable with abstraction in mathematics.  It demonstrates that mathematics can be found within many familiar objects in their everyday lives.

Students enjoy collaborating on this shiny, large-scale construction that they can display in their classroom to show their school how much fun math can be.  This workshop can be used as a starting point for a series of lessons on polyhedra or as a culminating activity.  It could be extended by drawing and building additional polyhedra.

This activity provides rich classroom material for teachers following the Common Core Standards for Mathematical Practice.  This lesson also provides cross-curricular connections to art and architecture.

# Detailed Instructions

• ## 1.5 hours for preparatory drawing (Part A)

• 1.5 hours for assembly (Part B)

## Materials

• Pencils
• Large blank paper
• 160 CDs
• 300 cable ties (6 to 8 inches long)
• Diagonal wire cutters
• String (to suspend approx. 3 pounds)

Optional Materials

• Geometric models of cube, truncated cube, icosahedron (e.g., paper, Zometool, wood, foam...)
• Soccer ball

Notes:

1. The sculpture consists of only 150 CDs, however a few might break, so have at least 160. These can be new blank CDs, or recycled (used) with mixed labels, or surplus from a printed run (all the same label). (Surplus CDs can sometimes be obtained from manufacturers, for art use, if they develop a new product version and discard their stock of the previous version.) Be cautious about using DVDs because some are constructed differently and are not as strong but others are fine.

2. The sculpture is held together with 180 cable ties, however when an assembly error is made or a CD is broken, some ties must be cut off to make the repair, so have 300 ties available.

3. Choose an appropriate place ahead of time to hang the structure from, because you will work on it while it is hanging.  It is light enough to clip on to a suspended ceiling.

4. This activity can be divided over several sessions.  The preparatory drawing is optional though highly recommended for deeper understanding.

Safety Note:

During the construction phase, always have a pair of wire cutters on hand in case a student places a cable tie around a body part.

Part A.  Preparatory Drawing

Note: the underlined numbers in this section are the answers to fill-in-the-blank questions that can be elicited from students.

1. Hide the soccer ball before the lesson starts so it will be a surprise when students discover it.  However, it will be helpful to show physical models of a cube, truncated cube, and icosahedron when they are drawing them.

2. Begin the lesson by distributing paper and pencils to students.  Ask students to make all their drawings large.  Ask them first to draw a cube. Elicit responses and let students share their approaches.  Using the board, sketch and discuss several ways to represent a cube.  (In early elementary grades, representing 3D shapes on paper might be a preparatory lesson in itself.)

Discuss dimensions and distortion: the drawing is 2D, so must distort lengths or angles compared to a 3D cube which has equal lengths and all 90-degree angles.  Often, drawn cubes will have some 90-degree angles and some that aren't. Similarly, the lengths of the drawn segments may be different.  This is important to discuss, especially in younger grades, when students are just becoming familiar with the properties of geometric shapes.  It is also essential to point out (to all grades) that a drawing is like a mathematical model in that it only incorporates certain aspects of an actual 3D shape.

3. After discussing several ways of representing a cube, direct students to follow the the "isometric" method by having them copy on their sheet what you draw on the board. (These drawings do not need to be very precise; a clear sketch is sufficient.)  As shown below, draw a regular hexagon. (Note: there are two vertical sides, no horizontal sides.) Connect the bottom vertex and two others to a point at the center. With dotted or lighter lines, connect the remaining three vertices to a point that is also at the center.

Emphasize modeling and visualization of the 3D object from the drawing. All the cube edges are identical, but by convention we can use dotted lines to represent edges that are hidden behind the back of a solid object.  Also note that the center of the drawing superimposes the near and far vertices, but that they are actually as far as possible away from each other in the 3D object. Young students take delight in the discovery of how to represent 3D objects on paper and should be given sufficient time to play with these ideas.

4. Now use the drawing (and physical model) to have the students count and answer questions about the number of components in a cube.  They should respond that a cube has 6 faces, 8 vertices, and 12 edges. At each vertex, 3 squares meet.

5. A "truncated cube" is a cube with the corners cut off.  Ideally a physical model can be shown, otherwise indicate with a cube the action of sawing off the corners.

The next step is to have students draw a truncated cube, as a way for them to more deeply understand the process of truncation.  Illustrate the following steps on the board and have the class follow on their own papers. Start by drawing the "isometric" cube as above, but lightly, and large enough to fill the paper. Mark two points on each of the cube’s edges [left figure below]. The points should divide the edges into three approximately equal pieces. Darken the middle third of each cube edge [middle figure below]. Around each cube vertex, connect three points to make a triangle with dark lines [right figure below].

You can erase the truncated parts and darken the triangles to help make the structure clear. In this diagram, the lines in the back are drawn lighter than the ones in the front.

6. With students, count and discuss the number of components.  The truncated cube has 14 faces, (namely: 6 octagons and 8 triangles).  It has 24 vertices and 36 edges. (Don’t count one-by-one; use addition and/or multiplication.) At each vertex, 1 triangle and 2 octagons meet.  The 6 octagons correspond to the original 6 faces of the cube.  The 8 triangles correspond to the original 8 vertices of the cube.

7. Now it is time to draw an icosahedron.  If you have time and a good physical model, you can first challenge students to find their own ways of drawing the icosahedron.  Demonstrate this method on the board and have students follow on their own papers:

Draw a regular pentagon—large . (Try to make it regular, with equal sides and angles. It shouldn’t look like a house.) Connect each vertex to the center. Add a dot just outside of each edge. Connect each dot to two nearby vertices of the pentagon. With dotted lines, connect the five dots in a cycle to make a dotted pentagon. With dotted lines, connect the five dots to the center.  Again you can point out that superimposed at the center are two points that are actually as far as possible away from each other in the 3D icosahedron.

8. Solicit the facts that the icosahedron has 20 faces, 12 vertices, and 30 edges. (Count these using groups of five.) At each vertex, 5 triangles meet.

9. Tell students to put together the process of truncation and their understanding of the icosahedron. With older students, you can challenge them to draw a truncated icosahedron without detailed instructions and some will discover that a truncated icosahedron is the shape of a soccer ball before you demonstrate the following. With younger students, simply demonstrate this method on the board as they follow:

Draw another icosahedron, as above but very large and leave out the dotted lines. (We are leaving out the back to keep things simple.) Mark two points on each edge of your icosahedron, dividing each edge into thirds. Darken the middle third of each edge. Around each icosahedron vertex, connect the five dots with dark lines, to make a pentagon. If you want it to look more like a soccer ball, color the six visible pentagons black.

10. Solicit the facts that the truncated icosahedron has 32 faces (namely: 12 pentagons and 20 hexagons).  It has 60 vertices and 90 edges. At each vertex, 1 pentagon and 2 hexagons meet.  Tell students that they’ll need to carefully keep track of pentagons and hexagons in what follows and to remember that each vertex has the same combination: one pentagon and two hexagons.

With middle and elementary school students, stop the drawing at this point and start the assembly (Part B). With high school students, continue with the next drawing step

11. Explain the concept of foreshortening. (Distances on paper do not correspond to distances in 3D, depending on the angle of the object.) In particular, a 3D circle, e.g., a CD, will appear in a drawing as an ellipse.  Hold up one CD as an example and point out that as you tilt it from face-on to edge-on, the view changes from a circle to a line segment with a range of ellipses in between:

12. Students will now draw a more detailed sketch of the construction.  The sculpture is based on the truncated icosahedron with one CD for each vertex and one CD for each edge, therefore they will draw an ellipse for each vertex and one for each edge.  Students should first lightly draw a truncated icosahedron either by the steps above or by tracing their first drawing. Make sure it is as large as possible. As shown on the left below, they will place a dark ellipse (representing a CD seen at an angle) over each vertex. The ellipses should almost touch at their edges. Because of foreshortening, the ones near the center (which are seen face-on) appear like circles but the ones near the sides (which are seen from an angle) appear like thin ellipses. Remind students to draw the CDs with more foreshortening as they get further from the center. Call these CDs the "vertex disks" because there is one for each vertex of the truncated icosahedron.

Then, as shown on the right above, indicate a CD for each edge of the truncated icosahedron with two arcs that represent an “edge disk” behind the “vertex disks”. The disks overlap and when constructed, will be connected by cable ties which are the “glue” to hold everything together. During parts of the construction below, when students are not assembling CD modules, they can continue working on their drawing.

13. This is a good time to ask students how many CDs are required for the entire construction.  (If this activity is divided into two sessions, this makes a good homework question for students to think about.)  Including edge disks and vertex disks, we’ll need a total of 150 CDs.

Wherever an edge disk overlaps a neighboring vertex disk, they will be tied together with a plastic cable tie that goes through the center hole of each. We’ll need 180 cable ties.

Part B.  Assembly Process

Preparatory notes:

Care is required. CDs are made of acrylic plastic, which is strong (the windows in storm doors are made of acrylic) but brittle. If bent too much, they crack and snap. One must be careful during the construction not to over-bend the CDs. If the sphere is rested on a table, the lower CDs will be bent from the weight above and will likely snap. So the sculpture must always be suspended as described below, so that its weight does not bend the CDs.

If a CD does snap, the sculpture can be repaired. The ties can be cut with wire cutters. The broken CD is easily removed and a replacement added. Similarly, when a mistake in assembly is made, you can cut the ties necessary to correct it.

1. Caution the class not to play with the cable ties:

Do not tie them around your fingers, wrists, or anywhere other than the CDs.”

2. Younger students are unfamiliar with cable ties and might need more help.  Even with older students, demonstrate how the tip of a cable tie goes into just one side of the locking part of the tie, not the other side.

3. Explain the following convention:  The vertex disks will be on the outside of the sphere, with their shiny side facing outwards. The edge disks will be on the inside, with their shiny side facing inwards. This means that whenever two CDs touch, they touch label-to-label. The shiny sides do not touch the shiny sides of other CDs.  You can show this by holding up two disks, overlapping them label-to-label.

4. Divide the class into six groups.  The class as a whole will make twelve pentagon modules. If the class is divided into six groups, each group can make two modules.  (For elementary students, this can be posed as a fun division problem.)

5. Hand out ten CDs to each group.  Do not hand out cable ties yet.

6. Ask students to make a “chain” of ten CDs on a flat table, as above, making sure no shiny sides touch. The shaded parts of the image represent the printed labels of the CD. The five CDs on the bottom have their shiny sides down. The five on top have their shiny sides up.

7. Walk around to each group to check they have the correct pattern.  If so, hand them ten cable ties. Explain that from disk to disk, the connection is made with a cable tie through two CD holes. They will use nine cable ties to make a straight chain then the tenth to close it into a cycle. When they curve the chain into a cycle and close the loop with the last tie, it makes a regular pentagon as shown:

You can position each tie so the free end sticks up out of the plane at either of the two holes that the tie passes through. The free ends will naturally point upwards. Pull all ties tight. You can point out that the alternating CDs correspond to vertex-edge-vertex-edge...  in the center pentagon of their drawing.

8. Choose one group's pentagon module to be at the “North Pole”. Place it on a flat table to be used for the construction. The other modules will be added to it.

9. Have five of the groups add an additional shiny-side-down CD to the bottom of their pentagon, making the shape above, using one more cable tie. Call this extra CD an “ear”. The ear will be the connector that joins their module to the center module.

10. Five one-eared modules will connect to the center module to make the shape above. Connect each module by its ear to the bottom of an upper CD of the center pentagon. Remind students that the rule still applies that the CDs must touch label-to-label. This means there are only five correct places for the ears to connect to the center module.  The angles are adjustable. For now, the angles need only be approximate; we will fine-tune them later.

11. While the students are working, cut five pieces of string each three feet long. Tie them together at one end so they form a bundle of five strings.

12. In this step, the five loose ends of the string bundle will connect to five location marked in the figure below. Each marked location in this image represents two cable ties. (The remaining cable ties are not shown.)

At each of these five locations, tie one string through (under) both of the cable ties, so the load is distributed twice as much.  Someone can hold the knotted end of the bundle in the air over the center of the structure while the five loose ends are tied to these five support locations. The knots should be made the same distance (say, two inches) from the free end on each of the five strings, so the load is balanced. Remember that each string goes through the two cable ties that are part of a pentagon, to distribute the load as much as possible.

13. We now go from 2D to 3D.  Gently raise the sculpture by the top of the string bundle and secure it to the ceiling. It will naturally curve to form a partial sphere. Don’t be afraid to bend the CDs slightly.

14. Add the five CDs that are shown above with dotted lines. Bend things gently to make the connections. They connect the five one-eared modules to each other. Notice that they complete five hexagonal openings. This completes the top half of the sphere. Again remind students that CDs always meet label to label.

15. Have some of the other students add two CDs shiny side down to the bottoms of five of the remaining modules. This makes two-eared pentagons, as shown above. The two ears go underneath two adjacent face-up CDs.

16. One at a time, connect the five two-eared modules as illustrated by the dotted lines above. These will form the bottom of the sphere. Again, as you connect the modules, bend the CDs gently. You are creating more hexagonal openings.

17. Add five CDs that connect module-to-module, as indicated by the dotted lines above. Again, you are making hexagonal openings. As you add these, you will gently bend the modules inwards, towards the south pole.

18. Add five ears shiny side down to the bottom of the remaining (twelfth) module. And insert it in the bottom of the sculpture. Connect one tie at a time, gently curving the CDs as necessary.

19. When complete, tighten all the cable ties and adjust the angles to regularize all the polygons. Make sure that each set of two ties that correspond to one truncated icosahedron edge line up in a straight line.

20. Snip off all the cable tie ends (unless you like the “hairy” look).

## Possible Extensions

Coloring Variations. Using different colors of cable ties will give color to the edges, for example, emphasizing the pentagons with a different color from the connecting edges.

Long-edge Egg Variation. Make pentagon modules as above, but connect them to each other with “ears” that are not just one CD but are three CDs long. Gravity will stretch the final sculpture into an egg shape. But if you spin it, it widens.

Tall Variation. An icosahedron can be seen as a pentagonal antiprism capped by two pentagonal pyramids. An elongated structure can be made as a stack of two or three pentagonal antiprisms, capped by two pyramids. Truncate such a structure to make a prolate form composed of pentagons and hexagons. You can make a Hamiltonian cycle on this form also.

Hamiltonian Cycle Variation. With 60 strips of masking tape (or strips cut from “post-its”) mark the edges of a Hamiltonian cycle on your truncated icosahedron. One at a time, replace the remaining ties with a different color. Exercises on paper with nets of polyhedra may be done first, to explain the concept of a Hamiltonian cycle.