This symmetric arrangement of six hollow
pentagons is a harder challenge after the four-triangles polylink and the six-squares polylink, which
introduce many prerequisite ideas. Having 5-way openings,
the 108-degree vertex angles of pentagons, and six polygons to
interweave, this is a rather challenging puzzle, but the result
is a lovely symmetric structure worth knowing. Again, we
present it as a puzzle and don't try to give detailed
step-by-step instructions. We believe that developing the
perseverance to find one's own solution is a big part of the
value of the activity.
Time Required: 60 Minutes
Thirty "jumbo craft sticks" per group, five in each of six
colors. These can be purchased pre-colored at craft stores
or dollar stores. (example)
Thirty small black binder clamps (example)
or mini-clothespins per group
Wood glue (example)
and scraps of cardboard to use as palettes
Small brushes to apply glue
1. This is the third and most difficult in a series of three polylink workshops. Do the others
first as warm-up.
2. You can use uncolored sticks if necessary, but the result
looks much better in six colors.
3. Working in groups of two is highly recommended.
Students will discover that having four hands is valuable when
assembling the parts.
4. It is helpful to have a model of an icosidodecahedron,
e.g., from Zometool, Polydron, or paper.
The 6-inch by 0.7-inch "jumbo craft sticks" we find in the North
America are just the right size for this construction, without any
need to trim them. Furthermore, they are available in packs of
six colors, exactly what is needed! If your sticks have a
different length-to-width ratio, they may need adjusting analogous
to the way the stick ends were trimmed for the triangle or square
Part A. Minds-On
1. Ask students to recall the discussion of how the four-triangle
polylink sits inside a cuboctahedron, shown at left above. One
way we could have derived that polylink is to start with a
cuboctahedron, notice it has four equatorial hexagons, and convert
each hexagon to a triangle by connecting every other vertex.
You can do this in a consistent manner so each of the original
vertices is used in exactly one triangle. In an analogous way,
if you start with an icosidodecahedron,
shown at right above, we can derive a new polylink.
2. Show this 1-minute video
which illustrates the derivation. Point out that the
icosidodecahedron has six equatorial decagons. (Two cross at
each of the 30 vertices.) We can convert them to six pentagons
by taking every other vertex in a consistent manner, with the edges
weaving over and under each other. Analogously to the four
triangles, every pair of pentagons is linked and the inner vertices
contact the edge midpoints. The triangular and pentagonal
openings of the polylink correspond to the faces of the original
icosidodecahedron. Ask students to try to visualize this and
plan how to build it from sticks.
Part B. Hands-On
1. Divide the class into groups of size two. Hand out thirty
sticks (five in each of six different colors) and thirty clamps to
2. Discuss how to build a regular pentagon. Somehow a 108
degree angle must be made. Students can do this accurately
enough without a protractor by doing a bit of adjustment and
checking. It is easy to see if a pentagon is regular: Use it
as a guide to make a copy of itself (as a second layer on top of
it), then rotate the copy a fifth of a revolution. If the two
pentagons always line up, all five angles must be equal. If
not, adjust them a bit and try again. Use one of these
pentagons as a guide for making the remaining pentagon angles.
3. Give students the puzzle of building the six-pentagon polylink
they saw in the video, with each pentagon a different color.
This will develop visualization, dexterity, teamwork, and
4. If requested, play the video again to help students understand
the desired structure.
5. It may take a while for students to find a correct
assembly, as shown above. When a group finishes it, ask them
to check: (1) every inner vertex touches some edge and every edge
touches some inner vertex. (2) The corner overlaps are
made in a consistent rotational direction. (3) Everything is
snug with no play between the pentagons.
6. Students who successfully finish their construction can help
7. The final step will be to glue the joints so the clamps can be
removed. First, ask students to snug up any looseness in their
model. If any movement is possible, a clamp can be removed,
the sticks that meet there can be adjusted to be closer together,
and the clamp can be replaced. Do this everywhere until all
the joints are snug and there is no relative motion possible between
8. Squirt some wood glue on a scrap of cardboard as a palette.
One at a time, remove a clamp, use a small brush to apply a thin
layer of glue in the joint, and replace the clamp making sure the
joint is snug. Tell students not to apply too much; it
shouldn't drip out. To keep track of what has and hasn't been glued,
it is easiest to do all joints of one color, then all of a second
9. Wait 20 minutes or more for the glue to dry before removing the
clamps. While the glue is drying, you can ask the class to
ponder whether all the structures in the room must be the same or
whether there might be more than one possible solution to the
Part C. Conclusion
Ask students to observe the structure and
discuss what they discover. This helps develop a deeper
understanding and the ability to visualize it. Some
observations you might elicit are:
1. The six pentagons are concentric.
2. Every pair of pentagons is linked.
3. The five inner vertices of any pentagon are filled with an edge
midpoint from the other five pentagons. The five edge
midpoints of any pentagon each sit inside an "elbow" from the other
4. You can stand it up on three vertices in twenty ways,
corresponding to the twenty triangles of an icosidodecahedron.
5. You can rest it on five vertices in twelve ways, corresponding to
the twelve faces of the icosidodecahedron. One of the
pentagons will be horizontal.
6. It is not the case that if you choose any set of three
colors and stand it on three vertices of those colors. Only
some of the triples appear together.
7. The structure can be made in either a left-handed or right-handed
manner. So ignoring color, there are two solutions.
8. For older students interested in combinatorics, one can show
there are 12 possible ways the colors could have been arranged, so
there are twelve different solutions if you count color for each of
the two geometric handednesses. To see this, note that using
six colors, for one cycle of five colors around a 5-way opening,
there are 6 ways to omit one color and 24 ways to make a cyclic
permutation of the five selected colors. So there are 144
different 5-way cycles possible. If a 5-way cycle in one model
matches any 5-way cycle in another, then the two models are
identical everywhere. Twelve different 5-way cycles of the 144
appear on each model. So there are 12 possible models.
9. By relating the shape to a dodecahedron, one can show that no
matter which pair of pentagons you choose, the angle between their
planes is always the same, roughly 62.5 degrees, which is the angle
between the planes of two adjacent faces in a dodecahedron.
1. Make the mirror-image version to the one you made.
2. Make a giant version from cardboard, thin wood, or other suitable
3. Find pictures of the ball used in the game sepak takraw---a
traditional sport popular in Asia, something like volleyball in that
there are two teams on opposite sides of a net and like soccer in
that you don't touch the ball with your hands. Note how the
sepak takraw ball design is isomorphic to the six pentagon
polylink. Weave a ball of this design from the material of
your choice. People have been weaving them for hundreds of
4. Many other polylinks are possible. Watch this
video survey and choose some others to build.