Time Required: 2 hours
- Cardstock printed with triangle
template (one sheet per two students)
- Cardstock printed with long
struts template (six sheets per group)
- Cardstock printed with short
strut template (five sheets per group) in a lighter
- glue sticks
- colored markers
- rulers (for making straight folds)
- optional clothespins
1. This small paper structure (21 inch diameter) is a
good pedagogical tool to introduce students to the geometric
properties of domes. It is not very rigid but is an
important foundation before making any of the larger, more stable
Part A: Minds-On
2. Students should work in pairs.
3. It is important to use two colors of paper.
4. Students should be familiar with the regular icosahedron.
This dome has two different lengths of struts and two different
types of vertices (five-fold and six-fold). Students need to
develop a clear mental model of the structure in order to fully
understand how to assemble the components. Students will first
build a regular icosahedron and divide its faces into smaller
triangles in order to have a visual understanding of the parts and
1. Hand out one sheet of triangles per pair of students, along with
scissors and tape. Ask students to cut out twenty triangles
and tape them together, five at a vertex, to make a regular
icosahedron. Ensure that the printed black lines are on the
outside of the model as in the image above.
2. Review the fact that the icosahedron has 20 faces, 12 vertices,
and 30 edges.
3. Ask students to use colored markers (not black) to draw lines
that divide each face into four identical equilateral
triangles. It is helpful to start by marking a dot at the
midpoint of the edges.
4. Let students generate questions and explore the structure they
have drawn. They should discover that there are 80 small
triangles. At the twelve original icosahedron vertices, the
small triangles meet in sets of five. Thirty new vertices have
been created (at midpoints of the original edges) at which the
triangles meet in sets of six. Point out that the 6-way
vertices are slightly closer to the center of the object than the
5. Continuing, notice that the colored edges (blue in the image
above) always go from a 6-fold vertex to another 6-fold
vertex. In contrast, the black edges of the small triangles
(each half of an original icosahedron edge) always connect a 6-fold
with a 5-fold vertex. This is important because although these
black and colored edges are the same length, there will be two types
of edge, of different lengths, in the next model.
6. Most geodesic domes are designed to have all the vertices
the same distance from the center. This can be achieved if we
imagine moving the 5-way vertices slightly towards the center.
As a consequence of this transformation, the black edges will become
slightly shorter than the blue edges.
7. Turn student's attention to the fact that a series of ten colored
edges form an "equator" dividing the model in half. (In fact
there are six different ways to do this. Why six?
Because you can choose two opposite vertices to be a "north pole"
and "south pole" in six different ways, since there are twelve
vertices.) Students will understand that a hemispherical dome
can be built by eliminating everything on one side of the equator.
8. Ask students to visualize one hemisphere and count how many of
the longer colored edges and how many of the shorter black edges are
required to build the dome. (Answer: 35 of the longer colored edges
and 30 of the shorter black edges.) In the next part, we will
curve these straight lines slightly to make a true hemisphere.
Part B: Hands-On
1. Hand out the printed colored paper to pairs of students and
ask them to cut out the curved bands on the solid lines. (Do
not cut on the dotted lines.) Each group needs 35 of the
longer (darker colored) bands and 30 of the shorter (lighter
colored) bands. These components will be the "struts" making
up a hemispherical dome as in the picture at the top of this page.
2. Instruct students to use a ruler to make a neat crisp fold on the
3. It is important that all the pieces be folded consistently.
As shown in the image above, when curved like a rainbow arc, the
right end is folded up and the left end is folded down. (Note
the interesting optical illusion in the above image: The darker
piece is actually longer, thought it appears shorter!)
4. Ask students to connect five of the shorter, lighter struts by
gluing the tab of one to the body of the next, in a cycle.
Align the pieces so the fold lines come together. Students
will discover that working in pairs makes this easier. (Never
glue a tab to a tab.)
5. Students can now add five of the longer, darker struts to make a
pentagon around the first five struts. There will be five
remaining tabs on the outside of the pentagon. This is a
6. Ask students to make a total of six pentagon modules. There
will be one for the "north pole" and five around it. There
will be five remaining long (darker) struts.
6. Ask students to connect two pentagon modules using one tab from
each. This makes one of the 6-way vertices.
7. Students should puzzle out how to connect the
modules. The five remaining struts are used to complete
8. It may be useful to improvise a support to
hold the shape while the glue dries. Clothes pins or other
clamps may also help.
Part C. Conclusion
Ask students to compare their paper dome with their marked
icosahedron and observe the analogous structures. (This
provides another example of an isomorphism, as introduced in the 12-Card Star
Students are now ready to replicate this structure using other
materials at a variety of scales.
1. Build a similar dome in other materials, e.g., straws for
struts with pipe cleaners inserted for connectors, or Zometool.
2. Make a complete sphere. How many of each length strut are
3. The design on this page is called a "frequency 2 icosahedron"
because the original icosahedron edge is divided by two. See
if you can design a frequency 3 or 4 icosahedral dome. There
will be more strut lengths required. How many of each