Design Your Own Sculpture
Geometric Sculpture V
Oftentimes, students don't realize that
mathematics can be used in creative ways. In this
workshop, they get to experience first-hand the process of
designing an original mathematical artwork. Based on an
understanding of geometric principles, they will propose a
design which satisfies mathematical constraints while also being
aesthetically pleasing. We have found that when students
are given the opportunity to take ownership of their work, they
become more confident and more fully invested in the learning
Time Required: 0.5 hour for design. 0.5 hour for
- The Autumn sculpture
- One copy of this rhombus
template per student on regular paper
- Card stock
- Access to photocopier
- Clear tape
1. We assume students have already built the Autumn sculpture, so are familiar
with its structure and how its parts connect.
Part A: Design
2. Allow time to make photocopies in the middle of this
3. The design and assembly can be naturally split over two days,
giving students more design time.
1. Looking at the Autumn sculpture,
invite students to recall how the sixty parts join at their straight
segments. Each part has two long edges of the same length and
two short edges of the same length. The long edges join other
long edges to make cycles of three parts. The short edges meet
other short edges to make cycles of five.
2. Point out how the part's shape fits inside of a rhombus, sharing
parts of the rhombus' edges. Those edges are important because
they are where the sculpture holds together. Each piece of the
sculpture is planar---it lies within a plane. The intersection
of two planes is a line, so where two parts meet must be a straight
line segment, not a curve.
3. Show students what the sculpture would look like if the pieces
were a complete rhombus. This shape is called a rhombic
hexecontahedron. It can be assembled from 60 rhombi if
their acute angle is 63.5 degrees. Explain to students that
the structure of the sculpture was designed by carving away parts of
4. Have a discussion with students about how much can be removed
from the rhombus before the sculpture falls apart. Students
should conclude that at least one segment from each of the four
edges is needed to hold the structure together, but it doesn't have
to be the particular long and short segments chosen in the design of
Autumn. Furthermore, segments on the left need to match
with segments on the right. The image above emphasizes the
matching edges for the example of the Autumn piece
shape. The upper half of the rhombus edge is retained in the
long edges; the middle third of the rhombus edge is retained in the
5. Hand out the paper template sheets. Project or draw the
image above of Autumn's matching segments on the
board. Ask students to first draw the mating edges on one of
the rhombus templates, making them thick and dark for clarity.
The template has ten grid lines per edge to measure the segment's
positions, allowing students to darken the upper half of the two
upper edges and the middle third of the two lower edges.
6. Ask students to sketch the Autumn part shape using their
segments as a guide and consider how it could be varied while still
connecting with the same matching segments. Have a discussion
in which they conclude that the curves could vary in many different
ways, as long as they connect the segments. The segments are a
mathematical constraint but the curves allow artistic freedom.
The hole in the Autumn part shape is just one example of
the freedom of design; it doesn't affect the connections.
7. Now students can start thinking of their own shape to build a
sculpture from. Instruct them to first draw (darken in) their
choice of segments where the pieces join. Give students the
following constraints when choosing their segments:
- Each of the four rhombus edges must contain a segment.
- The left and the right sides must match.
- Each of their segments should be at least 30% (three ticks) of
the rhombus edge so there is enough contact area for strength.
8. Students can then draw their own curves that connect the
segments. Suggest that they shade in the part to make clear
which regions, if any, are holes. Guide them to visualize
what their finished sculpture would look like if assembled from
sixty copies of the piece they designed. Point out that
there is also an engineering issue: the shape can not be too thin
or the part will flex and sculpture will not be rigid.
Part B: Construction
1. Ask students to display their designs and then choose one
design which the class will make into a paper sculpture.
2. Using a photocopier, make sixty copies of the chosen
template onto card stock.
3. Hand out the printed card stock, scissors, and tape. Ask
students to cut out several copies each, so there are sixty pieces
altogether for the class.
4. Direct students to tape the pieces together in the same manner as
Autumn, first making three-part modules using the segments in the
top half of the rhombus, then joining the modules in cycles of five
using the segments in the bottom half of the rhombus. The
result will be a unique original design.
5. Here is a student design in which the parts are shaped like a
curvy letter H. The top of this page shows a student design in
which the parts are shaped like the letter "Y". Note that this
has mirror symmetry while Autumn does not.
A. Each student can make their own complete paper
B. If you have access to a laser cutter, you can
fabricate your designs in wood. Using a drawing program, begin
with a rhombus that has a 63.5 degree angle, mark the matching
segments, then add your curves. You also have to add a hole large
enough for a cable tie near the midpoint of each segment.
Bevel the edges with the same angles as Autumn.
C. Because the rhombus is symmetrical, the piece can be
flipped around to make a different sculpture with the same part
shape. For example, in Autumn, the short edges could
be joined in groups of three and then the long edges could be joined
in cycles of five, as shown above. This gives a second
sculpture with the same part shape. Explore this idea to make
a pair of sculptures of your own design. If making them from
wood, note how the beveling angles are different.