This symmetric arrangement of four hollow
triangles was discovered around 1970 by the chemist Alan Holden.
It is surprisingly simple, yet difficult at the same time.
The goal of this workshop is for students to solve the problem
of building their own copy of the structure using colored wood
sticks. Along the way, they will come to appreciate some
of the beauty of the structure while learning that mathematics
is a living subject with new discoveries continuing in the
present. It is also great as a puzzle for developing both
spatial reasoning ability and problem solving persistence.
Students will feel a great "Aha!" moment when they complete it.
Time Required: 45 Minutes
Twelve "jumbo craft sticks" per group, three in each of four
colors. These can be purchased pre-colored at craft stores
or dollar stores. (example)
Twelve small black binder clamps (example)
per group, or twelve mini-clothespins
Strong scissors (or a saw) to cut the ends of the sticks
Wood glue (example)
and scraps of cardboard to use as palettes
Small brushes to apply glue
ruler to measure length during preparation
1. This is the first in a series of three
2. You can use uncolored sticks if necessary, but the result
looks much better in four colors.
3. With older students, you could have them mark and cut the
ends of the sticks using scissors. Here, we assume you will
prepare the parts before the workshop.
4. Working in groups of two is highly recommended.
Students will discover that having four hands is helpful when
assembling the parts.
1. The ratio of the length to the width of the sticks needs to be
fairly close to 7:1 for a snug fit. We have found that
different brands of craft sticks have somewhat different widths.
Measure the width of your sticks. Multiply by 7 to get the
desired length. (The U.S. "jumbo craft sticks" shown in these
pictures are 1.8 cm wide, so should be cut to 12.6 cm in length.)
2. Mark the ends of the sticks at 60 degrees to form trapezoids of
the desired length. (Watch out you don't make a
parallelogram!) After you cut one master-stick, you can just
trace its ends on to the others.
3. Cut the ends with strong scissors. (Or a woodworker can cut
a stack at a time with a saw.)
4. You need twelve cut sticks per group, three per color.
Part A. Minds-On
1. Remind students of the regular tetrahedron. It is a
polyhedron with four equilateral triangle faces. It has four
vertices and six edges. Ask if they can imagine another
simple, natural, symmetric, 3D structure made from four equilateral
triangles. They will quickly see that there is no other way to
arrange four triangles as a polyhedron, so they need to look beyond
the realm of polyhedra. Discuss ideas that the students
propose. As an illustration, one possibility they should be
able to visualize, shown above, is to remove four faces from a
regular octahedron, leaving four corner-connected triangles, but
that is not today's structure. (This octahedron is red on the
outside and yellow on the inside, so the image shows the inside of
the two rear faces.)
2. Tell students that there is an interesting solution, but even
though it is made of just four triangles and is very symmetric, it
is very difficult to visualize. In fact, mathematicians have
been thinking about the possible symmetric shapes made with
equilateral triangles for over two thousand years, but no one
conceived of the form we will make today until the 1970's. Emphasize
that mathematics is very much an active, creative field, with new
discoveries published every day. The challenge of this activity is
for students to discover this (relatively) new structure, given a
3. Tell students they will be making the structure from
trapezoid-shaped wooden sticks held together with small clamps (or
clothespins). Show them three of the sticks and three of the
clamps and ask them to visualize how they can build a
triangle. They will see that three sticks can be overlapped at
the ends to form a hollow triangle. Tell students that the
hollowness is important, because in this structure, each triangle
will pass through the hole of each of the other three triangles.
4. Point out that it will be helpful to have some vocabulary for
talking about these hollow triangles. There are three vertices
and three edges around the outside, just as in any
triangle. But we also want to name the three corners of the
hole, which we call inner vertices as indicated above.
In addition, we will find that the three points at the midpoints of
the outer edges are very important for understanding and talking
about this construction. One of the three edge midpoints
is indicated above. Having names for all these points helps
for keeping them in mind and when communicating with one's
Part B. Hands-On
1. Divide the class into groups of two. Hand out twelve
trapezoidal sticks (three in each of four different colors) and
twelve clamps to each group.
2. Ask each group to choose a color and make a triangle of that
color by overlapping the ends of three sticks and applying three
clamps. With younger students, there may be some confusion
about which way the sticks face. See if they can articulate
the observation that the longest edge goes on the outside.
3. Ask students to observe that if they want to be very symmetric,
they can make every overlap in the same way, e.g., always placing
the clockwise stick end over the counterclockwise one. This is
not crucial structurally, but it will look best if totally symmetric
with every triangle made the same way "like a spiral". Have
each group choose one consistent handedness and rebuild their
triangle if necessary. By doing this, they are creating 3-fold
rotational symmetry, which will be part of the overall symmetry of
the final result. With younger students, see if they can
articulate the observation that one end of the stick is on top and
the other end is on the bottom. Remembering this is a good way
to quickly notice if an errant stick has both ends on top or both
ends on the bottom.
4. As a nice visualization problem, ask students to predict what
their triangle will look like if they turned it over: Will the other
side be the same handedness or the opposite handedness? They
may be surprised to discover that the other side is not the
other handedness; both sides look the same.
5. Ask students to make a second triangle that spirals the same
way. They will naturally build it separately, as in the image
6. Tell students that in our construction any two triangles are
linked like two links of a chain. To link their two triangles,
they can remove one clip, slide the other triangle through the
little opening, and replace the clip. The image above shows
two triangles linked.
7. Remind students of the special points discussed earlier:
the inner vertices and the edge midpoints. Tell them that in
the structure they will build, for any pair of triangles, not only
are they linked, but they meet in a special way: an inner vertex of
each one touches an edge midpoint of the other. The image
above shows the red and blue triangle meeting like this. The
arrows in the image above indicate the two places where an inner
vertex meets an edge midpoint. In a sense, this is the
"formula" for solving this puzzle.
8. From this point on, it is difficult to give specific
instructions, so the groups should treat it as a puzzle. Can
they add a third triangle and then a fourth triangle so that for
every pair (whatever two colors they pick) an inner vertex of each
one touches an edge midpoint of the other. It is best to add one
stick at a time, visualizing where to add a triangle, instead of
making an entire triangle separately and trying to weave it
in. The image above shows a possible intermediate stage with a
third triangle positioned with the inner-vertex-to-edge-midpoint
9. A correct assembly of all four triangles is shown
above. It may take a while for students to find it. When
a group does, ask them to check that every inner vertex touches some
edge midpoint and every edge midpoint touches some inner vertex,
i.e. no inner vertex "elbow" is hollow.
10. Students who successfully finish their construction can help
11. 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
12. 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 three joints of one color, then
the three of a second color, etc.
13. 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 see. This helps develop a deeper
understanding and the ability to visualize it. Some
observations you might elicit are:
1. The triangles are concentric, meaning the centers of the
four triangles coincide.
2. No matter which pair of the four triangles you choose, you can
see they are linked like two links of a chain.
3. No matter which pair of the four triangles you choose, you can
see they touch twice. An edge midpoint of each sits inside an
"elbow" of the other at an inner vertex.
4. If you choose any triangle and look at its three inner vertices,
they touch edge midpoints of the other three triangles, i.e., the
other three colors. Similarly, for each triangle, the three
edge midpoints touch inner vertices of the three other colored
5. You can stand it up on three vertices. By symmetry, the
three points touching the ground form an equilateral triangle.
Three colors will touch the ground and the triangle of the fourth
color will be horizontal. There are eight ways to do this
because the triangle of any one of the four colors can be chosen to
be horizontal and it can be flipped over if desired.
6. You can stand it up on four vertices so there is a four-way
opening uppermost and lowermost. By symmetry, the four points
touching the ground form a square. All four colors touch the
ground. No triangle is horizontal. You can count that
there are six four-way openings, so there are six ways to do this.
7. The planes of the triangles are not perpendicular.
This is clear if you rest it on three points, so one plane is
horizontal and then observe that none of the other planes are
vertical. (If two planes are perpendicular and one is
horizontal, the other would have to be vertical.) You can see
this angle if you sight along a line which is the intersection of
two planes, as in the image above. The angle between the green
and red planes can be seen to be approximately 70.5 degrees.
It turns out that the angle between any two planes is the same as
the angle between any two faces of a regular tetrahedron.
If you have a small model of a regular tetrahedron handy, you can
align it to illustrate this relationship, with each face plane of
the tetrahedron parallel to one of the four triangle's planes.
See this video for an illustration of parallel relationship between
the regular tetrahedron faces and the four planes of this
8. There are two possible ways to link the triangles. They are
mirror images of each other. One would expect roughly half the
class to come up with one and half the class to come up with the
other. To understand the difference between the two solutions,
observe that in the image above, we are looking directly at a green
vertex, with a red edge near to us horizontally. The green
triangle is rotated slightly counterclockwise of vertical. In
the mirror image structure, the green plane would be slightly
clockwise of vertical. You can look at your structure in a
mirror to see what the other version would look like or look around
the classroom to see if some students have built the other version.
9. Imagine that for each of the eight ways of resting it on three
points we attach a paper equilateral triangle that stretches between
the three vertices that touch the ground. Also, for each of
the six ways of resting it on four points, attach a square that
stretches between the four contact points. The structure would
be completely covered by the six squares and eight triangles.
So it fits snugly inside a cuboctahedron,
the polyhedron shown above. Our structure has the same
vertices (the same positions of points in space) as a
cuboctahedron. Another way to think of this is that if you put
the structure inside a big balloon and let the balloon shrink until
it snugly surrounds the outermost vertices of the structure, the
balloon would have a cuboctahedron shape. (Mathematicians call
this the "convex hull" of a shape.)
10. You can show this video,
which shows how the structure fits inside a cuboctahedron.
11. Recall there are eight ways to rest it on three points. If
you choose any three of the four colors and list them in any order
you choose, e.g., green, yellow, blue, there is always exactly one
way to turn it so the three contact points, when listed in clockwise
order, match your choice of colors. For example, the image at
the top of this page shows a model positioned so it contacts the
table with blue, red, green points, in clockwise order. This
observation can be used as an introduction to combinatorics to
expose students to exciting branches of mathematics they may not
otherwise see in the curriculum.
12. Recall there are six ways to rest it on four points. If
you list all four colors in any order, there is always exactly one
way to rest it on four points so those points are colored in that
1. Make the mirror-image version of the one you first made.
2. Make a giant version from cardboard, thin wood, or other suitable
3. High school students should be able to prove that for a snug fit
(assuming the planes are of zero thickness) the inner triangle holes
must be exactly half the size of the outer triangles. From
this, they can show the ideal length-to-width ratio of the sticks
should be 4·sqrt, roughly 6.9282. Given the thickness and
imperfections of the wood, 7 is a reasonable approximation.