FIGURE 1315: Assembling a triangle
FIGURE 1318: Making the second
triangle
FIGURE 1316:. Tying off the triangle
FIGURE 1319: Adding the last piece
FIGURE 1317: Adding the next two sides
FIGURE 1320: The tetrahedron
FIGURE 1321: Trimming off the ends FIGURE 1322: Pushing the knots into
place.
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CUBE
Next, we are going to make a cube, and demonstrate how it and the tetrahedron we just made can
be combined into a truss (and why you want to do that). To get started, collect the twelve shorter (red)
straws and six pieces of cord. First, take four of the straws and run a cord through them to make a
square. Tie off the square. (Figure 13-23).
Now take a second piece of cord, and run it through three more straws, plus one side of the square we
just made. Tie it off to make a second square (Figure 13-24).
Now we will add two more sides. Run cord through the straws forming one long side of the two-square
polyhedron. Add another straw on each end. (Figure 13-25)
Next, tie together the two pieces you just added. The outside of the shape will become a hexagon, and
there will be three straws going to the center, creating something that looks like a projection drawing of
a cube (Figure 13-26).
Next, run a cord into one of the corners of the hexagon that doesn’t have a straw connecting it to the
center. Thread it through that straw and the next perimeter straw, so that it will come out of another
corner that is not connected to the center. Add two more pieces of straw (Figure 13-27).
Now, tie together the two pieces you just added. At this point, it may start to pull itself into a 3D shape,
and you’ll be able to hold it up to make a cube that is missing one edge. (Figure 3-28).
Your shape now has three straws meeting at each of its corners, except two, and you’ll use your last
100mm straw to connect those two. Your last two pieces of string will both go through this straw, with
each one making a loop around one of the two remaining faces. As with the tetrahedron, when you’re
done, every straw will have exactly two pieces of cord going through it. (Figure 13-29). This once again
makes certain that the joints are all equal.
Finally, tie off the loose strings. You will now have a cube (Figure 13-30). However, you will discover that
the cube doesn’t want to be a cube. It will just squish from side to side with no real strength whatsoever.
Cut off the excess cord and slide the knots inside the straws as before.
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FIGURE 1327: Adding two more edges to
the future cube.
FIGURE 1324: Adding the second square
FIGURE 1328: Tying together these two
pieces
FIGURE 1325: Adding two more sides FIGURE 1323: Making a first square
side.
FIGURE 1329: Adding the last piece
FIGURE 1326: Tying together two pieces
we just added.
FIGURE 1330: Our final cube
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COMBINE CUBE AND TETRAHEDRON
The cube really will not even stand up on its own. What happens, though, if
we were to combine the tetrahedron and the cube? If we think geometrically,
we see that the tetrahedron has six edges. The cube has six sides. What
happens if we try to line up the tetrahedron so that one edge is a diagonal of
each face of the cube?
First, take the cube and squish it into a four-pointed figure, like the inside of
a tetrahedron. Push it into the tetrahedron as shown in Figure 13-31 so that
one corner of the collapsed cube sticks out through each of the four sides of
the tetrahedron.
Now pull out the vertices of the cube so that cube vertices that are squished
into the center each come out and press out on a vertex of the tetrahedron
(Figure 13-32). Stretch the cord a bit as you do so, so that the cube’s corner
goes through the tetrahedron’s corner, to the outside of the shape. This will
link the two corners together.
Finally, you will see that the tetrahedron sides are now diagonal supports of
each side of the cube, just as we saw in Chapter 5 when we added a diagonal
support to a square (Figure 13-33).
We made the tetrahedron sides 1.4 times longer than the sides of the cube.
If you look at any triangle created by two cube edges and one tetrahedron
edge, they form a right triangle with two sides the same (let’s call that equal
to 1) and therefore the Pythagorean theorem says that the diagonal, the
hypotenuse of the triangle, (edge of the tetrahedron) will be the square root of
1 + 1, or , √
2
which is roughly 1.41.
FIGURE 1333: Finished cube FIGURE 1331: Putting the cube in the
tetrahedron
FIGURE 1332: Cube vertex joined with
tetrahedron vertex
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We can see that the cube is now very stable and very strong. Varia-
tions on this theme are called a box truss. Trusses can be made of
nothing but triangles. Here is an example of a complicated truss that
is made the same general way we just made our truss and tetrahe-
dron (Figure 13-34).
This truss is made using the same two straw lengths. In this case,
the tetrahedrons use the shorter length of straws, rather than the
longer one. The longer straws are the ones that run vertically in
Figure 13-34. Try some variations on this theme and see what kinds
of strong structures you can make!
People also do exercises where they make trusses out of
mini-marshmallow joints and dry spaghetti, or of course, there are
lots of construction toys that include, or allow you to make, trusses.
We encourage you to explore further!
THE GIFT SHOP
Our museum tour (and this book) has to come to a close at some
point, and there are always more things to look at than there is time
to cover. In this final section, we will give you a few more models, and
also some ideas and jumping-off points that you might want to use
as the basis for your own projects. Let’s wander around and see if we
can pick up a few things to play with at home.
THE TETRAHEDRON AND CUBE PUZZLE
Let’s start with a 3D printed pair of models that will give you another
way to think about the geometry of the cube and tetrahedron we just
created. We can create a tetrahedron and an appropriately-sized
hollow box, open on one side, and fit the tetrahedron diagonally into
the cube.
We created the two files puzzlebox.scad and puzzletetrahedron.
scad to make a cube and tetrahedron that will fit together, the way the
models of straws did. The edge length of the tetrahedron is slightly
less than times the inner dimension of the edge of the cube. The two
models allow for clearances and all that automatically as long as you
keep the variable size the same in both.
It looks like the tetrahedron can’t possibly fit in the cube, since its
FIGURE 1334: A bigger truss project
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