FIGURE 413: Now use a ruler to draw a line that intersects
the two places where the arcs you drew in the last two steps
cross each other. That line is the perpendicular bisector.
FIGURE 411: Draw an arc with the pencil point around the
needle point a bit beyond where you estimate the midpoint of
the line is — say about two-thirds of a circle.
FIGURE 410: Adjust the width of the compass by putting the
needle point at one end of the line you want to bisect, and the
pencil point at the other.
FIGURE 412: Leave the compass points the same width apart,
but now put the needle point on the other end of the line you
are bisecting. Again draw an arc until it intersects the one you
just created.
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FIGURE 422: A Reuleaux triangle (grey)
as constructed
FIGURE 414: First, set your compass
points a convenient distance apart.
FIGURE 415: This distance will be the
length of your triangle’s sides, and you
will leave them that way for the rest of
the process. Start by drawing a circle.
FIGURE 416: Put the needle point at
any point of the circle you just drew and
draw another, intersecting circle of the
same radius there.
FIGURE 417: Now put the needle point
at one of the places where these circles
intersect. Draw a third intersecting
circle.
FIGURE 420: Next, use a ruler to
connect two more vertices.
FIGURE 419: Use your ruler to connect
two vertices of this equilateral triangle.
FIGURE 418: You’ll see a triangle-like
shape in the middle. In the next section,
we’ll discover it is called a Reuleaux
triangle.
FIGURE 421: Connect the final two
vertices with a ruler.
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EQUILATERAL TRIANGLE
This construction allows you to create an equilateral triangle, which has all its sides (and, therefore, all
of its angles) the same. We’ll learn more about them in Chapter 5. Figures 4-14 through 4-21 show all
the steps.
You will now have an equilateral triangle. There are also three more possible equilateral triangles of
the same size in the overlapping circles you have drawn. Where are they? You might even observe that
there is another, larger equilateral triangle you could have drawn in this last step. Where is it, and what
is the ratio between its size and the triangle in Figure 4-21? See the “Answers” section at the end of the
chapter to check yourself.
Why this works: As in the previous constructions in this chapter, a circle’s radius is constant. Because
an equilateral triangle’s sides are all equal, each vertex is equidistant from the other two. We start the
process similar to the previous example, where we were creating a perpendicular bisector. The points
where the first two circles cross gives us two points that are equidistant from their respective centers,
and connecting the centers to each other and to one of those points gives us our triangle.
If we had been careful to mark those centers, or if we had drawn one line of the triangle first and used
points on that line as the centers, we wouldn’t even need to draw the third circle. Since those first
two points are the same distance from the third point, though, drawing the third circle, which passes
through both of their centers, will make it a little more clear where those centers were by intersecting
with the other circle at each center.
OTHER CONSTRUCTIONS TO TRY
You can find many other constructions if you want to try playing with them; the chapter summary sug-
gests more places to go. You might think about how to construct a hexagon next, and search on it to see
if you were right! How might you use constructions if you were an architect laying out a house? Or if you
were living in medieval times and wanted to measure out a building with a rope as your compass and
a taut string as your straightedge, how would you do that? Look ahead to Chapter 13 if you want to play
with some of these ideas.
REULEAUX TRIANGLES
If someone asked you what shape had the same diameter everywhere you would probably answer that
it was a circle, as we will explore in Chapter 7. However, there are other constant-diameter polygons.
Note that we say constant diameter, not radius, since it is possible to have constant-diameter objects
that are not symmetrical around a center point. One of the best-known ones is the Reuleaux triangle
(pronounced roo-low). A Reuleaux triangle is not a true triangle, in that its angles do not add up to 180°
like other triangles do, as we’ll see in Chapter 5. In fact, its internal angles are all 120°, and so they add
up to 360°.
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It’s named after the German engineer Franz Reuleaux, who was active in the
mid-to-late 1800s. He invented a lot of what we now call kinematics, which is
figuring out how a mechanism would work. His diagram-filled 1876 book, The
Kinematics of Machinery, is available at
https://en.wikisource.org/wiki/The_Kinematics_of_Machinery, if you want to
play with some of his other constructs.
We can get a Reuleaux triangle pretty easily as a byproduct of our equilateral
triangle construction. To make a Reuleaux triangle out of paper, start the
equilateral triangle construction in this chapter and go as far as drawing the
three circles. You will see something that looks like a rounded triangle in the
middle (Figure 4-22). That’s the Reuleaux triangle.
Use your compass to sketch out one of these on stiff cardboard and then cut
it out for the explorations to follow. If you have a 3D printer, you can use the
reuleaux-n-gon.scad model.
3D PRINTABLE MODEL
The 3D printable model has a few variables you can play with. It creates a
Reuleaux polygon (Reuleaux polygons exist for all odd numbers of sides) and
an enclosure. We’ll talk about that enclosure in a minute. Meanwhile, here
are the variables and their defaults.
width = 50;
Width of the polygon, mm
sides = 3;
Number of sides (has to be odd)
thick = 10;
How thick the polygon is when printed, in mm
wall = 1;
Wall thickness of the two pieces, in mm
base = 0.6;
Thickness of the solid base on each piece, in mm
We’d suggest leaving the defaults alone, other than changing the number of
sides. If you want to scale the model, however, do it by changing the width
variable, rather than by scaling in a slicer. Otherwise the side walls might get
too thin or be too thick for the rotation of one model in another (which we’ll
talk about next) to work correctly.
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ROLLING REULEAUX
We know that round wheels will roll, but it turns out that’s true of any constant-width shape like a Reu-
leaux triangle. The width is constant, but the distance to the center from all points on the perimeter is
not equal. This means that you could in principle make a wheel that is shaped like a Reuleaux triangle,
but it would need a complicated mechanism to work since the center of rotation has to move.
People have made bicycles with Reuleaux triangle wheels (search online to see examples, often spotted
riding around places like the Burning Man event) but they are pretty complex. Rather than rotating
around a fixed hub, the centers of the wheels are allowed to move up and down in a way that requires a
complex mechanical linkage. If you create a Reuleaux triangle, either by 3D printing or cutting one out
of cardboard, you can see that it rolls easily along a flat surface, but is not rotating around any one con-
stant point.
What happens if we confine a Reuleaux triangle in an appropriately-sized square (Figure 4-23)? It will
turn freely. Actually, if you were to make a Reuleaux triangle drill bit, it could be used to drill out a
square hole with slightly rounded corners. 3D print a Reuleaux triangle and its square enclosure, and
rotate the triangle. You’ll see it will go almost completely into the corners of the square.
FIGURE 423: A Reuleaux triangle inside its square enclosure.
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