Folding a Dollar Bill into a
Pyramid
This activity can be done with no preparation at
all, in the classroom or as an amusement anywhere, anytime. All you
need is a dollar bill! A crisp, new bill is best, and not only will
the trick amuse your audience, but there is some fascinating
mathematics in it. It would serve well as an introduction to
3-dimensional solids in a Geometry class, but will amuse students at
all grade levels. It is actually the easiest way to create a 3-D
model of a pyramid!
This activity can be done with no preparation at
all, in the classroom or as an amusement anywhere, anytime. All you
need is a dollar bill! A crisp, new bill is best, and not only will
the trick amuse your audience, but there is some fascinating
mathematics in it. It would serve well as an introduction to
3-dimensional solids in a Geometry class, but will amuse students at
all grade levels. It is actually the easiest way to create a 3-D
model of a pyramid.
In this activity, you begin with a dollar
bill . .
.
..
... and end up with a tetrahedron, using a lot of
math terminology along the way!
And on one of the faces
of the dollar bill tetrahedron, you will see a drawing of a
pyramid!
So, here are the
steps to follow:
Take a dollar bill, with
the side showing George Washington facing up as shown below, this
time with the vertices labeled:
Fold line DC across the
midline of rectangle ABCD as shown. Make a crisp crease on the
midsegment, EF:
Fold the upper half CDEF
back to it's original position. Then fold point D to the midline
EF:
Fold triangle GAD across
line GD:
Fold triangle GAH across
line AH:
Fold triangle ACG across
line AG:
Open up the figure. It
will form a tetrahedron, as shown below:
... and on one of the
faces of the dollar bill tetrahedron, you will see a drawing of a
pyramid!
Besides serving as a good example of a pyramid, it
is also a Tetrahedron; a pyramid with all faces and base congruent
equilateral triangles. There are a number of interesting things that
you will want to point out to your students. One face of the pyramid
(the "open" face) is formed by 2 congruent 30/60/90 triangles. That
is, each triangle has angles of 30 degrees, 60 degrees and 90
degrees. These 2 triangles have the numeral 1 on them, in perfect
reflection symmetry.
It is interesting to explain to the class that
this will not work with a piece of paper of any other ratio of
dimensions (length and width). Clearly the dollar bill, and all of
our paper currency, was designed in this exact ratio. And what is
this ratio? If we call the shorter dimension of the right triangle 1
unit then the longer leg of the 30/60/90 triangle is 1 times the
square root of 3, and the hypotenuse is 2. Therefore, the equilateral
triangle has sides 2 units long, and the ratio of the length of a
dollar bill to its width is the square root of 3 to 4.
If time permits, it is interesting and instructive
to give students a variety of slips of paper, each with different
proportions, and ask them to repeat the steps that they used to fold
the dollar bill, with their paper rectangle. If none of the paper
rectangles happens to be the same proportion as the dollar bill, the
class will see that although each student has done the same steps,
their piece of paper did not work as well as the dollar-bill shape;
there will be either extra or insufficient paper in the last
step.
It seems that the Treasury Department of the
United States of America must have a high regard for mathematics, to
have chosen this particular proportion!
We are usually convinced
more easily by reasons we have found ourselves than by those which
have occurred to others. Blaise Pascal (1623-1662)
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