Here is a trick for folding a U.S. dollar bill into an equilateral triangle and then into a tetrahedron.
Step 1: Fold a dollar bill in half lengthwise, crease it, and then unfold it and lay
it face up on a flat surface (see figure below).
Step 2: Fold the upper left hand corner down until it just touches the crease (see figure below).
Show that the triangle folded over is a 30-60-90 degree triangle.
Step 3: Fold the lower left-hand corner up until it just touches the top of the bill, as in the figure below.
Step 4: Fold what is now the upper left-hand corner down to touch the bottom of the bill
(see figure below). Remarkably, it will end up at or very near the lower right-hand corner.
Step 5: Fold the remaining tab over, to form an equilateral triangle (see figure below). This demonstrates that the ratio of the height of a dollar bill to its length is approxiamtely the ratio of the altitude of an equilateral triangle to two bases, an irrational ratio of
: 4.
Step 6: If all the creases have been sharp, you can now gently open up the dollar bill, and when
you put it down it will form a tetrahedron (with a picture of a pyramid on one side!). See
figure below.
