EXPERIMENTS: OCTAGON = PANTS
Experiment 1. Copy Figure A and cut out the octagon. Tape the identified sides together (the sides
that are supposed to be sewn), matching direction
arrows as shown. First attach the sides labeled
“a” together, then the “b” sides together. Notice
that the result has 3 holes. One hole is formed by
2 of the original octagon sides coming together for
the waist, and the other 2 holes are each formed
by 1 side curling around to make a leg hole. With
imagination, these are pants.
A little paper-folding helps to visualize this
octagon folded in half as pants. Fold the octagon’s
center point a bit more than halfway up to the
waist (Figure B1), and crease the middle third
of this fold horizontally. (Each blue arrow in
Figure B indicates an action transforming one
picture to the next.) Then fold the legs down
from the center of the octagon (Figure B2). This
causes the sides of the paper to bend up — don’t
worry, allow this to happen. Let the points of the
waist edges meet in the center, then crease what
would be the upper thighs if these really were
pants, as Figure B3 shows. Now look at the shape
of the pants (Figure B4).
Those of you who have been tenaciously shopping at world clothing stores located in college
towns will recognize this style, sold as Thai fisherman pants. However, even those pants don’t have
such a gathering of material at the crotch (where
the inside seams of regular pants meet). We would
rather have a result more like Thai wrap pants,
which are loose but have no excess crotch fabric.
Topologically (in terms of the number of holes),
we have successfully created pants. However,
geometrically (that is, in terms of proportions,
curviness, and distances), we have more work to
do. Our model shows that the problem is not the
sizing but the flatness of the original octagon. To
avoid flatness, we’ll work with curved surfaces.
Positive constant curvature yields a sphere, and
negative constant curvature yields a hyperbolic
plane or pseudosphere; we’ll approximate negative
constant curvature here with paper.
Our paper approximation will consist of a set of
triangles glued together. While each piece is flat,
the object made from them won’t be. Mathematicians say that such a surface is locally flat, but
globally curved. The global curvature will come
from a change in the angles around certain points
in the surface; ordinarily, adding the angles around
a point in a piece of paper gives 360°, but in the
next 2 experiments this won’t be the case.