by Dave Youngs
Puzzle Topic
Topology
Puzzle Question
How can you explain the apparent paradox of the double Möbius strips?
Materials
Scratch paper
Scissors
Tape
Student sheets
Puzzle Background
The Möbius loop is a topological surface first discovered by August Ferdinand
Möbius in 1858. Möbius was a mathematician and professor of astronomy
whose
work in topology revolutionized the field of non-Euclidean geometry. A Möbius
loop
can be constructed by connecting two ends of a strip of paper after giving one
end a
half twist. This results in a baffling surface which has only one side and one
edge.
The Möbius loop has been immortalized by artists like M.C. Escher, who
used it in his
print Moebius Strip II, which depicts ants marching in an endless
line around a Möbius
loop. It also has practical applications in the industrial world, where the
large belts in
some machinery have been designed with a half twist so that both sides get equal
wear.
This puzzle presents a fascinating variation of the Möbius loop in which
two
apparently disconnected loops turn out to be joined together. Students will
be challenged
to explain this phenomenon as they explore topology using the Möbius loop.
Puzzle Presentation
1. This puzzle works best if you construct a model in front of the class, move
the
pencil between the two loops to show that they are not connected, and
then try to
pull them apart, showing that they are, in fact, connected.
2. When moving the pencil between the two loops, you will find that after one
rotation the pencil will be facing the opposite direction than it was when you
started. It is necessary to make two complete rotations to return the pencil
to its
original orientation. This realization is an important part of explaining the
puzzle,
and students should be allowed to make the discovery for themselves without
having it pointed out to them.
3. Once you have demonstrated the puzzle for the class, give students the necessary
materials and have them construct their own version of the puzzle. It is better
if
the paper students are using is plain so that it is more of a challenge to
distinguish between front and back.
Activity Page I | Activity Page II
Solution Hint
Coloring each side of each strip of paper a different color before the
band is assembled can help students see which strips are being
attached to each other.