Today, Möbius' strip. The University of
Houston's College of Engineering presents this
series about the machines that make our
civilization run, and the people whose ingenuity
created them.

You've probably heard of
Möbius strips: Take a one-foot length of
cash-register tape, give it a half twist, and then
glue the ends together, forming a circle. Next,
draw a line down the middle of the strip, without
lifting your pen, until you close the line.

You'll find, to your surprise (if you didn't
already know) that the line is not on just one
side. Rather, there is no blank side. When you made
that half twist, you created a closed loop of paper
with no inside or outside. And, when you look at
it, you don't quite believe it.

So, who was the namesake of this strange thing?
August Ferdinand Möbius was born in 1790, in
what is, today, central Germany. He showed an early
talent in math, and he finished a doctorate at
Leipzig when he was 24. The rest of his life he
taught math and astronomy at Leipzig, and he died
there at 78.

No grand theories are named after Möbius. In
fact, he's famous for two ideas that weren't
uniquely his. Yet he was a fine mathematician in a
time when Germany excelled in math. Working near
him, were Gauss, Jacobi, Dirichlet, and someone you
older engineers remember, Bessel of the infamous
Bessel functions.

When he was fifty, Möbius gave a lecture in
which he posed an odd problem: You're the king, and
you must divide your kingdom among your five sons.
You want each region to touch all other regions.
That's easy enough to arrange with four regions.
But just try to do it with five.

A somewhat similar problem is proving that you can
color any map using only four colors. You can do
the coloring easily enough, but don't try to prove
that its possible! It finally took the computer to
convince people that they'd never find a map
arrangement that needed more than four colors. The
map problem is not really the same as the kingdom
problem, but Möbius gets wide credit for
inventing that four-color problem.

So what about the Möbius strip? In 1858, at
sixty-eight, he began his work on geometric solids.
He described his Möbius strip in a paper
published when he was seventy-five. However, his
note-books show that he'd formulated the idea just
after he began that work. Then we find that another
person, Johann Listing, discovered the Möbius
strip idea two months before Möbius did.

Well, that could hardly lessen one's regard for
this remarkable person. For he did everything: He
did his intuition-boggling geometry and he did
celestial mechanics. He wrote the mathematics of
musical intervals as well as of psychology.

Möbius' grandson Paul became a neurologist and
he eventually dug up Möbius skull. He wanted
to explain his grandfather's geometrical ability by
studying the shape of that skull. Well that bit of
topology failed, of course. But Möbius' strip
has been the mental driver for more valuable
applied mathematics than I could ever describe in
my three-minute allotment here.

I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.

(Theme music)

*Möbius and His Band: Mathematics and
Astronomy in Nineteenth-Century Germany:* (John
Fauvel, Raymond Flood, and Robin Wilson, eds.) New
York: Oxford University Press, 1993.
M. J. Crowe, Mobius, August Ferdinand, Dictionary
of Scientific Biography (C.C. Gilespie, ed.) New
York: Charles Scribner's Sons, 1970-1980.

Möbius strip with a half twist on the left. On
the right is similar strip with a full twist.

Try splitting both of these strips down the middle
as shown above.

Slice the Möbius strip down the middle and you
get the continuous (but twisted) loop on the left.
Slice the full twist strip and you get the
remarkable result on the right -- two twisted loops
form a two-link chain.

(All photos by John Lienhard)

The Engines of Our Ingenuity is
Copyright © 1988-2003 by John H.
Lienhard.