Möbius band 339

in Schulpforta, Saxony (now Germany), scholar Aug-

ust Ferdinand Möbius is remembered in mathematics

for his work in

GEOMETRY

and

TOPOLOGY

, and his

conception of the one-sided surface today called a

MÖBIUS BAND

.

Möbius entered the University of Leipzig in 1809

to study law but soon changed interests and took up a

study of mathematics, astronomy, and physics. In 1813

he traveled to Göttingen to work with C

ARL

F

RIED

-

RICH

G

AUSS

(1777–1855), the finest mathematician of

the day, and also director of the astronomical observa-

tory in Göttingen. Möbius completed a doctoral thesis

in 1815 in the topic of astronomy and soon afterward

completed a habilitation thesis on the topic of trigono-

metric equations. In 1816 he was appointed chair of

astronomy and mechanics at the University of Leipzig.

He stayed at Leipzig the remainder of his life.

Möbius published an influential work, Der barycen-

trische Calcül (The calculus of barycenters), on analytic

geometry in 1827. In it he outlined a number of signifi-

cant, and original, advances in the fields of

PROJECTIVE

GEOMETRY

and

AFFINE GEOMETRY

. He also studied the

geometry of the plane of

COMPLEX NUMBERS

, and

showed that any complex function of the form:

with za complex number, and a, b, c, and dreal num-

bers satisfying ad – bc ≠0, transforms straight lines and

circles in the complex plane into new straight lines and

circles. Complex functions of the form above are today

called Möbius functions.

In 1831 Möbius published important results in

NUM

-

BER THEORY

. He is best known in this field for his discov-

ery of the Möbius inversion formula, which can be

described as follows. Consider a function µdefined on

the set of positive integers as follows: µ(1) = 1; µ(p) = –1

if pis

PRIME

; µ(n) = (–1)kif nhas kprimes factors, all dis-

tinct; and µ(n) = 0 if nhas a prime factorization that

includes repeated primes. (Thus, for instance, µ(70) = µ(2

· 5 · 7) = –1 and µ(20) = µ(2 · 2 · 5) = 0.) If fis any func-

tion on the positive integers, and, from it, a new function

Fis defined as F(n) = Σd|nf(d),a sum over all the factors d

of n, then Möbius’s inversion formula states that one can

recover from Fthe original function fvia the rule:

Möbius wrote about the one-sided surface that

bears his name in an 1858 piece discovered only after

his death. His interest in the object was motivated by a

general question on the geometry of polyhedra that had

caught his interest. Although Möbius was not the first

to describe the one-sided surface (German mathemati-

cian Johann Listing had considered the object just a

few years earlier), his mathematical analysis of the sur-

face was deep and significant.

Möbius also published important works in the fields

of astronomy, celestial mechanics, and statics. He died in

Leipzig, Germany, on September 26, 1868. His work on

number theory has had a profound effect on the nature

of research in the subject today. For instance, the concept

of the Möbius function has found natural applications to

generalized abstract settings in algebraic geometry, com-

binatorics, and partially ordered sets, thereby providing

new insights into the study of numbers.

Möbius band (Möbius strip) The one-sided surface

obtained by gluing together the two ends of a long rect-

angular strip twisted 180°so as to produce a half-twist

in the resulting object is called a Möbius band. This

surface has the property that if one paints the surface

all the way round, one finds that both sides of the orig-

inal strip are colored. This shows that the surface is

indeed one-sided. Also, tracing one’s finger along the

edge of a Möbius band covers every possible point on

the edge of the object, thus showing, in addition, that

the surface has only one edge.

In some sense it is impossible to cut a Möbius band

in half. For instance, cutting the object along a central

line parallel to the edge produces a single connected

two-sided object. The reason for this can be seen in the

diagram on the following page. (Note that the top half

of the original strip is connected to the bottom portion

via the segments labeled athat are glued together, and

again for the segments labeled b.) In general, if a band

of paper with nhalf-twists is cut in half along the cen-

tral line, one piece will result if nis odd, and two pieces

are produced if nis even. Also, if one attempts to cut a

Möbius band into thirds (along two parallel lines that

divide the strip in thirds), then two interlocking rings

result, one twice the length of the other. The shorter of

the two is another Möbius band.

Gluing together two Möbius bands along their

edges produces a surface called a K

LEIN BOTTLE

. One