in Schulpforta, Saxony (now Germany), scholar August 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 CARL FRIEDRICH GAUSS (1777–1855), the finest mathematician of
the day, and also director of the astronomical observatory 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 trigonometric 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 barycentrische Calcül (The calculus of barycenters), on analytic
geometry in 1827. In it he outlined a number of significant, 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 z a complex number, and a, b, c, and d real numbers 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 NUMBER THEORY. He is best known in this field for his discovery of the Möbius inversion formula, which can be
described as follows. Consider a function µ defined on
the set of positive integers as follows: µ(1) = 1; µ(p) = –1
if p is PRIME; µ(n) = (–1)
k if n has k primes factors, all distinct; and µ(n) = 0 if n has 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 f is any function on the positive integers, and, from it, a new function
F is defined as F(n) = Σ d |n f(d), a sum over all the factors d
of n, then Möbius’s inversion formula states that one can
recover from F the original function f via 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 mathematician Johann Listing had considered the object just a
few years earlier), his mathematical analysis of the surface 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, combinatorics, 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 rectangular 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 original 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 a that are glued together, and
again for the segments labeled b.) In general, if a band
of paper with n half-twists is cut in half along the central line, one piece will result if n is odd, and two pieces
are produced if n is 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 KLEIN BOTTLE. One
f n
F d
n
d
d n
( )
( ) ( )
|
= ∑
µ
f z
az b
cz d
( ) =
+
+
Möbius band 339
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 CARL FRIEDRICH GAUSS (1777–1855), the finest mathematician of
the day, and also director of the astronomical observatory 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 trigonometric 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 barycentrische Calcül (The calculus of barycenters), on analytic
geometry in 1827. In it he outlined a number of significant, 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 z a complex number, and a, b, c, and d real numbers 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 NUMBER THEORY. He is best known in this field for his discovery of the Möbius inversion formula, which can be
described as follows. Consider a function µ defined on
the set of positive integers as follows: µ(1) = 1; µ(p) = –1
if p is PRIME; µ(n) = (–1)
k if n has k primes factors, all distinct; and µ(n) = 0 if n has 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 f is any function on the positive integers, and, from it, a new function
F is defined as F(n) = Σ d |n f(d), a sum over all the factors d
of n, then Möbius’s inversion formula states that one can
recover from F the original function f via 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 mathematician Johann Listing had considered the object just a
few years earlier), his mathematical analysis of the surface 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, combinatorics, 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 rectangular 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 original 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 a that are glued together, and
again for the segments labeled b.) In general, if a band
of paper with n half-twists is cut in half along the central line, one piece will result if n is odd, and two pieces
are produced if n is 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 KLEIN BOTTLE. One
f n
F d
n
d
d n
( )
( ) ( )
|
= ∑
µ
f z
az b
cz d
( ) =
+
+
Möbius band 339