By the 10th century, Spain was under Islamic con-
trol, and European scholars interested in the intellectual
culture of the Islamic world traveled to Spain to study
the Arabic texts. It was there that Al-Khw–
arizm
–
ı’s
works were discovered and translated into Latin.
Al-Khw–
arizm
–
ı also wrote texts on astronomy, the
sundial, the Jewish calendar, and on geography. He
computed the latitudes and longitudes of over 2,400
specific localities in preparation for the construction of
an accurate world map.
The word algorithm is believed to be derived from
al-Khw–
arizm
–
ı’s name. Medieval European scholars,
attempting to translate the Arab scholar’s name into
Latin, called the practice of using Hindu-Arabic
numerals “algorism,” from which, in turn, any general
practice or procedure became known as an “algo-
rithm.” That a version of his name became part of our
Western vocabulary illustrates the extent to which
al-Khw–
arizm
–
ı’s work influenced the development of
arithmetic and algebra in Europe.
Klein, Felix Christian (1849–1925) German Abstract
algebra, Geometry, Topology Born on April 25, 1849,
scholar Felix Klein is remembered for uniting the dis-
parate fields of
GEOMETRY
and
ALGEBRA
through the
study of
GEOMETRIC TRANSFORMATION
s. He showed that
the abstract analysis of the algebra of transformations
through
GROUP THEORY
leads to a clear understanding
of both E
UCLIDEAN GEOMETRY
and
NON
-E
UCLIDEAN
GEOMETRY
. In particular, he proved that ordinary
Euclidean geometry can be proved
CONSISTENT
, that is,
free of contradiction, if, and only if, non-Euclidean
geometry is consistent. This demonstrated, for the first
time, that the controversial non-Euclidean theories of
geometry were of equal importance to the theory of
ordinary Euclidean geometry. In topology, the K
LEIN
BOTTLE
is named in his honor.
Klein received a doctorate in mathematics from
the University of Bonn in 1868 after completing a the-
sis exploring applications of geometry to mechanics.
After a short period of military service, Klein accepted
a position as professor of mathematics at Erlangen, in
Bavaria, in 1872. It was here that he began his work
on the analysis of geometric transformations and the
unification of Euclidean and non-Euclidean geome-
tries. His approach to the subject with a focus on
“group invariants” was extremely influential, and
work in this field continues today. (It is called Klein’s
“Erlangen program.”)
In 1886 Klein accepted the position as department
chair at the University of Göttingen, where he remained
until his retirement in 1913. He tutored students from
his home in the years that followed.
In addition to conducting high-level research in
mathematics, Klein also wrote mathematical pieces
intended for the general public and founded a mathemat-
ical encyclopedia that he supervised until his death. Klein
was honored with election to the R
OYAL
S
OCIETY
in
1885 and received the Copley Medal from the Society in
1912. He died on June 22, 1925, in Göttingen, Germany.
See also K
LEIN
-
FOUR GROUP
.
Klein bottle The M
ÖBIUS BAND
is a three-dimensional
object possessing just one surface and just one edge. A
SPHERE
and a
TORUS
(donut shape), on the other hand,
are objects with two surfaces—an outside surface and
an inside surface—possessing no edges. The Klein bottle
is an alternative three-dimensional object with only one
surface, akin to that of a Möbius band, but possessing
no edges.
One can easily model these surfaces mentioned
with a pair of trousers. For example, sewing together
the two leg openings produces a circular tube with
a hole for the waist. This hole can be patched with a
piece of material to produce a complete model of a
torus. A Klein bottle is produced by bringing one trouser
leg up, over, and through the waist of the trousers and
pushing it down through the tube of the second leg
before sewing the two leg openings together. Again,
the hole for the waistband represents a hole in the
surface, but the object produced is nonetheless a
(punctured) Klein bottle. Unfortunately it is no longer
physically possible to patch the hole with a piece of
material. This shows that the Klein bottle does not
properly exist in a three-dimensional universe. (One
can imagine, however, that the material that would
make a patch for the hole of the waist has been
“plucked” up into the fourth-dimension. In this
sense, the Klein bottle is a valid mathematical object
in four-dimensional space.)
One can check with this model that the ideal sur-
face of a Klein bottle would indeed be one-sided: an
ant crawling on one side of trouser material could
reach any other part of the trousers, on either side of
294 Klein, Felix Christian