1. Each town is connected to one Steiner point
2. Each Steiner point has three roads emanating from
it, equally spaced 120°apart.
See also
ISOPERIMETRIC PROBLEM
.
stem-and-leaf plot (stem plot) See
STATISTICS
:
DESCRIPTIVE
.
stereographic projection Consider a
PLANE
tangent
to a
SPHERE
touching the sphere at its south pole S.
Then one can map points on the surface of the sphere
to points on the plane by drawing straight lines from
the north pole Nof the sphere through points on the
surface, and continuing them until they intercept the
plane. Every point, except the north pole itself, is thus
mapped to a point in the plane. This geometrical trans-
formation of a sphere onto a plane is called the stereo-
graphic projection.
Every point on the plane corresponds to a unique
point on the sphere, with points farther and farther
away from Son the plane matching points closer and
closer to Non the sphere. In this sense, the sphere
can be regarded as topologically equivalent to a plane
with a single well-defined additional point of infinity
attached to it. If the plane is taken to be a representa-
tion of the plane of
COMPLEX NUMBERS
, then the
sphere in this construct is usually called a Riemann
sphere to honor the work of German mathematician
G
EORG
F
RIEDRICH
B
ERNHARD
R
IEMANN
(1826–66) in
this field.
A stereographic projection is a
CONFORMAL MAP
-
PING
. This means that it preserves angles between
intersecting curves on the surface of the sphere. Great
circles on the sphere, not through N, are mapped to
circles in the plane, and great circles through Nare
mapped to straight lines. For this reason, geometers
often deem it appropriate to regard straight lines as
special types of circles.
A gnomonic projection maps points on the south-
ern hemisphere of a sphere onto a plane tangent to Sby
drawing straight lines from the center of the sphere
through points on the surface, and continuing them
until they intercept the plane. In this model, each point
on the equator of the sphere represents a different
“point of infinity” attached to the plane.
See also M
ERCATOR
’
S PROJECTION
;
PROJECTION
.
Stevin, Simon (1548–1620) Flemish Arithmetic, Engi-
neering Born in Flanders, now Belgium, in 1548 (his
exact birth date is not known), Simon Stevin is best
remembered for The Tenth, his influential 1585 text
that advocates and explains the use of decimals in all
of mathematics and accounting. Although he did not
invent the decimal system (it had been used by the
Arabs two centuries before), his expository piece on
the subject convinced scholars of the time of its merit
as an approach to manipulating fractions and real
numbers.
Stevin started his career as a bookkeeper and a tax
office clerk before entering the University of Leiden at
the age of 35. His work on mechanics and engineering
garnered him note as an expert in hydrostatics and its
related mathematics. Stevin wrote 11 texts in all
throughout his academic career, covering topics in
arithmetic, algebra, trigonometry, geography, and navi-
gation. In his text Principles of the Art of Weighing,
Stevin analyzed the geometric addition of forces, devel-
oping an approach that simplified the mathematics of
mechanics. This approach also formed the basis for the
theory of
VECTOR
analysis developed 200 years later—
Stevin had essentially correctly defined vector addition.
Stevin also recognized that the distance an object falls
in a fixed amount of time is independent of the object’s
weight. This discovery is normally attributed to the
scholar G
ALILEO
G
ALILEI
(1564–1642), but Stevin had
reached the same conclusion 3 years before Galileo
reported his findings.
By the 15th century, mathematicians in Persia,
China, and India were using the decimal system to
represent fractions. Stevin recognized the advantages
of the system and argued, in his piece The Tenth, that
adding decimal fractions is just as easy as adding
whole numbers. (Summing 0.73 and 0.25, for
instance, is no more difficult than adding 73 to 25).
He did not, however, use a decimal point in his work,
choosing to write 34.875, for instance, as 340817253,
circling the digits 0, 1, 2, and 3 to indicate the digits
to the left are multiplied by those powers of one-10th.
(Scottish mathematician J
OHN
N
APIER
(1550–1617)
is responsible for popularizing the use of the decimal
point.)
Writing all numbers as decimal fractions had the
profound psychological effect of placing all numbers on
an equal footing, as it were. The number π, for exam-
ple, written as 3.141 (at least as an approximation)
Stevin, Simon 485