398 Playfair’s axiom

2

It is worth mentioning that, like the Platonic solids,

there are only a finite number of semiregular polyhe-

dra, that is, polyhedra with two types of faces, each a

regular polygon with each vertex alike. For example,

the classical pattern of 12 pentagons and 20 hexagons

on a soccer ball represents a semiregular polyhedron.

A

RCHIMEDES OF

S

YRACUSE

(287–212

B

.

C

.

E

.) proved

that there are only 13 semiregular polyhedra. For this

reason, they are today called the Archimedean solids.

See also

CONGRUENT FIGURES

;

POLYGON

.

Playfair’s axiom See

PARALLEL

;

PARALLEL POSTULATE

.

plot The act of specifying the location of a point on a

set of coordinate axes is called “plotting” the point. If

a collection of points are specified by an equation of

the form y= f(x),then a graph of the equation is some-

times called a plot.

See also C

ARTESIAN COORDINATES

;

COORDINATES

;

GRAPH OF A FUNCTION

;

SCATTER DIAGRAM

.

plus The symbol used to denote

ADDITION

is the plus

sign, +. For example, the number 4 increased by the

addition of two more units is written 4+2. In general,

any operation that is to be interpreted as analogous to

addition is denoted with a plus sign. For example, the

GROUP

operation in an abstract Abelian group is usu-

ally denoted +. As an adjective, the + symbol is used to

describe a quantity of positive value. For example, +4

is a quantity 4 units greater than zero.

The symbol is believed to be derived as an abbrevi-

ation of the Latin word et for “and,” which was often

used to describe addition: 4 and 2 make 6, for instance.

A 1489 Latin manuscript on arithmetic written by

Johannes Widman contains the first known printed use

of the symbol. It also contains the symbol “–” for

SUB

-

TRACTION

, which is believed to have already been in

common use in Germany for several decades.

The plus/minus symbol, ±, is used to denote a quan-

tity that which should be both added and subtracted.

For example, the two solutions to the

QUADRATIC

equation x2– x– 1 = 0 can be written . If the

choice of operation is important, mathematicians will

introduce the symbol . For example, the expression

a±bcindicates that the second operation is to be

different from the first. (Thus the expression can be

interpreted as either a+ b– cor a– b+ c, but not a+ b

+ cor a– b– c.) Matters are confusing, however. If the

symbol is not used, then all choices of operations are

deemed permissible. The expression a±b±c, for

instance, could be any one of the four possibilities.

There is no special symbol to indicate that both opera-

tions must be the same.

The plus/minus symbol was first used by French

mathematician Albert Girard in his 1621 text Tables.

Poincaré, Jules-Henri (1854–1912) French Analysis,

Topology, Mechanics Born on April 29, 1854, in

Nancy, France, mathematician Jules-Henri Poincaré is

remembered as one of the great geniuses of all time who

was active in almost every area of mathematics and

physics. Poincaré founded the field of algebraic

TOPOL

-

OGY

(the application of algebraic techniques to solve

problems about space, shape, and form) and discovered

an important class of functions called automorphic

functions. (These functions, defined in the field of

COM

-

PLEX NUMBERS

, are ratios of linear functions and have

the property of being invariant under various symme-

tries of the complex plane.) In applied mathematics, he

is remembered for his substantial work in the theory of

celestial mechanics and for his work in optics, electro-

magnetism, thermodynamics, quantum theory, and the

development of the theory of special relativity. (He is

considered a codiscoverer of the special theory with

A

LBERT

E

INSTEIN

and Hendrik Lorentz.) Poincaré pub-

lished over 500 memoirs during his lifetime, as well as a

number of popular books on the philosophy of mathe-

matics and science.

After working as a mining engineer at Vesoul for

several years, Poincaré received a doctoral degree in

mathematics from the University of Paris in 1879. His

thesis examined applications of

DIFFERENTIAL EQUA

-

TION

s and served as a springboard for his later work in

topology, automorphic functions, and in physics.

Poincaré had a talent for recognizing connections

between disparate topics of study, allowing him to

solve problems from many different perspectives.

Poincaré published his first piece on the topic of

topology in 1895 with his text Analysis situs (Analysis

of position). At the same time he also outlined his

method of homotopy theory based, essentially, on the

±

±

±