any polyhedron with six faces. There is no polyhedron
with fewer than four faces.
A convex polyhedron is called regular if all of its
faces are congruent regular polygons. The geometer
E
UCLID
(ca. 300–260
B
.
C
.
E
.) proved, in his final volume
of T
HE
E
LEMENTS
, that there are only five regular poly-
hedra: the regular tetrahedron (with four triangular
faces), the cube (with six square faces), the octahedron
(with eight triangular faces), the dodecahedron (with 12
pentagonal faces), and the icosahedron (with 20 trian-
gular faces). These five regular solids played an impor-
tant role in the Greek study of geometry, and much
mystic significance was ascribed to the figures. Philoso-
pher P
LATO
(ca. 428–348
B
.
C
.
E
.) studied and wrote
extensively about the five regular polyhedra, and for
this reason they are today called the P
LATONIC SOLIDS
.
A polyhedron is called semiregular if it is composed
of two different types of polygonal faces combined
together in the same way at each vertex of the polyhe-
dron. For example, the classical pattern drawn on a soc-
cer ball describes a semiregular polyhedron composed
of 12 pentagons and 20 hexagons arranged so that each
vertex of the figure is surrounded by one pentagon and
two hexagons. A
RCHIMEDES OF
S
YRACUSE
(ca. 287–212
B
.
C
.
E
.) proved that there exist only 13 semiregular poly-
hedra, today called the Archimedean solids.
A polyhedron is said to be stellated if it is built
from a regular polyhedron by attaching pyramids to its
faces, or faceted if it is formed with these pyramids
turned inward.
Swiss mathematician L
EONHARD
E
ULER
(1707–83)
discovered a remarkable formula, E
ULER
’
S THEOREM
,
relating the number of vertices v, edges e, and faces f,
of any polyhedron free from holes akin to the hole of a
TORUS
. We have:
v– e+ f= 2
This formula proves, for instance, that any polyhedron
composed of pentagonal and hexagonal faces, not nec-
essarily regular, with three edges meeting at each ver-
tex, must contain precisely 12 five-sided faces. (To see
this, let vbe the number of vertices of the polyhedron,
ethe number of edges, pthe number of pentagonal
faces, and hthe number of hexagonal faces. Then v– e
+ p+ h= 2. Also, since three edges meet at each vertex,
each pentagonal face has five edges, and each hexago-
nal face six edges, we have 3v= 2e= 5p+ 6h. These
equations force pto have value 12, no matter what
value hmay adopt.)
See also
ALTITUDE
;
BASE OF A POLYGON
/
POLYHE
-
DRON
;
CONCAVE
/
CONVEX
;
CONE
;
CYLINDER
;
FACE
;
FRUSTUM
;
HYPERCUBE
;
NET
;
PARALLELEPIPED
;
PRISM
;
PYRAMID
.
polynomial A sum of multiples of positive integer
powers of a variable is called a polynomial. The gen-
eral form of a polynomial is an expression of the form:
anxn+ an–1xn–1+…+ a1x+ a0
where an, an–1,…,a1,a0are numbers, called the
COEFFI
-
CIENT
s of the polynomial, with leading coefficient an
assumed to be different from zero. The number a0,
which may be zero, is called the constant term of the
polynomial. The highest power nthat appears in the
expression with nonzero coefficient is called the
DEGREE
OF THE POLYNOMIAL
. For example, √
–
2x3– 5x+ 8 and 5
are polynomials of degrees three and zero, respectively.
A polynomial of degree two is called a
QUADRATIC
, of
degree three a cubic, of degree four a quartic, and of
degree five a quintic.
One can add two polynomials to produce a new
polynomial by collecting like terms, and one can multi-
ply two polynomials using the process of
EXPANDING
BRACKETS
. For example:
(x3+ 2x2– 3x+ 4) + (x2+ 4x– 2) = x3+ 3x2+ x+ 2
and
(x2+ 2x+ 1)(3x+ 5) = 3x3+ 6x2+ 3x+ 5x2+ 10x+ 5
= 3x3+ 11x2+ 13x+ 5
If one thinks of the variable xas an unspecified
BASE OF A NUMBER SYSTEM
, then one can perform the
same arithmetical operations on polynomials (such as
E
LIZABETHAN MULTIPLICATION
and
LONG DIVISION
) as
for ordinary numbers. The ratio of two polynomials
produces a
RATIONAL FUNCTION
.
A polynomial function is a function whose outputs
are given by a polynomial expression. For example,
f(x) = 5x3– 7x– 3 is a cubic function. It is not possible
for a nonzero polynomial function to produce zero as
an output for all inputs x. We have:
404 polynomial