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
EUCLID (ca. 300–260 B.C.E.) proved, in his final volume
of THE ELEMENTS, that there are only five regular polyhedra: 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 triangular faces). These five regular solids played an important role in the Greek study of geometry, and much
mystic significance was ascribed to the figures. Philosopher PLATO (ca. 428–348 B.C.E.) studied and wrote
extensively about the five regular polyhedra, and for
this reason they are today called the PLATONIC 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 polyhedron. For example, the classical pattern drawn on a soccer 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. ARCHIMEDES OF SYRACUSE (ca. 287–212
B.C.E.) proved that there exist only 13 semiregular polyhedra, 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 LEONHARD EULER (1707–83)
discovered a remarkable formula, EULER’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 necessarily regular, with three edges meeting at each vertex, must contain precisely 12 five-sided faces. (To see
this, let v be the number of vertices of the polyhedron,
e the number of edges, p the number of pentagonal
faces, and h the 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 hexagonal face six edges, we have 3v = 2e = 5p + 6h. These
equations force p to have value 12, no matter what
value h may adopt.)
See also ALTITUDE; BASE OF A POLYGON/POLYHEDRON; 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 general form of a polynomial is an expression of the form:
a n x
n + a n–1 x
n–1 +…+ a 1 x + a 0
where a n , a n–1 ,…,a 1 ,a 0 are numbers, called the COEFFICIENTs of the polynomial, with leading coefficient a n
assumed to be different from zero. The number a 0 ,
which may be zero, is called the constant term of the
polynomial. The highest power n that appears in the
expression with nonzero coefficient is called the DEGREE
OF THE POLYNOMIAL. For example, √
–
2x
3 – 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 multiply two polynomials using the process of EXPANDING
BRACKETS. For example:
(x
3 + 2x
2 – 3x + 4) + (x
2 + 4x – 2) = x
3 + 3x
2 + x + 2
and
(x
2 + 2x + 1)(3x + 5) = 3x
3 + 6x
2 + 3x + 5x
2 + 10x + 5
= 3x
3 + 11x
2 + 13x + 5
If one thinks of the variable x as an unspecified
BASE OF A NUMBER SYSTEM, then one can perform the
same arithmetical operations on polynomials (such as
ELIZABETHAN 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) = 5x
3 – 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
with fewer than four faces.
A convex polyhedron is called regular if all of its
faces are congruent regular polygons. The geometer
EUCLID (ca. 300–260 B.C.E.) proved, in his final volume
of THE ELEMENTS, that there are only five regular polyhedra: 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 triangular faces). These five regular solids played an important role in the Greek study of geometry, and much
mystic significance was ascribed to the figures. Philosopher PLATO (ca. 428–348 B.C.E.) studied and wrote
extensively about the five regular polyhedra, and for
this reason they are today called the PLATONIC 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 polyhedron. For example, the classical pattern drawn on a soccer 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. ARCHIMEDES OF SYRACUSE (ca. 287–212
B.C.E.) proved that there exist only 13 semiregular polyhedra, 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 LEONHARD EULER (1707–83)
discovered a remarkable formula, EULER’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 necessarily regular, with three edges meeting at each vertex, must contain precisely 12 five-sided faces. (To see
this, let v be the number of vertices of the polyhedron,
e the number of edges, p the number of pentagonal
faces, and h the 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 hexagonal face six edges, we have 3v = 2e = 5p + 6h. These
equations force p to have value 12, no matter what
value h may adopt.)
See also ALTITUDE; BASE OF A POLYGON/POLYHEDRON; 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 general form of a polynomial is an expression of the form:
a n x
n + a n–1 x
n–1 +…+ a 1 x + a 0
where a n , a n–1 ,…,a 1 ,a 0 are numbers, called the COEFFICIENTs of the polynomial, with leading coefficient a n
assumed to be different from zero. The number a 0 ,
which may be zero, is called the constant term of the
polynomial. The highest power n that appears in the
expression with nonzero coefficient is called the DEGREE
OF THE POLYNOMIAL. For example, √
–
2x
3 – 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 multiply two polynomials using the process of EXPANDING
BRACKETS. For example:
(x
3 + 2x
2 – 3x + 4) + (x
2 + 4x – 2) = x
3 + 3x
2 + x + 2
and
(x
2 + 2x + 1)(3x + 5) = 3x
3 + 6x
2 + 3x + 5x
2 + 10x + 5
= 3x
3 + 11x
2 + 13x + 5
If one thinks of the variable x as an unspecified
BASE OF A NUMBER SYSTEM, then one can perform the
same arithmetical operations on polynomials (such as
ELIZABETHAN 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) = 5x
3 – 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