496 tensor

(Noting that calculators only display eight or 10 deci-

mal places, one need only use the first few terms of the

Taylor series to obtain adequately accurate answers.)

To approximate a function near a value x= adif-

ferent from zero, one uses a polynomial of the form:

a0+ a1(x– a) + a2(x– a)2+ a3(x– a)3+… Using the

same technique of differentiating and substituting in

x= ayields:

This is called a Taylor series “centered at x= a.” Taylor

series, as discussed above, centered at x= 0 are some-

times called Maclaurin series. In 1742 Scottish mathe-

matician C

OLIN

M

ACLAURIN

(1698–1746) wrote an

influential text in which he described Taylor’s methods.

Although Maclaurin made no pretense of having dis-

covered these series centered at zero (he himself

acknowledged that they are nothing more than a spe-

cial case of Taylor’s general results), scholars honor his

work nonetheless by associating his name with these

special series.

Talyor conducted his work in the early 1700s, but

it is known that other scholars, such as J

AMES

G

RE

-

GORY

(1638–75), used power series in the same way

decades earlier.

See also

DIFFERENTIAL EQUATION

; M

ERCATOR

’

S

EXPANSION

.

tensor Just as a

VECTOR

is a mathematical quantity

that describes translations in two- or three-dimensional

space, a tensor is a mathematical quantity used to

describe general transformations in n-dimensional

space. Precisely, if the locations of points in n-dimen-

sional space are given in one coordinate system by

(x1,x2,…,xn) and in a transformed coordinate system

by (y1, y2,…,yn) (it is convenient to use superscripts

rather than subscripts), then a “rank 1 contravariant

tensor” is a quantity T, with single components, that

transforms according to the rule:

(The coefficients here are

PARTIAL DERIVATIVE

s.) Thus,

for example, if the change of coordinates is a transla-

tion, yi= xi+ aifor some numbers ai, it follows that

Tinew = Ti. This means that the quantity Tis a quantity,

with ncomponents, unchanged by translations. That is,

Tis indeed a vector.

More complicated transformation rules are permit-

ted for quantities with components given by two or

more superscripts (or even subscripts). A

LBERT

E

IN

-

STEIN

(1879–1955) used tensor analysis in his general

theory of relativity.

tessellation (covering, tiling) A covering of the

PLANE

with geometric shapes is called a tessellation or a tiling.

Usually the shapes, called tiles, are

POLYGON

s, and the

pattern produced is, in some sense, repetitive. Every

point in the plane is to be covered by a tile, and two

tiles may intersect only along their edges. A location

where three or more edges meet at a point is called a

vertex of the tessellation. (It is usually assumed that

neighboring tiles meet along the full length of their com-

mon edge.)

A regular tessellation uses congruent regular poly-

gons, all of one type, as tiles. For example, square tile

can be used to cover a plane, as shown below. Four

edges meet at each vertex of the tessellation. (Note that

four angles of 90°add to a total of 360°around that

vertex.) An equilateral triangle also tiles the plane (six

triangles, containing angles of 60°, fit around one ver-

tex), as does the regular hexagon. (Three hexagons, con-

taining angles of 120°, fit about a vertex.) As no other

regular polygon has appropriate angle values to fit about

a vertex, these are the only regular tilings of the plane.

A semiregular tessellation uses congruent regular

polygons of more than one kind, arranged so that the

arrangement of polygons about every vertex of the tes-

sellation is identical. For example, one can tile the

plane with regular hexagons and equilateral triangles

so that each hexagon is surrounded by six triangles and

each triangle by three hexagons to produce a semiregu-

lar tessellation. Mathematicians have proved that there

are only eight semiregular tilings of the plane. (If one

abandons the restriction that the arrangement of poly-

gons about each vertex be identical, then there are

infinitely many such tilings.)

A monohedral tessellation is a tessellation that uses

congruent copies of only one type of tile (not necessarily