390 perigon

perigon (round angle) An angle of 360°is called a

perigon. It represents one full turn.

perimeter The length of the boundary of a plane fig-

ure is called its perimeter. For example, the perimeter of

a rectangle is twice its length plus twice its width. The

perimeter of a circle is its circumference. Surprisingly, it

is possible for a plane figure of finite

AREA

to have an

infinite perimeter.

See also

FRACTAL

.

period See

PERIODIC FUNCTION

.

period doubling See

DYNAMICAL SYSTEM

.

periodic function A function is periodic if it repeats

itself at regular intervals of the variable. For example,

the function sin(x) is periodic because it cycles every

360°, precisely: sin(x) = sin(x+ 360°) for all values of

x. We also have that sin(x) = sin(x+ 720°) and sin(x) =

sin(x+ 1080°), for example. The smallest positive

value pfor which a periodic function fsatisfies f(x) =

f(x+ p) for all values xis called the period of the func-

tion. The function sin(x) has period 360°, as does the

function cos(x).(The tangent function tan(x),however,

has period 180°.)

In physics, any phenomenon that repeats itself at

regular intervals, such as the swinging of a pendulum,

the vibration of a source of a sound, or the rotation

of the Earth, is called periodic, and the time it takes

for one complete cycle of the phenomenon is called

its period. This idea extends to other branches of

mathematics as well. For example, the repeating deci-

mal 45.76185185185185185… = 45.76185

—

has pe-

riod 3, since three digits are being repeated, and a

rotation of 60°about a point in the plane has period 6,

since six rotations of this type return points to their

original locations.

See also

DYNAMICAL SYSTEM

; F

OURIER SERIES

.

permutation (arrangement, order) A specific ordered

arrangement of a given collection of objects is called a

permutation. For example, a selection of a winner, a

first runner-up, and a second runner-up from a group of

three finalists in a competition would be a permutation

of those three participants. The lists ADBC and BDCA

are two permutations of the letters A, B, C, and D.

The

MULTIPLICATION PRINCIPLE

shows that ndis-

tinct objects can be ordered n! different ways. There

are thus 4! = 24 different permutations in all of the let-

ters A, B, C, and D. (See

FACTORIAL

.)

The number of ways to arrange just robjects from

a collection of ndistinguishable objects (r≤n) is

denoted Pn

r. These are called “permutations taken rat a

time.” For example, there are 12 permutations of

A,B,C,D taken two at a time: AB, AC, AD, BA, BC,

BD, CA, CB, CD, DA, DB, DC. Thus P4

2= 12. Again,

the multiplication principle shows that

In particular Pn

n= n!.

Permutations of the full set of numbers {1,2,…,n}

can be classified as either even or odd by counting the

number of times a large number appears to the left of a

smaller number. For example, in the permutation

25143 of the first five counting numbers, the number 2

appears to the left of 1 ; 5 appears to the left of 1, 4,

and 3; and 4 appears to the left of 3. In total, a large

number appears to the left of a small number an odd

number of times. This permutation is odd.

A transposition is a permutation that interchanges

just two objects. For example, 14325 represents a

transposition (the 4 and the 2 have switched places).

One can check that it is possible to return the entries of

any even permutation back to their original order by

applying an even number of transpositions, and the

entries of an odd permutation by an odd number of

transpositions. This is often taken as the definition of

what it means to say that a permutation is even or odd.

A permutation in which no element appears in its

original location is called a derangement. For exam-

ple, BCAED is a derangement of A, B, C, D, E. There

are precisely

derangements of nobjects.