fractal 203

1

–

72

1

–

52

1

–

32

π

2

–

8

f′are not continuous for a finite number of locations in

the interval [–π,π]. This shows, for example, that func-

tions that zig-zag like a sawtooth, or jump up and

down in value like a staircase, for example, can still be

well approximated by a sum of trigonometric func-

tions. For example, take f(x) to be the V-shaped func-

tion f(x) = |x| on the interval [–π,π], with this section of

graph repeated over the entire number line to produce

the picture of a sawtooth. One checks:

giving:

at least on the interval [–π,π]. Note, as a curiosity, that

if we place x= 0 into this formula we obtain the

remarkable identity:

= 1 + + + +…

See also

ZETA FUNCTION

.

fractal If we

SCALE

the picture of a geometric object

by a factor k, then its size changes accordingly: any line

of length abecomes a line of length ka, any planar

region of area Abecomes a planar region of area k2A,

and any solid of volume Vis replaced by a solid of vol-

ume k3V. An object can thus be described as d-dimen-

sional if its “size” scales according to the rule:

new size = kd×old size

At the turn of the 20th century, mathematicians discov-

ered geometric objects that are of fractional dimension.

These objects are called fractals. One such object is Sier-

pinski’s triangle, devised by Polish mathematician Vaclav

Sierpinski (1882–1969). Beginnning with an equilateral

triangle, one constructs it by successively removing cen-

tral triangles ad infinitum. The final result is an object

possessing “self-similarity,” meaning that the entire fig-

ure is composed of three copies of itself, in this case each

at one-half scale. If the dimension of the object is dand

the size of the entire object is S, then according to the

scaling rule above, the size of each scaled piece

is . As the entire figure is composed of three

of these smaller figures, we have . This

tells us that 2d= 3, yielding d= ≈1.58. Thus the

Sierpinski triangle is a geometric construct that lies some-

where between being a length and an area.

In 1904 Swedish mathematician Nils Fabian Helge

von Koch (1870–1924) described a fractal curve con-

structed in a similar manner. Beginning with a line seg-

ment, one draws on its middle third two sides of an

equilateral triangle of matching size and repeats this

construction ad infinitum on all line the segments that

appear. The result is called the Koch curve. It too is

self-similar: the entire figure is composed of four copies

of itself, each at one-third scale. The object has fractal

dimension d= ≈1.26.

The Cantor set, invented by German mathemati-

cian G

EORG

C

ANTOR

(1845–1918), is constructed from

a single line segment, by removing its middle third and

the middle thirds of all the line segments that subse-

quently appear. The result is a geometric construct,

resembling nothing more than a set of points, but again

with the same self-similarity property: the entire con-

struct is composed of two copies of itself, each at one-

third scale. The Cantor set is a fractal of dimension

d= ≈0.63. It is not large enough to be considered

one-dimensional, but it is certainly “more” than a dis-

connected set of isolated points.

Fractals also arise in the theory of

CHAOS

and the

study of

DYNAMICAL SYSTEM

s. French mathematician

Gaston Maurice Julia (1892–1978) considered the

iterations of functions fthat take

COMPLEX NUMBERS

as inputs and give complex numbers as outputs. If zis

a complex number and the set of points f(z), f(f(z)),

f(f(f(z))),… are all plotted on a graph, then two pos-

sibilities may occur: either the sequence is unbounded,

or the points jump about in a bounded region. The set

ln2

––

ln3

ln3

––

ln2

ln3

––

ln2