Peano’s curve 385

Peano’s space-filling curve

For example, in the triangle shown on previous

page, the number 20 is one greater than the sum of

numbers bounded in the parallelogram with 6 as the

lowest corner. (We have 6 + 3 + 3 + 1 + 2 + 1 + 1 + 1

+ 1 = 19.) This property can be proved by noting

that the number Mis the sum of the two numbers

above it, which, in turn, are each the sum of two

numbers in the previous row, and so forth. One can

match this backward tabulation with the entries of

the described parallelogram (except for a single 1).

See also C

ATALAN NUMBERS

.

Peano, Giuseppe (1858–1932) Italian Foundations of

mathematics Born on August 27, 1858, in Cuneo, Italy,

logician Giuseppe Peano is remembered for his influential

work in mathematical logic and in the

FOUNDATIONS OF

MATHEMATICS

. In 1889 he published a famous set of

axioms (revised in 1899), today called P

EANO

’

S POSTU

-

LATES

, which defined the

NATURAL NUMBERS

in terms of

sets. A year later he invented space-filling curves, such as

P

EANO

’

S CURVE

, which were thought impossible at the

time. These curves show, in some sense, that there are

just as many points on a line as there are in a plane.

Peano graduated from the University of Turin in

1880 with a doctorate in mathematics and his habilita-

tion degree in 1884. He remained at the university as a

professor of mathematics throughout his career.

Peano’s early work was in the field of

DIFFERENTIAL

EQUATION

s, where he studied and established significant

results on the problem of classifying those equations for

which solutions are guaranteed to exist. But his interests

changed toward mathematical logic in 1888 with the

publication of his text Geometric Calculus, the opening

chapter of which was devoted to the topic. Peano pub-

lished his famous axioms defining the natural numbers

1 year later in the small pamphlet Arithmetices prin-

cipia, nova methodo exposita, (Arithmetic principles,

exposition of a new method), which he wrote entirely in

Latin to the surprise of his colleagues. Soon after this,

he presented his work on his famous curve.

Mathematical historians feel that Peano’s contribu-

tions to mathematics dwindled after this. In 1892 Peano

began work on an enormous undertaking to collate all

known results in mathematics, essentially as a giant list.

This project, which he called Formulario mathematico

(Mathematical forms), consumed his working hours for

a full 16 years. He completed the project in 1908, but

received little attention for it. Although the text con-

tained a great deal of valuable information, it was diffi-

cult to read, not only because of its dry nature, but also

because Peano chose to write it in a new “universal lan-

guage” he invented based on a simplified version of

Latin. He hoped to develop a universal culture of math-

ematical exploration united by a common language. His

dream was never realized.

Peano died in Turin, Italy, on April 20, 1932.

Much of his mathematical work, although significant

at the time, is chiefly of historic interest today.

Peano’s curve In 1890 Italian mathematician G

IUSEPPE

P

EANO

(1858–1932) described a curve that could pass

through every point of a square. The curve is constructed

by an iterative process. One begins by drawing the diago-

nal of a square and then breaking the square into nine

subsquares and drawing certain diagonals of those sub-

squares. At the next stage, each subsquare is divided into

nine sub-subsquares and the pattern is repeated. The

Peano curve is the curve that results when this procedure

is repeated indefinitely. One can intuitively see that

Peano’s construct is an object that passes through every

point of the square. (It passes through some points more

than once.) For this reason, Peano’s curve is described as

a space-filling curve. His construct shows, in some sense,

that the set of points inside a square is no more infinite

than the set of points on a curve.

To make the definition of Peano’s curve mathemat-

ically precise, for each number tin the interval [0,1] let

Pn(t) denote the point in the square that is tunits

along the length of the curve that is produced in the

nth step of the above procedure. (Notice, for example,

that for all n.) Then define P(t) to be

the

LIMIT

of these points: