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# Peano's Curve

DOI link for Peano's Curve

Peano's Curve book

# Peano's Curve

DOI link for Peano's Curve

Peano's Curve book

## ABSTRACT

In 1890, Peano constructed the beautiful example of a continuous map of an interval whose image is a square. The construction is the following. Decompose an interval A into four equal intervals A i l , i l = 1,2,3,4 indexed from left to right. Then decompose a square A into four equal squares A i l , i l = 1,2,3,4, enumerating these small squares in such a manner that consecutive squares have a mutual side. We call the constructed decomposition a first step. Let us assign to each interval Ail the square A,, and denote this correspondence by f i , i.e. A,, = , f l (A i , ) . Decompose every interval A,, in a similar way into four equal intervals AiIi2. We also decompose every square A,, into four equal squares here we choose the index i2 in such a way that the Ail,, have a mutual side with AiIi, ,I and AiI4 have a mutual side with Ai l+ l l . We call this decomposition a second step. Let us assign to each interval A,,,, the square Ai,;* and denote this correspondence by f 2 , i.e. A,,;, =,f2(AiIi2). We continue the decomposition in this way and denote intervals of the n-th step by A,, . . . , and squares by A,, ...i,, . If two intervals of the n-th step have a mutual point, then the corresponding

squares of the n-th step have a mutual side. The correspondencef, between intervals and squares has the following property: if Ail...;,, c Ail...;,,_l, then fn(Ai ,... ,,) c j;,-,(Ail. Let us assign to every point of A a point of the square A in the following manner. Each point t E A belongs to an infinite sequence of enclosed intervals A i l , A; ,,,, . . . , A; ,... , , , . . . . The squares corresponding to these intervals form an infinite sequence of enclosed squares:

Because the lengths of the sides of the squares converge to 0, there exists a unique point P E A, belonging to all squares of the sequence. Thus we define the map f : A 4 A, assigning to every point t E A the corresponding point P = f ( t ) . It follows from the definition that if t E Ail ...i,, then f ( t ) E , fn(Ai ,... i,,).