Question

Find a Peano Curve which fills out the unit square in E^2

Answer 1

Answer on Question #43676, Math, Topology

Problem. Find a Peano Curve which fills out the unit square in $\mathsf{E}^{\wedge}2$

Solution. The Peano curve is the first example of a space-filling curve.

We present this example.

We will define the sequence of the sets of squares $i$ . Let $S_0$ is the single unit square. The set $i$ is defined by partition all squares of $i - 1$ into 9 equals squares. Let $P_i$ be the set of the centers of the squares from the set $S_i$ .

For each $i$ we will define curve which will connect the points from the set $P_{i}$ with curve. The limit of such curve will be the Peano curve which fills out the unit square. For each $i$ we define the curve which connect left bottom corner and right top corner. For each $i$ we could also obtain curve which connect left top corner and right bottom corner by rotation. The curve for $S_{1}$ is shown in the following figure (blue line)

By red arrow we will denote the connection between corners.

Suppose that we have constructed curve for $i - 1$ , the curve for $i$ we will construct by the following scheme

The order of the arrows is enumerated with the numbers.

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