Mathematical coloring book - instructions

A typical mathematical coloring picture is assigned like this:

On the left we see a grey canvas with a prominent border, in this case the canvas is a square. We also see the given coordinate system, in this case it is the Carthesian system whose origin is in the lower-left corner of the canvas. Draw this canvas on your paper, unless you decided to try the interactive application. In the full version we show the canvas white so that you can print it. The aim is to put the right colors on this canvas.

On the right we see the specifications that determine colors in the picture. In order to find the color of a specific point of the canvas, we substitute the coordinates of this point into the given coloring function ƒ, then we find the resulting value on the color chart and this determines the right color for this point. For instance, substituting the point (0,3) we find that ƒ(0,3) = |-3| = 3, according to the chart we then color this point red.

The basic strategy is based on the hope that the coloring function is reasonable, and thus the given canvas partitions into regions of reasonable shape with each having its own color. In order to determine the color of a certain region we simply pick some point from its interior, find its color in the way described above and then color the whole region in this color.

The basic version

The assignment for the basic version of a picture also indicated the borders between the colored regions:

Thus it is enough to determine the color for each of the regions. In the middle strip we can pick, say, the point (1,1). We compute ƒ(1,1) = |0| = 0, the color chart indicates white. If you do not feel like painting with colored pencils on paper, you can click on the appropriate color in the chart (its border becomes decorated to be prominent) and then click somewhere in the region; it will then be colored in the chosen color.

In the lower-right triangle we find, say, the point (3,1/2). We evaluate ƒ(3,1/2) = |-2.5| = 2.5, the chart tells us that this means red. We click on the red in the chart and then on the triangle to color it.

In the upper-left triangle we may try the point (1/2,3). We find that ƒ(1/2,3) = |2.5| = 2.5, The chart shows red.

We get the picture

Trick: The red lies at the end of the color chart, so we can imagine that it extends further. This allows us to choose a point that lies on the outer border of the triangle (but not on the border between the triangle and the neighboring strip), for instance the point (4,0). The we evaluate ƒ(4,0) = 4, which is a bit more pleasant. ¨

The full version

The full version only shows the canvas, determining the borders between regions is up to us. The starting points is the borders between colors in the color chart. In our example we see just one border, namely between white and red, and it happens when the value of the function ƒ is equal to two. This sets up the equation of a curve ƒ = 2, that we will investigate closer.

| y - x | = 2
y - x = ±2
y = x + 2   or   y = x - 2

We obtained equations determining two straight lines, exactly the lines that we have seen marked in the basic version. If there are more colors in the chart, then we also have more borders and more curves to draw.

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