NCERT Book Class 8 Mathematics (Part II) Chapter 4 Exploring Some Geometric Themes

EXPLORING SOME
4 GEOMETRIC THEMES

In this chapter, we will explore two geometric themes. We will study
fractals which are self-similar shapes. They exhibit the same or similar
pattern over and over again — but at smaller and smaller scales. We will
then look at different ways of visualising solids.

4.1 Fractals
One of the most beautiful examples of a fractal that also
occurs in nature is the fern. The fern is seen to have smaller
copies of itself as its leaves, and these in turn have even
smaller copies of themselves in their sub-leaves, and so on!
Similar phenomena of self-similarity occur in trees
(where a trunk has limbs, and a limb has branches, and the
branches have branchlets, and so on), clouds, coastlines,
mountains, lightning, and many other objects in nature.
Other mathematical fractals can also be very beautiful.
We will explore some of them here.

Sierpinski Carpet
The Polish mathematician Sierpinski discovered a type of fractal known
as the Sierpinski Carpet. It is made by taking a square, breaking it into
9 smaller squares, and then removing the central square (see the figure
below); the same procedure is then repeated on the remaining 8 squares,
and so on. One then sees the same pattern at smaller and smaller scales.

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