NCERT Class 8 Mathematics Ganita Prakash Part 2 Chapter 04 Exploring Some Geometric Themes PDF Download

Chapter 4: Exploring Some 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.

[Figure: Fern, See in your textbook]

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.

[Figure: Sierpinski Carpet construction showing Step 0, Step 1, and Step 2, See in your textbook]

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Carpet.

By its construction, each step in the sequence has

• (i) squares of the same size that remain in the figure, and the size of these squares becomes smaller and smaller as the step number increases, and
• (ii) square holes that are formed by removing square pieces.

Do you see any pattern in the number of holes and squares that remain at each step?

Let \( R_n \) represent the number of remaining squares at the \( n \)th step, and \( H_n \) represent the number of holes at the \( n \)th step.

Let us understand how these numbers grow by analysing how the holes and squares that remain are generated from the previous step. Every square that remains at a given step, say Step \( n \), gives rise to 8 squares that remain at the \( (n + 1) \)th step. Thus, we have

\[ R_{n+1} = 8 R_n \]

Can this be used to get a formula for \( R_n \)?

We have

\[ R_0 = 1 \] \[ R_1 = 8 \times 1 = 8 \] \[ R_2 = 8 \times 8 = 8^2 \]

In general, \( R_n = 8^n \).

Teacher's Note

Notice how the recurrence relation \( R_{n+1} = 8 R_n \) tells you the rule for moving from one step to the next. To find a direct formula, trace back: \( R_n = 8 \times R_{n-1} = 8 \times 8 \times R_{n-2} = \ldots \) until you reach \( R_0 = 1 \), giving \( R_n = 8^n \). This pattern - starting with a recurrence, then finding the explicit formula - appears often in fractal problems.

Similarly, how do we find the number of holes at a given step?

Every square that remains at the \( n \)th step gives rise to a hole in the \( (n + 1) \)th step. All the holes present at the \( n \)th step remain in the \( (n + 1) \)th step as well. Thus,

\[ H_{n+1} = H_n + R_n \]

So we have

\[ R_0 = 1 \quad H_0 = 0 \] \[ R_1 = 8 \quad H_1 = 1 \] \[ R_2 = 8^2 \quad H_2 = 1 + 8 \] \[ R_3 = 8^3 \quad H_3 = 1 + 8 + 8^2 \]

Sierpinski Gasket

Sierpinski came up with another fractal made in a similar way. An equilateral triangle is broken up into 4 identical equilateral triangles by joining the midpoints of the bigger triangle, and then the central triangle is removed. This procedure is repeated on the 3 remaining triangles, and so on.

[Figure: Sierpinski Gasket construction showing Step 0, Step 1, and Step 2, See in your textbook]

Show that by joining the midpoints of an equilateral triangle, we divide it into 4 identical equilateral triangles.

[Hint: Note that the corner triangles are isosceles.]

This fractal is called the Sierpinski Triangle/Gasket.

[Figure: Sierpinski Triangle showing intricate pattern of triangular holes, See in your textbook]

Figure it Out

1. Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle.
2. Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.
3. Find the area of the region remaining at the \( n \)th step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit.

Teacher's Note

When finding the area remaining at step \( n \), remember that each time you remove holes or a central piece, you are subtracting area. For the Sierpinski Triangle, at each step you remove one-quarter of the area. Track the area left after each removal: at step 1 you have \( \frac{3}{4} \), at step 2 you have \( \left(\frac{3}{4}\right)^2 \), and so on. Always work with fractions of the starting area.

Key Points

• A fractal is a shape that shows self-similarity: the same pattern repeats at smaller and smaller scales, such as in ferns, trees, and coastlines.
• The Sierpinski Carpet is made by dividing a square into 9 equal squares and removing the centre, then repeating this process on the remaining 8 squares; the number of remaining squares at step \( n \) follows the formula \( R_n = 8^n \).
• The Sierpinski Triangle is made by joining the midpoints of an equilateral triangle to create 4 smaller identical triangles and removing the centre one, then repeating on the 3 remaining triangles.
• Both Sierpinski fractals can be studied by tracking two quantities at each step: the number of shapes that remain and the number of holes that are created.

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