A Shape Whose Boundary Grows Longer at Every Stage

Almost. In the Koch Snowflake the boundary is multiplied by $\frac{4}{3}$ at every stage, so it grows without limit, while the area it encloses stays bounded.

Shapes like this are called fractals, and they are built by repeating one rule for ever. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: self-similarity and drawing the next stage, counting pieces as powers, area and boundary length, and fractals in art and architecture.

A shape is self-similar when a small part of it, magnified, looks like the whole. Fractals are built by applying the same rule to every piece, again and again.

Sierpinski Carpet. Start with a square.

- Stage 1 — divide it into $9$ equal smaller squares and remove the middle one
- Stage 2 — do exactly the same to each of the $8$ squares that remain
- Stage 3 — repeat on each of those $64$ squares

Sierpinski Gasket (Triangle). Start with an equilateral triangle.

- Stage 1 — join the midpoints of the three sides, making $4$ smaller triangles, and remove the middle one
- Stage 2 — repeat on each of the $3$ triangles that remain

Koch Snowflake. Start with an equilateral triangle.

- Stage 1 — divide each side into $3$ equal parts and replace the middle part with two sides of an outward-pointing equilateral triangle, forming a spike
- Stage 2 — do the same to every one of the new, shorter segments

To draw the next stage, apply the rule to every piece produced by the last one — never just to one of them.

That repetition is what makes the shape self-similar. Zoom in on any remaining square of a Sierpinski Carpet and you find a smaller Sierpinski Carpet, identical in pattern to the whole — which is why the rule has to be applied everywhere at once rather than in one corner.

Each stage multiplies the count by the same factor, so the total at stage $n$ is that factor raised to the power $n$.

Sierpinski Carpet — each square becomes $8$ squares:

- Stage 1: $8^1=8$ squares remain
- Stage 2: $8^2=64$
- Stage 3: $8^3=512$
- Stage 4: $8^4=4096$
- **Stage $n$: $8^n$ squares remain

The number removed** at each stage is the number of squares that existed before it. So $1$ square is removed at stage 1, $8$ at stage 2, $64$ at stage 3 — and the removed counts are $8^{n-1}$.

Sierpinski Gasket — each triangle becomes $3$ triangles:

- Stage 1: $3^1=3$ triangles remain
- Stage 2: $3^2=9$
- Stage 3: $3^3=27$
- Stage 4: $3^4=81$
- **Stage $n$: $3^n$ triangles remain

Koch Snowflake** — each segment becomes $4$ segments, starting from $3$ sides:

- Stage 0: $3$ segments
- Stage 1: $3\times 4=12$
- Stage 2: $3\times 4^2=48$
- Stage 3: $3\times 4^3=192$
- **Stage $n$: $3\times 4^n$ segments

The growth is exponential**, which is why the numbers climb so fast — a Sierpinski Carpet at stage 5 already holds $8^5=32768$ squares.

Writing the count as a power rather than listing is the point of this section. Once you know the multiplying factor — 8, 3 or 4 — any stage can be answered immediately, without drawing anything.

Track the fraction kept at each stage, then raise it to the power of the stage number.

Sierpinski Carpet. Of $9$ squares, $8$ are kept, so each stage keeps $\frac{8}{9}$ of the area:

- Stage 1: $\frac{8}{9}$ of the original area remains
- Stage 2: ${\left(\frac{8}{9}\right)}^2=\frac{64}{81}$
- Stage 3: ${\left(\frac{8}{9}\right)}^3=\frac{512}{729}$

With a starting square of area $81$ cm², stage 2 leaves

$$\frac{64}{81}\times 81=64{\text{ cm}}^2$$

Sierpinski Gasket. Of $4$ triangles, $3$ are kept, so each stage keeps $\frac{3}{4}$:

- Stage 1: $\frac{3}{4}$
- Stage 2: $\frac{9}{16}$
- Stage 3: $\frac{27}{64}$

With a starting triangle of area $64$ cm², stage 3 leaves $27$ cm².

So the area shrinks towards zero as the stages continue.

Koch Snowflake boundary. Each segment of length $\ell$ becomes $4$ segments of length $\frac{\ell }{3}$, so the boundary is multiplied by $\frac{4}{3}$ each stage.

Starting with an equilateral triangle of side $9$ cm, the perimeter is $27$ cm:

- Stage 1: each side becomes $4$ segments of $3$ cm, so the perimeter is $3\times 4\times 3=36$ cm. And $27\times \frac{4}{3}=36$
- Stage 2: $36\times \frac{4}{3}=48$ cm
- Stage 3: $48\times \frac{4}{3}=64$ cm

The boundary grows without limit, since it is multiplied by a factor greater than 1 every time.

Those two directions together are the striking result. In the Sierpinski shapes the area falls towards zero; in the Koch Snowflake the boundary rises for ever while the area stays finite — which is exactly why fractals need their own way of measuring size.

In decoration and design that repeats a motif at several sizes at once.

Kandariya Mahadev Temple. Its towering shikhara is built from many smaller replicas of the tower's own shape, clustered around and up the main spire. Each smaller turret echoes the outline of the whole tower, and smaller ones echo those — the same self-similar idea as a Sierpinski construction, carried out in stone.

Fulani wedding blanket. Its woven bands carry a pattern that repeats at more than one scale, with the arrangement of small motifs inside a band mirroring the arrangement of the bands themselves.

Other examples worth recognising:

- Rangoli and kolam designs, where a motif is repeated around a centre and within itself
- Jaali screens and temple carvings, with patterns nested inside patterns
- Romanesco broccoli and cauliflower, where each floret resembles the whole head
- A fern leaf, whose leaflets repeat the shape of the full frond
- Branching in a tree, where each branch divides like the trunk did
- Coastlines and river networks, which look similarly ragged at every scale

A cauliflower broken apart in the kitchen is the easiest of these to check by hand — each piece looks like a miniature of what it came from.

The difference from a mathematical fractal is worth stating. Natural and artistic examples are self-similar over only a few scales, because a stone carving or a plant cannot keep subdividing for ever. A mathematical fractal repeats its rule without limit, which is what allows its boundary to grow beyond any length.