Fractals and self-similarity

Zoom into a fern leaf and each small frond looks like a miniature copy of the whole leaf. Zoom into a coastline map and the jagged inlets at one scale echo the jagged inlets at a much smaller scale. Nature is full of shapes that repeat themselves inside themselves, and mathematicians call this property self-similarity. A fractal is a pattern built by applying one simple geometric rule again and again, at smaller and smaller scale, so that a magnified piece of the final pattern looks like the whole pattern. This article builds three classic fractals by hand — the Sierpiński triangle, the Sierpiński carpet, and the Koch snowflake — tracks how the number of pieces grows stage by stage, and shows how to count kept and removed pieces after several rounds of the same repeating rule.

The core insight

A fractal is made by repeating one fixed rule at every scale forever, so a magnified piece of the pattern looks like the whole, and the number of surviving pieces multiplies by a fixed factor at every stage.

The core idea

A shape is self-similar when a small part of it, suitably magnified, looks like the entire shape. Fractals are built by taking a starting shape (Stage 0) and applying one fixed transformation rule to it; applying that same rule again to the result gives Stage 2, and so on. Each single application of the rule is called an iteration, and the process can in principle be repeated without end, revealing finer detail at every scale.

Sierpiński triangle. Start with a solid equilateral triangle. Join the midpoints of its three sides, splitting it into 4 smaller triangles, and remove the upside-down middle one, keeping the 3 corner triangles. Repeat this same "split into 4, remove the middle" rule on every triangle that remains.

Sierpiński carpet. Start with a solid square. Cut it into a 3×33\times3 grid of 9 equal squares and remove the central one, keeping 8. Repeat the same rule on each of those 8 squares.

Koch snowflake. Start with an equilateral triangle. On every edge, remove the middle third of the segment and replace it with two outward-pointing segments of the same length, turning each edge into 4 shorter segments. Repeat the rule on every new segment. In every case the rule never changes and is applied to every surviving piece, at every scale — this is what makes the pattern self-similar rather than merely decorative. Because each stage multiplies the piece-count by a fixed number, the growth in pieces follows a clean pattern:

Nn=r nN_n = r^{\,n}

where: NnN_n is the number of surviving (or newly created) pieces after nn iterations of the rule, rr is the fixed multiplying factor of that particular rule (kept pieces per old piece), and nn is the stage number, starting from Stage 0. For the Sierpiński triangle r=3r = 3 (3 corner triangles survive from each old triangle); for the Sierpiński carpet r=8r = 8 (8 squares survive from each old square); for the Koch snowflake's segment count r=4r = 4 (1 segment becomes 4 segments).

Figure
Three stages of building the Sierpinski triangle by repeatedly removing the middle upside down triangle, showing Stage 0 a solid triangle, Stage 1 with the centre triangle removed, and Stage 2 with the centre removed again from each of the three remaining triangles
Sierpinski triangle: one rule, applied again and again Stage 0 Stage 1 — 3 kept Stage 2 — 9 kept Kept triangles multiply by 3 every stage: 1, 3, 9, 27, ... — the rule at every scale is identical
Derivation ACounting pieces after repeated iterations of a fractal rule

The goal here is to prove why the number of surviving (or newly created) pieces in a fractal built by repeating one fixed rule follows the simple power formula Nn=r nN_n = r^{\,n}, and then use that formula to predict the piece count after any number of stages without having to draw every step by hand.

Step 1. Look at what one application of the rule does to a single piece. Whatever the shape, one iteration always turns 1 old piece into a fixed number rr of new pieces of the same type — for the Sierpiński triangle, 1 triangle becomes 3 surviving triangles; for the carpet, 1 square becomes 8 surviving squares; for the Koch snowflake, 1 segment becomes 4 segments.

Step 2. Track Stage 1. Starting from Stage 0 with exactly 1 piece, applying the rule once gives N1=rN_1 = r pieces, by the definition in Step 1.

Step 3. Track Stage 2 by applying the rule to every piece from Stage 1. Each of the rr pieces from Stage 1 independently turns into rr new pieces, so the total becomes N2=r×r=r2N_2 = r \times r = r^2. The key point is that the rule acts identically and independently on every surviving piece — this is exactly what "self-similar at every scale" means.

Step 4. Generalise by repeating Step 3's reasoning. Each further iteration multiplies the current count by rr again, so after nn iterations Nn=r×r×⋯×rN_n = r \times r \times \cdots \times r (nn factors), giving Nn=r nN_n = r^{\,n}. This matches the Sierpiński triangle's known counts of 31=33^1 = 3, 32=93^2 = 9, 33=273^3 = 27 triangles kept at Stages 1, 2 and 3.

Worked example

Question. A Sierpiński triangle starts (Stage 0) as one solid equilateral triangle, and a Sierpiński carpet starts (Stage 0) as one solid square. (a) How many small triangles are kept at Stage 4 of the Sierpiński triangle? (b) How many small squares are kept at Stage 2 of the Sierpiński carpet? (c) For the triangle, what fraction of the finest-level pieces at Stage 4 have been removed in total?

Step 1. Apply Nn=rnN_n = r^n to the Sierpiński triangle with r=3r = 3 and n=4n = 4: N4=34=81N_4 = 3^4 = 81 triangles kept at Stage 4.

Step 2. Apply Nn=rnN_n = r^n to the Sierpiński carpet with r=8r = 8 and n=2n = 2: N2=82=64N_2 = 8^2 = 64 squares kept at Stage 2.

Step 3. For the removed fraction, note that at Stage 4 every original piece has been split into 44=2564^4 = 256 smallest triangles in total (since each stage quarters every triangle into 4, keeping 3), of which 8181 survive (from Step 1). So the number removed is 256−81=175256 - 81 = 175, out of 256256 smallest triangles.

Answer: 81 triangles are kept at Stage 4 of the Sierpiński triangle, 64 squares are kept at Stage 2 of the Sierpiński carpet, and 175 out of 256 smallest triangles (a little over two-thirds) have been removed by Stage 4.

Competitive stretch

For the Koch snowflake, one edge that starts as a single segment becomes 4n4^n segments after nn iterations, by the same formula with r=4r = 4. Starting from a 9 cm edge (as in the Pass-1 article), after 2 steps there are 42=164^2 = 16 segments of length 1 cm each — exactly matching the earlier count, now obtained directly from the formula instead of by tracking each step separately.

Common traps

  • Thinking a fractal pattern is random — it follows one exact rule applied identically to every piece, at every scale, with no exceptions.
  • Forgetting that a removed piece never returns at a later stage; the rule is applied only to the pieces still present.
  • Mixing up the multiplying factor rr: it is 3 (not 4) for the Sierpiński triangle since the middle triangle is discarded, 8 for the Sierpiński carpet, but 4 for the Koch snowflake since the middle segment is replaced rather than simply discarded.
  • Confusing self-similarity with ordinary symmetry — a figure can have line or rotational symmetry without a smaller part resembling the whole; self-similarity specifically requires that a magnified piece looks like the entire pattern.

Ask the tutor: Want to try predicting the piece count for a fractal rule of your own design, or see how Nn=rnN_n = r^n changes if two different pieces are removed instead of one?

Why does the Sierpiński carpet use 8 while the Sierpiński triangle uses 3?

The multiplying factor rr in Nn=rnN_n = r^n simply counts how many pieces survive from each old piece after one application of that shape's own rule. The triangle rule splits a triangle into 4 and removes 1 middle one, so 3 survive. The carpet rule splits a square into a 3×33\times3 grid of 9 and removes only the 1 central square, so 8 survive. Different starting shapes and different removal rules naturally give different values of rr; the formula Nn=rnN_n = r^n itself works the same way for any of them once rr is known.

Does a real fractal actually go on forever, or do we stop after a few stages?

Mathematically, a true fractal is defined by imagining the rule repeated infinitely many times, producing detail at every possible scale, however small. In practice, at school level and even in computer drawings, we can only carry out a finite number of iterations because each stage has more, smaller pieces than the last and eventually they become too small to draw or see. The examples in this article stop at a few stages precisely because that is enough to see the pattern and predict later stages using Nn=rnN_n = r^n, without needing to draw infinitely many triangles.

Are the ferns and cauliflower florets people talk about really fractals like these?

Not in the strict mathematical sense of following one perfectly repeating rule forever — real ferns and cauliflowers are only approximately self-similar, and the repetition eventually stops because the plant runs out of scale to keep branching. But they are excellent everyday examples of the same underlying idea: a small frond or floret looks roughly like a smaller copy of the whole. Mathematical fractals like the Sierpiński triangle are the idealised, perfectly repeating version of this same natural pattern, which is why the two are always mentioned together.

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