A Shape Made from Smaller Versions of Itself, math that surprises you lesson

Zoom in on a circle and its edge soon looks like a straight line. Now picture a triangle with a triangular hole in its middle, in which each of the three corner triangles that remain has a hole of its own, and each of their corners has one too, without end. Zooming in on this shape never makes it simpler, because every piece holds the same arrangement again. This page explains what such a shape is, how one rule repeated over and over builds it, and why no drawing, however detailed, shows the finished shape.

Key terms

Fractal

Definition The word fractal was introduced by Benoit Mandelbrot in 1975, and it has no single precise definition that all mathematicians use. It names sets of points, on a line, in the plane or in space, that have detail at arbitrarily small scales. Magnifying the region around a point of such a set keeps showing new structure, while a smooth curve or surface looks flatter and flatter under magnification. Sets called fractals usually have some further features as well. Most are too irregular to be described by the familiar shapes of geometry, most have a fractal dimension, made precise on a later page, that is larger than their ordinary dimension, and many are produced by a short rule repeated without end. Many contain exact smaller copies of themselves, as in the example below, but that is a feature of the first examples people meet and not part of the meaning of the word. A fractal can keep showing structure at every magnification because it is the limit of an endless construction, and no finished drawing, which contains only finitely many pieces, can do the same.

In plain words A fractal is a shape that does not get simpler when you zoom in. A circle seen up close looks like a straight line, but a fractal keeps showing holes, bumps or branches at every level of magnification.

Example Start with a filled equilateral triangle whose sides have length $1$ . Remove the inside of the middle triangle, the one whose corners are the midpoints of the three sides, and keep its edges. This leaves three filled corner triangles with sides of length $\tfrac12$ that touch at their corners. Do the same to each of those three triangles, then to each of the nine triangles this leaves, and so on without end. After $n$ steps there are $3^n$ filled triangles with sides of length $2^{-n}$ . The shape that remains in the limit is called the Sierpinski triangle, and it has its own page in Sierpinski’s Triangle, Holes Inside Holes.

Compare a circle of radius $1$ seen through a small square window of height $2h$ centered at the point $(1, 0)$ . The arc inside the window, between the heights $-h$ and $h$ , strays from the vertical tangent line $x = 1$ by at most

1 - \sqrt{1 - h^2}.

For $h = 0.1$ this is about $0.005$ , roughly $2.5$ percent of the window’s height $0.2$ , and for $h = 0.01$ it is about $0.00005$ , roughly $0.25$ percent of the height $0.02$ . Each tenfold magnification makes the arc look about ten times straighter. A close-up of the triangle shape at any of its points keeps showing holes instead, because every small triangle that remains gets a hole cut out of its middle at the next step.

Self-similarity

Definition A set is exactly self-similar when it is made of at least two pieces, each of which is a copy of the whole set scaled down by a factor strictly between $0$ and $1$ , possibly also rotated, reflected or moved. More precisely, a nonempty bounded set $K$ is exactly self-similar if

\begin{aligned} K &= S_1(K) \cup S_2(K) \\[0.4em] &\quad \cup \cdots \cup S_m(K) \end{aligned}

for some whole number $m \ge 2$ and some maps $S_1, \dots, S_m$ called similarities, where each $S_i$ multiplies every distance by the same factor $r_i$ with $0 < r_i < 1$ . The pieces may touch or overlap. Self-similarity alone does not make a set a fractal, since a line segment is the union of its two halves and a filled square is the union of its four quarters. Many fractals are self-similar only in a looser sense. In a self-affine set the smaller copies are stretched by different factors in different directions, and in a statistically self-similar set, such as a model of a rough coastline, the parts resemble the whole only in their average roughness over a range of scales.

In plain words Pick out one of the right pieces of a self-similar shape, enlarge it by the right factor and move it into place, and you get the whole shape back exactly. The same arrangement appears again at a smaller scale, and again inside that.

Example Put the corners of the starting triangle at $v_1 = (0, 0)$ , $v_2 = (1, 0)$ and $v_3 = \bigl(\tfrac12, \tfrac{\sqrt{3}}{2}\bigr)$ . Let $S_i$ send each point $p$ to the midpoint of $p$ and $v_i$ ,

S_i(p) = \tfrac12 \left(p + v_i\right).

Each of these maps halves every distance and pulls the plane toward one corner. The limit shape $K$ satisfies $K = S_1(K) \cup S_2(K) \cup S_3(K)$ , so its upper corner piece, enlarged by a factor of $2$ about the top corner, is exactly the whole. Choosing the upper piece three times in a row, the upper piece of the upper piece of the upper piece, selects a region $\tfrac18$ as wide as the original, and enlarging it $8$ times gives the whole again. The exact match holds only for the limit shape. After $4$ steps the upper corner piece contains $27$ small triangles, and doubling it gives the picture after $3$ steps, not after $4$ . This is also why a magnified view needs a scale bar. A view can fill the screen and look just like the whole while showing only a tiny part of it.

Iteration

Definition Iteration means applying the same rule again and again, each time to the result of the previous application. Given a starting object $S_0$ and a rule $F$ that turns an object of some kind into another object of the same kind, iteration produces the sequence $S_0, S_1, S_2, \dots$ defined by

S_{n+1} = F(S_n)

for every $n \ge 0$ , so $S_n$ is the result of applying $F$ exactly $n$ times to $S_0$ . The rule does not change from step to step. The objects can be shapes, as on this page, or numbers, as in later pages where one equation is applied to its own output.

In plain words One instruction is used over and over, each time on what the last round produced. The instruction is short, but the pattern it builds after many rounds can be far more intricate than the instruction itself.

Example For the triangle shape, the rule $F$ says to remove the inside of the middle triangle from every filled triangle. In terms of the maps above it reads $F(S) = S_1(S) \cup S_2(S) \cup S_3(S)$ , which gives exactly the same stages. Each step replaces every filled triangle by three triangles with half its side length. A triangle with half the side length has one quarter of the area, so every step multiplies the number of triangles by $3$ , the side length by $\tfrac12$ and the filled area by $\tfrac34$ .

Step $n$	Triangles	Side	Share of the area still filled
$0$	$1$	$1$	$1$
$1$	$3$	$\tfrac12$	$\tfrac34$
$2$	$9$	$\tfrac14$	$\tfrac{9}{16}$
$3$	$27$	$\tfrac18$	$\tfrac{27}{64}$
$4$	$81$	$\tfrac{1}{16}$	$\tfrac{81}{256}$

Finite approximation

Definition A finite approximation of a fractal built by iteration is one of its stages $S_n$ , the result of finitely many steps. When every step only removes material, as here, each stage lies inside the one before it, and the limit is the set of points that belong to every stage,

K = S_0 \cap S_1 \cap S_2 \cap \cdots.

A stage approximates the limit closely. For the triangle shape, $K$ lies inside $S_n$ , and every point of $S_n$ is within distance $2^{-n}/\sqrt{3}$ of a point of $K$ , since every point of a triangle with side $2^{-n}$ is that close to one of its corners and the corners are never removed. Still, a stage is not the limit. A stage is made of finitely many filled triangles, all of one smallest size, with a positive total area. The limit contains no filled triangle at all, has holes of every size, and has area zero, as the example below shows. A drawing is a further approximation, because a screen has finitely many pixels and cannot show detail smaller than one of them. Statements about the limit, such as exact self-similarity, are proved by reasoning about every stage at once and are never read off a picture.

In plain words Keep three things apart. The rule says what to do next, a stage is what has actually been built after some number of steps, and the limit is what the endless repetition defines. Zooming into a stage eventually shows plain filled triangles with no holes, and only carrying out more steps adds more detail.

Example Stage $4$ of the triangle shape has $81$ filled triangles with sides $\tfrac{1}{16}$ , which can be counted and stored as a list of corner coordinates. Magnifying one of them $16$ times shows a plain filled triangle with no hole, because the hole inside it is only made at step $5$ . The limit $K$ keeps every corner of every stage, so it contains infinitely many points. Yet it lies inside every stage, and the filled area of stage $n$ is $(3/4)^n$ times the original area, which approaches $0$ , so $K$ has area zero. Every stage has a positive area, so having zero area is a property of the limit that no stage shares.

Check yourself

How many filled triangles are there after $6$ steps, how long are their sides, and what share of the original area is still filled?

Show the answer

Each step multiplies the number of triangles by $3$ and halves the side, so after $6$ steps there are $3^6 = 729$ triangles with sides of length

2^{-6} = \frac{1}{64}.

Each of them has $\bigl(\tfrac{1}{64}\bigr)^2 = \tfrac{1}{4096}$ of the original area, because area scales with the square of the side. Together they fill $\tfrac{729}{4096}$ of the original area, about $0.178$ , which agrees with $(3/4)^6$ .

Change the rule so that each filled triangle is cut into its four half-size triangles and all four are kept, the middle one included. Is the limit self-similar? Is it a fractal?

Show the answer

After $n$ steps there are $4^n$ triangles with side $2^{-n}$ , each with $\tfrac{1}{4^n}$ of the original area, so together they always fill the whole triangle. Every stage, and therefore the limit, is just the filled triangle. It is exactly self-similar, since it is the union of four copies of itself scaled by $\tfrac12$ , three of them moved toward the corners and the middle one turned half a turn as well.

It is not a fractal. Magnifying any part of it shows a flat filled region with no new structure, and its area is the full area of the triangle. The original rule makes a fractal because it cuts a hole out of every piece at every step, not merely because the pieces are copies of the whole.

A screen draws the starting triangle with sides $1000$ pixels long. After how many steps are the smallest triangles less than one pixel across, and how many triangles are there then? What does this say about drawing the limit?

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After $n$ steps the sides are $1000/2^n$ pixels long. After $9$ steps this is $1000/512 \approx 1.95$ pixels, and after $10$ steps it is

\frac{1000}{1024} \approx 0.98

pixels, so $10$ steps are needed. At that stage there are $3^{10} = 59049$ triangles. Every later step only cuts holes smaller than a pixel, which the screen cannot show as separate shapes.

To show the holes made at step $20$ even one pixel wide, one would have to magnify a small part of the picture about $2^{10} = 1024$ times and draw that part again with more steps. The limit has holes at every scale and no smallest triangle, so no screen of any resolution shows all of it. A picture always shows a stage.