The Sierpinski Triangle and Fractal Area

A pattern can contain more and more pieces while covering less and less of the page. This may sound contradictory until we ask how large the pieces are. The Sierpinski Triangle gives a clear way to see the difference between counting pieces and measuring their area.

You already know the carpet rule: keep eight of nine equal parts. The triangular rule will keep three of four equal parts. We will build both an arithmetic record and a shaded picture, using each to check the other.

The Sierpinski Gasket

Start with an equilateral triangle: all three sides have the same length and all three angles are 60°. Locate the midpoint of each side. Join the three midpoints to form a smaller upside-down triangle in the centre. Together with the three corner triangles, it divides the original into four parts.

Each joining segment is parallel to one original side and half as long. Consequently all four small triangles are equilateral with half the original side length. They are congruent: one can be moved or turned to fit exactly over another. The central triangle being upside down does not change its size or area.

Here is a direct reason for the midpoint result. At an original corner, the two half-sides are equal, so the small corner triangle is isosceles. Its corner angle is 60°, leaving 120° for its two equal base angles: each is 60°. It is therefore equilateral. The same argument works at all three corners. Their inner sides form the central triangle, so that triangle also has three equal half-length sides. This explains the four equal pieces without simply relying on how the picture looks.

Remove the central triangle to make step 1. Three corner triangles remain. At step 2, repeat the midpoint construction and removal inside all three. Nine smallest retained triangles result. At every stage work only inside triangles that still contain material.

Counting Triangles and Holes

One retained triangle becomes three retained triangles at the next stage. Begin at step 0 with one triangle and multiply by 3 for each completed removal step. If Tₙ is the count at step n, this gives Tₙ = 3ⁿ.

Hole counts follow a different rule. Step 1 adds one hole, step 2 adds three, and step 3 adds nine. Older holes remain, so the cumulative totals are 0, 1, 4, 13, and 40. Adding the new holes, rather than multiplying the old hole count, explains these totals.

The Area That Remains

Area measures how much surface is covered. In the first triangular step, removing one of four equal parts leaves three-quarters of the starting area. In the next step we retain three-quarters of that remaining amount, because the same fraction is kept in every surviving triangle.

Suppose the original area is A₀. The area after one step is A₀ × 3/4. After two steps it is A₀ × 3/4 × 3/4. If Aₙ denotes the area after n steps, repeated multiplication gives the rule below. The units remain square units throughout.

We can check step 2 another way. There are nine small triangles, and each has one-sixteenth of the original area because its side is one-quarter as long. Nine pieces each of area A₀/16 give 9A₀/16, agreeing with A₀ × 3/4 × 3/4. Counting pieces and measuring one piece lead to the same total.

Comparing the Carpet and Triangle

For the carpet, each operation keeps eight of nine equal square parts. Its remaining area is therefore multiplied by 8/9 at every step. The triangle keeps 3/4. To compare fairly, give both starting shapes the same area and perform the same number of steps.

Since 3/4 is less than 8/9, the triangle keeps a smaller fraction at each step. Its remaining fraction falls faster. This comparison concerns fractions of equal starting areas; it does not say that every physical triangular drawing must be smaller than every carpet drawing.

After any finite number of steps, a positive amount of material remains. Continuing the ideal rule makes the retained fraction smaller and smaller. We can describe this trend without treating a practical paper model as if it had infinitely many cuts. The construction is an idea that drawings can approach.

Compare Two Equal Areas

Use fractions to compare patterns even though the original shapes look different. A table helps keep counts and areas separate.

1. Take a triangle and a square, each imagined to have area 1296 square units.
2. For the triangle, calculate the areas after steps 1 and 2 by multiplying by 3/4 twice.
3. For the carpet, calculate the same stages by multiplying by 8/9 twice.
4. Find the total removed area in each case and explain the difference.

The triangle retains 972 then 729 square units; the carpet retains 1152 then 1024. After two steps, 567 square units have been removed from the triangle and 272 from the carpet. Equal starting area makes this comparison meaningful.