Concepts: Exploring Some Geometric Themes (part 1)

1.1 Self-similar shapes

A fractal is a shape that repeats its own pattern at smaller and smaller scales. A fern leaf carries small copies of the whole frond, each of those carries smaller copies, and the process continues. Trees, coastlines, clouds, mountains and lightning show the same habit.

The mathematical version applies a rule to a shape and then applies the same rule again to every new piece. Step 0 is the starting shape and each later step is one application of the rule. The sequence approaches a limit shape, the fractal. Because the rule never stops, the limit is drawn by convention with a few steps, not by listing every piece.

Fractals also appear in art. The Kandariya Mahadev temple at Khajuraho, completed around 1025 CE, repeats its tower form in smaller copies. Patterns on Nigerian Fulani wedding blankets repeat diamonds inside diamonds, and M.C. Escher's print 'Smaller and Smaller' repeats lizards at shrinking scales.

1.2 Sierpinski carpet

Start with a square of area 1. Divide it into a 3 by 3 grid of nine smaller squares and remove the central one. Eight squares remain. Repeat the same division and removal on each remaining square, and keep repeating.

Count the remaining squares. Each square at one step gives 8 squares at the next step, so if Rₙ is the number of remaining squares at step n, then Rₙ₊₁ = 8Rₙ with R₀ = 1. Therefore Rₙ = 8ⁿ. At step 1 there are 8 squares, at step 2 there are 64, and at step 3 there are 512.

Count the holes. Every remaining square produces one new hole in the next step, and all older holes stay. Thus Hₙ₊₁ = Hₙ + Rₙ with H₀ = 0. So Hₙ = 1 + 8 + 8² + ... + 8ⁿ⁻¹ = (8ⁿ - 1)/7. At step 1 there is 1 hole, at step 2 there are 9, and at step 3 there are 73.

The area also shrinks by a fixed factor. Each remaining square keeps 8 of its 9 equal parts, so its area is multiplied by 8/9 at every step. Starting from area 1, the area after n steps is (8/9)ⁿ. The area becomes very small but never reaches zero for any finite step, which is one of the strange features of a fractal.

1.3 Sierpinski gasket

Start with an equilateral triangle of area 1. Join the midpoints of its three sides to make four identical equilateral triangles, then remove the central one. Three triangles remain. Repeat on each remaining triangle.

The counting is similar. Each remaining triangle gives 3 triangles at the next step, so Rₙ = 3ⁿ. Each remaining triangle also produces one new hole and all old holes stay, so Hₙ = 1 + 3 + 3² + ... + 3ⁿ⁻¹ = (3ⁿ - 1)/2. The area keeps 3 of every 4 equal parts, so Aₙ = (3/4)ⁿ. Step 1 has area 3/4, step 2 has area 9/16, and the limit area is 0.

Why are the four small triangles identical? The corner triangles have one side half of the original side and the two adjacent angles 60°, so they are equilateral. Each small triangle has side half of the original and area one quarter. The central triangle is also equilateral and congruent to the other three.

Worked example 1. For the gasket, find the number of remaining triangles and holes at step 4. R_4 = 3⁴ = 81. H_4 = (81 - 1)/2 = 40. The area is (3/4)⁴ = 81/256 square units. These three numbers all follow from the same step rule, so a counting question can be answered without drawing every step.

1.4 Koch snowflake

Start with an equilateral triangle of side 1. On each side, divide the side into three equal parts, build an equilateral triangle outward on the middle third, and remove the middle third. The straight side is replaced by a bump made of four segments, each one third of the old side.

Every side becomes four sides, so the number of sides after n steps is Sₙ = 3 × 4ⁿ. At step 0 there are 3 sides, at step 1 there are 12, and at step 2 there are 48. Each new segment has one third of the length of the segment it replaced, so the side length at step n is (1/3)ⁿ. The perimeter is therefore Pₙ = Sₙ × (1/3)ⁿ = 3 × (4/3)ⁿ. The perimeter grows without bound even though the shape stays inside a bounded region, another strange fractal property.

1.5 Why the limit is a fractal

At every finite step the shape is an ordinary polygon, but the rule keeps producing smaller copies of the same pattern. The limit is not drawn by any single step; it is the shape that the sequence approaches. That is why a fractal is described by its rule and its step number rather than by a finished drawing, and why questions about step n can be answered with formulas instead of pictures.