Exploring Some Geometric Themes · Lesson 3 of 13
The Koch Snowflake and Fractals in Art
“Construct a growing boundary, calculate its perimeter, and recognise repeated shapes in art and architecture.”
Describe and draw the replacement rule for a Koch Snowflake. Calculate the number and length of boundary segments at a given step. Explain why the perimeter increases even though each segment becomes shorter. Identify nested motifs in architecture, textiles, and artwork.
In the carpet and gasket, repeated operations made holes inside a shape. Now imagine changing the boundary itself. Begin with a simple triangular outline and replace every straight part with a small triangular bump. Repeating the rule produces a boundary with more and more detail.
This construction is called the Koch Snowflake. It also prepares us to notice geometric repetition in human designs, where smaller towers, diamonds, or animal shapes can echo a larger arrangement.
Constructing the Koch Snowflake
Draw an equilateral triangle and call it step 0. Divide each side into three equal lengths. On the middle third, construct a smaller equilateral triangle pointing outwards. Remove the middle-third segment from the boundary, leaving the other two sides of the new small triangle in its place.
Follow one original side from end to end. You now travel along the first third, up one side of the bump, down the other side, and along the final third. Four equal segments replace the original single segment. Each new segment is one-third as long as the old one.
At the next step, apply this replacement to every segment of the new boundary, including the sloping sides of the bumps. “Outwards” always means away from the interior of the current shape. Keeping the old middle segment would leave an internal line in the drawing; it is no longer part of the perimeter.
Counting Boundary Segments
Step 0 has three segments because the starting outline is a triangle. Every segment produces four at the next step, so the counts are 3, 12, 48, 192, and 768. The number 4 multiplies the old count; it is not the number added to it.
Let Nₙ represent the number of boundary segments after n steps. Starting with three and multiplying by four n times gives Nₙ = 3 × 4ⁿ. If the original side length is a, the length of each small segment is a divided by 3ⁿ.
Problem
A snowflake starts from side 9 cm. Find the segment count and individual segment length after two steps.
- 1.Step 0 has 3 segments; step 1 has 12; step 2 has 48.
- 2.The individual lengths are 9 cm, then 3 cm, then 1 cm.
- 3.Thus step 2 consists of 48 boundary segments, each 1 cm long. The number 48 is a count, while 1 cm is a length.
Why the Perimeter Grows
Perimeter is the total length around the outside boundary. Replacing a segment of length a by four segments each of length a/3 gives a new length 4a/3. It is longer than a by a/3. The same change occurs on every segment, so the whole perimeter is multiplied by 4/3 each time.
The starting perimeter is 3a. After n steps it becomes 3a multiplied by 4/3 n times. When a = 1 unit, the starting perimeter is 3 units and the first few values are 3, 4, 16/3, and 64/9 units. Fractions keep these values exact.
| Step | Segments | Segment length when a = 9 cm | Perimeter |
|---|---|---|---|
| 0 | 3 | 9 cm | 27 cm |
| 1 | 12 | 3 cm | 36 cm |
| 2 | 48 | 1 cm | 48 cm |
| 3 | 192 | 1/3 cm | 64 cm |
Problem
A snowflake begins with an equilateral triangle of side 12 cm. Find the perimeter after two steps.
- 1.The initial perimeter is 3 × 12 = 36 cm.
- 2.After one step it is 36 × 4/3 = 48 cm.
- 3.After two steps it is 48 × 4/3 = 64 cm.
- 4.Check with pieces: 48 segments each of length 12/9 = 4/3 cm give 48 × 4/3 = 64 cm.
Although the segments become shorter, their number grows quickly enough to increase the total boundary length. There is no contradiction: a long journey can be split into many short steps. Under endless repetition, the ideal perimeter has no finite upper bound. A paper drawing, however, always stops after finitely many stages.
The old middle-third line is removed from the boundary. Counting it as well would incorrectly give five segments for each old one. Also, a larger perimeter does not by itself tell you how much area a shape encloses.
Problem
A student says the perimeter becomes one-third as large because every new segment is one-third as long. What is missing?
- 1.The statement tracks the length of one segment but ignores the segment count.
- 2.Every old segment creates four new ones, so total length changes by 4 × 1/3 = 4/3.
- 3.The perimeter grows by one-third of its previous value at each step; it does not shrink to one-third.
Fractals in Art
Geometry is also a way to describe a design. In the Kandariya Mahadev Temple, groups of smaller tower forms echo the shape of the larger tower. Their sizes differ, yet the repeated form makes the structure feel connected.
In the Fulani wedding-blanket example, diamond patterns contain smaller diamond patterns. The important observation is nesting: a motif reappears inside a larger version of itself. A row of equally sized diamonds shows repetition, but nesting across sizes more directly illustrates self-similarity.
M. C. Escher’s work “Smaller and Smaller” explores a repeated lizard motif at diminishing scales. Looking for a repeated unit, its scale changes, and the way copies fit together gives us a mathematical language for describing the artwork. Real art contains finitely many visible details; it can suggest the idea of continued repetition.
Design a Repeating Motif
Make a design whose geometric rule another student can follow. The rule matters more than elaborate decoration.
- Choose a triangle, square, or diamond as your starting motif.
- Describe where smaller copies will go and what fraction of the original side each copy will have.
- Draw two or three stages and record the number of new copies at each stage.
- Exchange only the written rule with a partner and compare the drawing they produce.
If both drawings have the same arrangement, your rule communicates the construction clearly. If the drawings differ, locate the missing instruction: size, position, orientation, or which parts should receive the next step.
Quiz
One segment is replaced by how many in the Koch rule?
How many segments occur at step 3, starting from a triangle at step 0?
What multiplies the perimeter at each step?
For original side 9 cm, what is the perimeter at step 2?
Which feature most directly illustrates self-similarity across scales?
Practice Problems
- For original side 1 unit, find the count, segment length, and perimeter at step 3.
- A snowflake has perimeter 48 cm at a certain stage. Find the next two perimeters.
- At what stage is the segment count 768?
- Explain the difference between one segment becoming shorter and the entire boundary becoming shorter.
- Describe a two-step nested-diamond design precisely enough for someone else to draw.
1. The count is 3 × 4³ = 192. 2. Each segment is 1/27 unit long. 3. The perimeter is 192/27 = 64/9 units.
Key Takeaways
Each Koch step replaces one segment by four segments one-third as long. The segment count is 3 × 4ⁿ when the starting boundary is a triangle. Perimeter grows by a factor of 4/3 per step. Track both number and size when measuring a repeated construction. Nested motifs in architecture, textiles, and art can illustrate self-similarity.