Big Maths Ideas · Beta

Fractal geometry

Build a Koch snowflake or Sierpiński triangle one stage at a time.

Move through finite construction stages and compare their area and boundary.

P_n=3\left(\frac43\right)^n

Each old segment becomes four segments, each one third as long.

Segments: 48
Perimeter (initial side = 1): 5.33333
Area / initial triangle area: 1.48148

The idea behind the experiment

Every stage shown is finite. In the limiting Koch snowflake, the boundary has infinite length while area approaches 8/5 of the initial triangle’s area. Its boundary repeats the same construction at smaller scales.

Harvey Mudd on the Koch snowflake

Predict, test and explain

Step 1 of 3

Predict

What happens to the Koch snowflake perimeter in one construction step?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

What happens to the Koch snowflake perimeter in one construction step?

$P_2=3\left(\frac43\right)^2$

Hint

Track both the number of segments and their length.

Reveal answer

Each segment is replaced by four segments, each one third as long.

$\frac{16}{3}$

Build an explanation

Hint 1

Track both the number of segments and their length.

Hint 2

A finite construction stage is not the limiting fractal.

Hint 3

Area and perimeter can behave differently.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the perimeter after two stages if the initial triangle has side 1.

The initial perimeter is 3.
Each step multiplies perimeter by 4/3.
Calculate 3 × (4/3)².

Answer: 16/3.

Connect this idea

Watch out: The segments shrink but their number grows fourfold.