The Koch Snowflake
Raise a bump on every side, again and again.
A bump on each side
The Koch Snowflake starts with an equilateral triangle. Cut each side into 3 equal parts. Raise a small equilateral triangle over the middle part, and then take that middle part away.
The triangle has 3 sides. After this is done on every side, how many sides will the new shape have?
What this lesson covers
The idea
The Koch Snowflake starts from an equilateral triangle: each side is divided into 3 equal parts, an equilateral triangle is raised over the middle part and that part removed, and this is repeated on every side of the new shape.
A bump on each side
The Koch Snowflake starts with an equilateral triangle. Cut each side into 3 equal parts. Raise a small equilateral triangle over the middle part, and then take that middle part away.
The triangle has 3 sides. After this is done on every side, how many sides will the new shape have?
- 6
- 9
- 12
Raise the bumps
Tap a side of the triangle. Do it to all 3 sides. Then use the button to do it on all the new sides, again and again.
A bump replaces every side
The Koch Snowflake starts from an equilateral triangle: each side is divided into 3 equal parts, an equilateral triangle is raised over the middle part and that part removed, and this is repeated on every side of the new shape.
Each side is replaced by a bump made of 4 shorter sides, each one third as long. So the number of sides goes 3 → 12 → 48 → 192. The Swedish mathematician Von Koch first described it in 1904.
Notes
Koch Snowflake: start from an equilateral triangle. Divide each side into 3 equal parts, raise an equilateral triangle over the middle part and remove that part. Repeat on every side of the new shape.
Check yourself
The triangle has sides of 9 cm. After step 1, how long is each side of the snowflake shape, in cm?
Answer: 3 cm
9 cm ÷ 3 = 3 cm. At step 2 the sides are 1 cm.
Step 1 has 12 sides. Each side gets a bump of 4 shorter sides. How many sides does step 2 have?
Answer: 48 sides
12 × 4 = 48 sides.
This is step 1. What do we do to get step 2?
What happens to the middle part of a side?
- Raise a triangle over each corner. The bumps are raised over the middle part of every side, not over the corners.
- Do the same on all 12 sides of this shape — correct. Yes! The same procedure is repeated on every side of the new shape.
- Do it only on the 3 old sides. The new shape has 12 sides, and every one of them gets a bump.
- A triangle is raised over it, and then it is removed — correct. Yes! The raised triangle takes its place, so the side becomes a bump.
- It stays, and the triangle is raised over the whole side. The triangle is raised over the middle third only, and the middle part is removed.
- It is cut in half and both halves stay. The middle part is removed, not cut in half.