On this web page you can investigate the Sierpinski triangle and its properties.
Sketch
The buttons in the sketch here show the first few stages in the construction of the Sierpinski triangle. Start with a colored triangle, a Stage 0 Sierpinski triangle. To create a Stage 1 triangle, connect the midpoints of the sides to form four smaller triangles; color the three outer triangles and make the inner one white. Do this again to each of the shaded triangles in the Stage 1 triangle to get the Stage 2 triangle. Continue in this way. You can drag any of the vertices of the original triangle to change its shape. Press Start Over to return to the equilateral Stage 0 triangle.
Investigate
- Does the Sierpinski triangle have the property of self-similarity? Explain.
- How many reduced copies of a Stage 0 triangle would you need to make a Stage 1 triangle? How many reduced copies of Stage 1 are needed to make Stage 2?
- Think about what happens to the number of shaded triangles in successive stages of the Sierpinski triangle. How many copies of a Stage n triangle make up a Stage n + 1 triangle?
- Copy and complete this table of the number of triangles in each successive stage of the Sierpinski triangle. Can you find a pattern that helps you to fill in the table?
| Stage number |
0 |
1 |
2 |
3 |
... |
n |
... |
50 |
| Number of shaded triangles |
1 |
3 |
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- What would happen to the number of triangles if you could infinitely increase the stage number?
- Think about what happens to the shaded area in each successive stage of the Sierpinski triangle. Suppose the Stage 0 triangle has area 1. Find the shaded area in Stage 1. Copy and complete this table by looking at the areas of Stages 2 and 3, and finding a pattern.
| Stage number |
0 |
1 |
2 |
3 |
... |
n |
... |
50 |
| Shaded area |
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- What would happen to the shaded area if you could infinitely increase the stage number?
- Think about what happens to the perimeter in each successive stage of the Sierpinski triangle. Suppose the Stage 0 triangle has perimeter 6. Find the perimeter of the shaded area in Stage 1. Copy and complete this table by looking at the perimeters for the next two stages and finding a pattern.
| Stage number |
0 |
1 |
2 |
3 |
... |
n |
... |
50 |
| Perimeter |
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- What would happen to the perimeter if you could infinitely increase the stage number?