Sierpinski triangle

Definition

The Sierpinski triangle is a fractal formed by recursively removing equilateral triangles from a larger equilateral triangle. It exhibits self-similarity and has a dimension that is not an integer.

5 Must Know Facts For Your Next Test

1. The Sierpinski triangle is created through an iterative process of removing triangular sections, starting with a single equilateral triangle.
2. It is an example of a fractal, meaning it shows self-similarity at different scales.
3. The area of the Sierpinski triangle approaches zero as the number of iterations approaches infinity.
4. Its Hausdorff dimension is given by $\frac{\log 3}{\log 2} \approx 1.585$.
5. In terms of infinite series, the creation process can be related to geometric series, where each iteration reduces the area by a factor.

Review Questions

• How is the Sierpinski triangle constructed?
• What property does the Sierpinski triangle exhibit that makes it a fractal?
• What happens to the area of the Sierpinski triangle as the number of iterations increases indefinitely?

Related terms

Fractal: A complex geometric shape that can be split into parts, each of which is a reduced-scale copy of the whole.

Geometric Series: A series with a constant ratio between successive terms.

\textbackslash{}textit\{\textbackslash{}Hausdorff Dimension\}: \textbackslash{}textit\{\textbackslash{}A measure of how completely a fractal appears to fill space, calculated using logarithmic functions.\}