Sierpiński triangle

The Sierpiński triangle is a fractal formed by repeatedly removing the central triangle from each remaining equilateral triangle. Its parts are scaled copies of the whole, producing detail at every scale.

Iterated function system: A finite collection of maps whose repeated application generates a set or image. Three contractions map the triangle onto its corner copies, generating the fractal.
Wacław Sierpiński: A Polish mathematician known for foundational work in set theory, number theory, and topology. The triangle bears his name; he studied the set in the early twentieth century.
Equilateral triangle: A triangle with three equal sides and three equal angles. The standard construction begins with this shape and subdivides it into four congruent triangles.
Fractal antenna: An antenna shaped with repeating geometric patterns to support multiple resonant frequencies or compact designs. Sierpiński-like branching geometries have been used in multiband antenna designs.
Affine transformation: A map that preserves points, straight lines, and parallelism while allowing scaling, rotation, and translation. Each recursive step uses scaled and translated copies of the original triangle.
Sierpiński carpet: A planar fractal made by repeatedly removing the central square from each remaining square. This related construction was introduced by Sierpiński alongside his work on triangular fractals.
Midpoint: The point halfway between the endpoints of a line segment. Joining the side midpoints identifies the central triangle removed at each stage.
Fractal compression: An image-compression method that represents images through transformations mapping regions onto similar regions. The triangle illustrates how recursive rules can encode detailed images compactly.
Self-similarity: A property in which an object resembles all or part of itself across different scales. Every surviving corner region reproduces the same triangular pattern at a smaller scale.
Sierpiński space: A two-point topological space with one open singleton and one non-open singleton. Sierpiński introduced this distinct topological construction in the same period as his fractal work.

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