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The Barnsley fern is a famous example of a fractal, a mathematical object that exhibits self-similarity at different scales. It was introduced by mathematician Michael Barnsley in 1988 as a way to demonstrate how simple iterative rules can generate incredibly complex and natural-looking shapes. Unlike traditional geometric figures, the Barnsley fern is created using an iterated function system (IFS)—a set of affine transformations applied repeatedly to points on a plane. With just a handful of e

The Klein bottle is a fascinating mathematical surface that challenges our everyday understanding of geometry and space. It is a non-orientable surface, meaning it has no distinct “inside” or “outside,” much like the Möbius strip. However, while a Möbius strip has an edge, the Klein bottle does not—it’s a closed surface without boundaries. To construct it properly, you’d need four dimensions, because in three-dimensional space, the surface must pass through itself, which isn’t truly accurate to