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 From the DNA in our cells to the fabric of spacetime, reality is tied in knots. Discover the branch of mathematics that finds profound order in the simple act of a tangled loop. Introduction We are all intimately familiar with knots. We tie them in our shoelaces every morning, secure boats to docks with them, and spend frustrating minutes untangling them from headphone cords. In our everyday world, a knot is a temporary state a useful tangle in a piece of string that can, with some effort, be undone. But what if you took that knotted shoelace and fused the ends together, creating a single, closed loop? Now, the knot is no longer temporary. It’s a permanent feature of the loop itself. You can stretch it, twist it, and deform it, but you can never untie it without cutting the string. In that simple act of closing the loop, you’ve just stepped out of the everyday world and into the fascinating realm of mathematical knot theory. This is a branch of topology that studies these closed, ...

  If Voronoi diagrams are about dividing space into territories, Delaunay triangulation is about connecting it. Discover the elegant, powerful relationship that turns a scatter of points into the most natural network imaginable. Introduction ​In a previous exploration, we delved into the world of Voronoi diagrams—the beautiful geometric patterns that arise from a simple rule of proximity. We imagined a city full of coffee shops and saw how Voronoi diagrams perfectly partition the city into zones, where every location in a zone is closest to its designated coffee shop. It’s a geometry of separation, of defining territories. ​But what if we asked the opposite question? Instead of dividing the city, what if we wanted to connect those coffee shops to form the most natural and efficient network of paths? What if we wanted to create a mesh of triangles linking them together, but not just any random set of triangles the best possible set? ​It turns out that the answer to this question is ...