The Geometry of Optimal Triangles

Boris Delaunay introduced his triangulation criterion in 1934: given a set of points in the plane, connect them into triangles such that no point lies inside the circumscribed circle of any triangle. This single constraint produces the "best-shaped" triangulation possible, maximizing the minimum angle across all triangles. Thin slivers and degenerate shapes are avoided naturally. The result is a mesh of well-proportioned triangles that covers the convex hull of the point set with mathematical elegance.

Low-Poly Art and the Faceted Aesthetic

The low-poly art movement treats computational geometry as a creative medium. Random points produce an organic, unpredictable triangle mesh. Color each triangle based on its position, and the rigid geometric facets create an impressionistic image that reads as both abstract and structured. The appeal lies in the tension between random point placement (chaotic, natural) and Delaunay triangulation (optimal, mathematical). Chaos provides the input. Mathematics provides the structure. Art emerges at the intersection.

The Circumcircle Property

Every triangle has exactly one circumscribed circle passing through its three vertices. Delaunay's criterion states that for any triangle in the triangulation, no other point from the set falls inside this circumcircle. This property ensures that triangles are as "equilateral" as the point distribution allows. The practical consequence: smoother interpolation, fewer rendering artifacts, and a more aesthetically pleasing mesh. The Bowyer-Watson algorithm, used in this tool, enforces this property incrementally as each point is added to the triangulation.

The Voronoi Dual

Every Delaunay triangulation has a mathematical dual: the Voronoi tessellation. Connect the circumcenters of adjacent Delaunay triangles and you get Voronoi cell boundaries. This duality means the same set of random points simultaneously defines two distinct geometric partitions. Explore the dual at /voronoi. Both structures emerge from the same points, yet create visually and structurally different patterns.

In the Classroom

Delaunay triangulation brings computational geometry to life. Have students visit /delaunay/10 and observe how 10 points produce roughly 12-16 triangles. Then try /delaunay/50 and /delaunay/200. The relationship between point count and triangle count follows a predictable formula: approximately 2n - 5 triangles for n points. Students can verify this by comparing the statistics panel across different counts. Each generation produces a unique piece of art they can download as SVG for further exploration.

Private by Architecture

Every point coordinate, every triangle, and every color originates from your browser's Web Cryptography API. The triangulation algorithm runs entirely in JavaScript on your device. The server delivers this page and is finished. Generated artwork, download history, and generation counts stay in your browser.