Voronoi Art Patterns and the Beauty of Efficient Seeds

Voronoi Art Patterns in Nature

I have been playing around with Voronoi concepts this week. I have found that Voronoi art patterns fit in well with our tessellation and tiling work. Like many of the tools on this site, they are beautiful and mathematically efficient. They mirror the natural world, and like all things in our universe, are the reward of order flourishing.

Voronoi diagrams were first described by Georgy Voronoy, a mathematician from Ukraine. He was looking to investigate number theory and the geometry of quadratic forms, using them to divide space into regions that were as efficient as possible in terms of distance. He was hoping to mathematically describe a lattice and the distribution of its midpoints in space. Sadly, he died at 40, soon after publishing a paper on how to divide an area using quadratic forms.

Later, Alfred Thiessen used similar methods to divide landmasses into areas for weather prediction. Today, we know that Voronoi algorithms can be used for geography, computer science, robotics, and more. On this site, we are using them as art, and they are beautiful.

If you have ever admired a giraffe’s skin patterns, dragonfly eyes, or soap bubbles, you have seen Voronoi art patterns. They simply are the best way to arrange objects efficiently within a matrix. Think of Voronoi as each region having a seed at its center — each seed drawn as close as possible to its own cell, and equidistant from its neighboring seeds.

Natural systems exist in a world of finite dimension, and the natural rule is to organize in a way that minimizes distance. It is, for example, why bones are so strong: the matrix inside a bone cavity, filled with a spongy lattice of bone, maximizes the distance from the center to the edge within each cell. Where the edges meet lattice-wise, they are equidistant from the center. It repeats, with the outer edges forming a thin layer of bone that reinforces the structure’s integrity. The same idea can be easily visualized with bubbles, which, by nature, tend to form the most stable structures.

But what is nearest? The question is how you conceptualize the cells within our space. Euclidean distance moves as the crow flies — no barriers to efficient movement — and gives us what we see in normal life. But if you can only move in 90-degree steps, you get Manhattan distance, as if driving the square streets of a city, and the geometry of the cells becomes blocky and jagged. In Chebyshev, only the longest axis moves, like a king on a chessboard, and the cells snap into squares. The seed influences movement within those constraints.

Our cells have many points or seeds in them; the average position is called the centroid. Lloyd’s relaxation moves each seed to that centroid and then redraws the cells, order upon order. Things settle to stillness. This is all mathematical, but the result is beautiful.

Our tool lets you add any number of points, or seeds, and watch how moving them shifts the whole diagram. Every boundary and vertex recalculates in real time — move one seed and the geometry of every neighboring cell responds. Put seeds close together and watch the cells shrink; place them far apart, and the cells expand to fill the space.