# Voronoi art

Voronoi diagrams are commonly thought of in terms of a set of points. Given
a set of points {P_{i}}, you cover the plane with disjoint convex
tiles {T_{i}} such that T_{k} contains all the points in the
plane which are closer to p_{k} than any other p_{i}. This
has many practical applications, and there are well-known algorithms for
finding the Voronoi diagram of a set of points.
My interest is in applying and extending this concept to produce attractive
ornamental designs. It turns out that taking the Voronoi diagram of an
interesting set of points tends to yield an interesting tiling of the plane!
I've written a program that renders Voronoi diagrams of sets of points
to show you what I mean. For each pair of illustrations below, the
diagram on the left is a schematic showing the set of points used to
generate the rendered tiling on the right. Note that in each case, the
tilings were produced using only overlapping square lattices.

But this concept can be made more general. In particular, the p_{i}
don't have to be points. They can be arbitrary subsets of the plane. The
mathematical construction is the same: T_{k} still contains all the
points closer to p_{k} than any other p_{i}. You just need
to be more intelligent about computing distances from objects. As a
canonical example, consider the Voronoi diagram induced by a point and a line:

This will induce two disjoint regions in the plane. If you remember the
construction of conic sections, a parabola can be defined as the locus of
points equidistant from a focus and a line. Sure enough, we get the following
Voronoi diagram:

I've extended the basic renderer mentioned above to implement this more general
Voronoi diagram drawing concept. It can create some beautiful, varied
and often unexpected pictures. Here's my favorite:

It was created by taking the Voronoi diagram of coloured rings, interlocked in
a hexagonal pattern. Here's a schematic, showing the elements {p_{i}}
that induced the above picture.

## Gallery

All images are copyright 2000 by Craig S. Kaplan. You are
free to use them for **personal** and **non-commercial** purposes.
Please check with me about any other uses.

## Papers

## Other resources

- Jos Leys has adapted the
ideas presented here, creating a Voronoi diagram renderer in
Ultrafractal. I invite you to visit
his gallery
for many more examples like the ones given here.