What do giraffe spots, garlic bulbs, tree canopies, and ice formations all have in common? They are just a few gorgeous examples of Voronoi patterns, also known as Thiessen polygons or Dirichlet tessellations.
I’ve been thinking about Voronoi patterns a lot lately because they have inspired our architecture friends in their visualizations of the science centre we hope to build one day – a building that is both inspired by nature and powered by nature. It doesn’t take much for me to fall down a fascinating rabbit hole and now I’m finding these shapes everywhere it seems!
For example, last week I was in Kingston to visit with physics colleagues at Queen’s University and they put me up in a lovely hotel right beside Lake Ontario. When I woke up on that chilly Friday morning in early March and looked down from my balcony, a thin layer of ice had formed on the water in the harbour overnight, creating a lovely display of Thiessen polygons to greet me.
And I thought: “of course it creates a Voronoi pattern!”
Ice forms through a process called nucleation – the molecules in the cold liquid start to cluster together and the resulting ice expands outward at a steady pace from each starting cluster. If many nucleation sites are randomly scattered across the surface of the water, individual ice sheets will grow outward from each seed point until they reach the edges of neighbouring ice sheets. By definition, the resulting pattern consists of Theissen polygons as illustrated in this excellent gif from Wikipedia:
The ice fragments shown above aren’t perfect Thiessen polygons because there are wind and water currents jostling the pieces around, but you can definitely see the overall tendency. Each polygon surrounding a seed point encloses all the locations on the 2D plane that are closer to that seed point than any other in the region. The edges of each polygon are locations that are equidistant to two neighbouring seed points.
You can explore this phenomenon by drawing your own Voronoi pattern, with a little patience. First randomly scatter a few seed points on a piece of paper. (The more points you have, the longer it will take to draw.) Then draw lines connecting all dots to their neighbours – I used dashed lines here. The final step is to draw a (solid) line across each connecting/dashed line that is at the midway point between two dots – this is part of the boundary of the resulting polygon. Here’s what my shapes look like for three central dots in my random scatter:
For giraffe spots, their signature look comes from a random scattering of the cells that produce melanin, the dark skin pigment, in the skin of the giraffe embryo. As the giraffe grows, the dark spots expand outward from those random seed points to fill the available space with Thiessen polygons, and the resulting pattern is largely unique to the individual. This study does a lovely job of modeling giraffe spots with the mathematics of Voronoi patterns, resulting in some pretty realistic looking creatures! On the left is an image of real giraffe skin and on the right is an image that was computer generated by the authors.
Individual garlic cloves in a bulb are a great 3D version of the same process – each clove grows radially outward at a steady rate from the central random seed point, eventually pushing up against adjacent cloves to fill the space with Thiessen polygons.
Bubbles on a 2D surface, cells in a leaf, the outward spreading canopies of individual trees in a dense forest … they all tend to fill the available space with these closely packed, irregular-shaped, polygons.
The hexagonal shape of honeycombs (as discussed in a previous blog post) is actually just a special case of these patterns. With honeycombs, bees are creating a Voronoi pattern with regularly-spaced seed points rather than randomly distributed starting points. When we start with a structured pattern of seed points, as shown in my sketch here, we end up with hexagons surrounding each point. Those honeybees do like their symmetry!
And, although my tendency is to focus on examples from nature, there are lots of other scenarios where we encounter Voronoi patterns too, such as this incredible map of the globe that has been divided up into Thiessen polygons based on a location’s proximity to a regional airport:
Or this map of the 1854 cholera outbreak in London, England, where the white “P” on blue circle indicates the location of a shared public water pump. Dr. John Snow, a local physician, created the original version of this map to dramatically show the connection between proximity to a specific contaminated water pump (the Broad Street pump) and the highest density of cholera deaths, which was contrary to the commonly held belief at the time that the disease was spread through the air. I’ve superimposed the Thiessen polygon (lightly shaded yellow) that you get when you enclose all the locations in London that are closer to the Broad Street pump than any other pump in the neighbourhood, and it clearly encompasses almost all of the cholera hotspots shown in pink/red.
Someday, I hope you’ll be able to wander through our exhibit space in the science centre that might look a little like this:
But, until then, keep your eyes open for Thiessen polygons all around you!
Giraffe spots – shutterstock.com
Ice formation on Lake Ontario (2 images) – author
Voronoi pattern development gif – Wikipedia
Sketch of drawing your own Voronoi pattern – author
Images of real and computed giraffe patterns – Figure 4 (parts b and c) from https://doi.org/10.1145/383259.383294
Garlic bulb – shutterstock.com
Slideshow: bubbles – shutterstock.com; tree canopy from below – shutterstock.com; leaf close up – shutterstock.com; honeycomb by justus.thane – flickr.com (licensed under CC BY-NC-SA 2.0)
Sketch of drawing hexagons as a special case Voronoi pattern – author
Tessellated map of the globe based on proximity to an airport – https://www.jasondavies.com/maps/voronoi/airports/
Coloured map of the 1854 cholera outbreak in London – https://www.arcgis.com/apps/instant/sidebar/index.html?appid=18b0e940adde49b198f2a93454a15351 (modified slightly by the author)
Student Works: Cellular Tessellation pavilion lights the way in Sydney – Dec 30, 2014 by Amelia Taylor-Hochberg in Archinet; Photos by Patrick Boland photography
Integrating shape and pattern in mammalian models 2001 Marcelo Walter, Alain Fournier, & Daniel Menevaux SIGGRAPH ’01: Proceedings of the 28th annual conference on Computer graphics and interactive techniques Pages 317–326 https://doi.org/10.1145/383259.383294
Mapping the 1854 Broad Street Pump Outbreak – https://www.ph.ucla.edu/epi/snow/mapsbroadstreet.html and https://plus.maths.org/content/uncovering-cause-cholera
The Physics of Ice: It All Begins with Nucleation (acceleratingscience.com)
The fascinating world of Voronoi diagrams | by Francesco Bellelli | Towards Data Science
Voronoi Tessellations and Scutoids Are Everywhere – Scientific American Blog Network