Voronoi Patterns

Facilitator(s): Sarah Lonberg-Lew
Date of Meeting: September 11, 2020
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Whole Group

We started this meeting with a Which One Doesn’t Belong which also included talking about what the four pictures have in common:

Here are some of the observations people made:

  • The honeycomb is the only one that appears to have a regular pattern.
  • The leaf is the only picture without an animal in it.
  • The turtle is the only one under water.
  • In the leaf, the borders are a similar color to the shapes, but in the other pictures they are not.
  • The honeycomb pattern looks more three-dimensional than the other patterns.
  • The turtle picture has several patterns in it, not just on the turtle.
  • All the pictures have a pattern involving shapes.
  • The shapes in the patterns mostly seem to have four, five, or six sides.
  • Even though there is irregularity in the patterns, the shapes seem to have some uniformity of size.

I then revealed that the pattern in all four pictures is the same kind of pattern generated by following certain rules and shared another example of the same kind of pattern generated by a computer:

(Pattern generated by http://alexbeutel.com/webgl/voronoi.html)

In the context of noticing and wondering, these are some of the ideas the group came up with:

  • I notice that there are not very many triangles, but lots of shapes with 4, 5, or 6 sides.
  • I notice that it feels like there are some reflections going on.
  • I notice that it feels like a map coloring problem.
  • I notice that the dots are not placed randomly.
  • I notice that the dots are sometimes on the edge and sometimes closer to the middle of their own shape.
  • I notice that all of the polygons are irregular (the sides and angles are not the same).
  • I notice that it looks like county seats.
  • I notice that there is folding symmetry with the dots between adjacent shapes.
  • I notice that there tends to be only three colors touching at an intersection.
  • I wonder if this is connected to where post offices or fire stations should be.
  • I wonder what the deal is with the dots.
  • I wonder how you figure out how to place the dots.
  • I wonder why the dot is where it is in each shape.
  • I wonder if it is possible for more than three colors to come together at a point.
  • I wonder what the rule is behind this pattern.

After a solid round of noticing and wondering, I shared that this pattern is called a Voronoi pattern and that the dots are called “seeds.” I shared a couple resources before letting people loose to play in breakout rooms.

This website offers an interactive element where you can click to place seeds and generate your own Voronoi pattern: http://alexbeutel.com/webgl/voronoi.html

This website has a game based on the Voronoi pattern: http://cfbrasz.github.io/VoronoiColoring.html

I also provided a link to a document with some push and support cards (note that there wasn’t a specific task, but the support cards are around figuring out the rules for the pattern).

With only the instruction to play and have fun, we broke into groups for the next hour, using a communal Jamboard to record some of our work.

Small Groups

All the groups explored the rules for the pattern by playing with placing seeds, noticing relationships, and trying to predict the results of placing new seeds. Each group also explored specific questions that arose during their play.

Patricia, Eric, and Audrey used annotation to explore the relationships between lines between seeds and lines connecting seeds.

They later moved on to annotating directly on a giraffe (!) and asking the (still open) question of how to locate the seeds if they are not in the picture. (Side note: Annotating on a giraffe is my favorite thing that I have seen happen over Zoom.)

Jeniah, Maggie, Sophie, and Mark started by investigating the Voronoi game and used their exploration to come up with theories about how the pattern is generated from the seeds. They made predictions about what would happen before placing each seed.

They also pondered why this pattern shows up in the natural world and speculated about whether it had to do with the nuclei of cells. They played with adjusting some of the options on the Voronoi generator, including animations, and made guesses and discoveries about the effects of changing the parameters.

Usha, Lynda, Kevin, and Amy played with the interactive Voronoi generator, placed a few seeds, and then challenged themselves to predict the results of placing a new seed. They challenged themselves to see if they could place the seeds in such a way as to make a perfect square. Here is one of their attempts:

They also wondered whether it was possible to have more than three regions coming together at one point and attempted to place seeds to create a honeycomb pattern:

We had a lot of fun exploring Voronoi patterns and ended the meeting with people feeling like they wanted to keep exploring. For lots more information, including applications beyond what we talked about at the meeting, google “Voronoi.” Here are a few more links that I found interesting:

Article with some interesting variations: https://library.fridoverweij.com/docs/voronoi.html 

Some animations and images: https://library.fridoverweij.com/codelab/voronoi/index.html 

Voronoi patterns superimposed on a melon: http://cfbrasz.github.io/VoronoiColoringSavePNG.html

List of applications of Voronoi patterns in different fields: https://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html

In attendance: Sophie, Lynda, Jeniah, Eric, Patricia, Amy, Nadia, Mark, Usha, Audrey, Maggie, Kevin

Note: Sarah Lonberg-Lew is a math teacher and professional developer in Gloucester, MA. She is also the treasurer of the Adult Numeracy Network. Sarah (and many others from around the country) started attending our CAMI meetings when we went online during COVID. Thank you for leading us and sharing these fascinating explorations, Sarah! -Eric


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