I have something to say about that…

Tessellations: a math metaphor for team working

When I was in fifth grade, our math teacher introduced us to tessellations. Here’s the brief rundown:

A tessellation is a repeated pattern of 1 closed shape, arrayed one after the next, with no gaps or overlaps. At their simplest, a grid of squares or a pattern of hexagon tiles on a bathroom floor are tessellations.

We were taught that if you start with a tessellating shape (like a square) and remove a section of it from one edge but add that same section to another edge, the shape will still tessellate. (This preserves the surface area of the shape. See diagrams below for examples.) We cut these tiles out of construction paper and glued 4×4 squares of them of them to a black background, learning from the 20-some examples in our class that this process always works.

Recently, this memory has been tapping me on the shoulder and it’s taken me a few days to figure out why. In constructing a project team, I’ve been conscious of allowing space for conflicting personalities and filling skills gaps. But the team members need go grow together: for the team to successfully accomplish its goals, each member should be adaptable, ready to interact with any of the other members at any time. They need to allow for personality differences and varieties of skills across the group. Tessellations make an interesting metaphor for this adaptation. (Each shape represents one person)

Tessellating squares, tiled one after another
A group of colleagues working in the same space, but on different projects. (Can you see the outlines of cubicles forming?)


Tessellated figures that aren't square, but still fit perfectly together in an infinite pattern
A close team, as you might find in a small company. Each team member is adapting for the others, compensating for gaps and allowing for preferences. This team doesn’t necessarily work much with other teams, so it’s allowed to have “rough edges”.


Four tessellated shapes that themselves fit perfectly into a larger square
A close team within a larger company of other teams. Each team then has to tessellate with the other teams. Smooth outside edges make this easier (on the assumption that the other teams will also have smooth outside edges, to collaborate with whomever they need).

These are ideals for a team, and generally take quite a while to build. In my experience, when you initially bring a team together, it looks something like this:

Four mismatched irregular shapes, which won't fit together
A newly-formed team

The “team-building process” is one of learning to identify the shape of your colleagues and adapt yours to accommodate them (and vice versa). With a bit of work, you can end up a bit more like this:

The four mismatched shapes are starting to form common boundaries, melding into a closer fit
The newly-formed team is adapting.

This is a model (and I admit I like the uniformity of a tessellation) and I’ll acknowledge its limitations. For the sake of argument, we should note that each team member doesn’t generally tessellate to the roles of their counterparts in the team. (Because each person brings differing experiences and personalities to the situation, they may each end up with a shape of their own, as opposed to being one repeating uniform shape labelled “team member”.) But there is one benefit to approaching the metaphor with the concept of uniformity:

When you remove one team member and replace them with someone outside, the clearer that tessellating shape is (ie., what it means to be a member of this team), the easier it will be for that new person to adapt.

Take these two teams as examples. The first is tessellating, each member has adopted a uniform method of interacting and set of expectations for their team members. (Therefore each has the same shape.) Along comes the new person, who can learn from any of the original members how to fit in and what is expected of them.

Three perfectly fitting shapes and one new one that doesn't fit with the three
A well-organised team with clear ways of working together and expectations for each other. While the new person, in black, will have to adapt to fit into the team, they should have a fairly easy time of it due to the clarity in the existing team.

In contrast, this picture below shows a chaotic team with no coherence. The new person has no idea how the members are working together, let alone how to join them. The best he/she can do is to pick an outside edge (anybody’s!) and start there, try to connect with everyone in due course and build up a shape of their own to fit the odd structure already there. If you’re the new person, it’s a much less fun scenario.

Three disparate and unrelated shapes, and a fourth new shape that also doesn't fit
A team with no coherence or organisation, attempting to incorporate a new person. This scenario is hard for the newcomer, as the team’s ways of working are unclear and even communication between the existing members doesn’t appear to be guaranteed. How should the newcomer adapt themselves?

These visual images provide a model for the ways team members work together. Though I’ve found that the shape of each tessellation changes from team to team (and even changes when a new person joins a team), it’s important to have a fundamental model. Every team member I’ve worked with has indicated that they are happier and more productive when they see the whole of which they are a part. Understanding the shape (or role) they are to take is a big step towards seeing the bigger picture — the entire team, the company, or even the entire industry.

Four complex but perfectly tessellated shapes, fitting into one neat square