Emergent Hexagons Origami Tessellation

Serendipity

There are some things that you can't know in advance.

In the garden I can know what I planted and where, but not when it'll come up or how many plants will survive.

In tessellations I can know my symmetry pattern, know what twists I'm using and in what spacing, but still not know what will emerge from the backlighting.

In both cases, the end results are often more beautiful than we could have planned for.

In both cases, I like to space things close together to see what happens.

I'm often pleasantly surprised in the garden, just like with my Emergent Hexagons tessellation, at how well the plants grow even when they're tightly spaced.

Clustering

Let's take a look at the details - how are the twists connected in this tessellation?

Look for a closed triangle twist, then see that it's connected on the same side of the paper to three open triangle neighbors.

Each hexagon structure is centered on six open triangle twists that alternate sides of the paper.

Can you see that the open triangles on the back also cluster around closed triangles on the back?

This cluster of three open triangles around a closed triangle is the dominant structural motif of this Emergent Hexagons tessellation.

The next thing to look at is spacing - could these clusters be put closer together? further apart?

This is actually the densest possible spacing of these clusters (on a grid) and the only spacing that gives rise to the shadow hexagons seen here. 

Isoarea

One of the coolest things in tessellations is the ability to fold something that's the same on both sides - aka isoarea patterns.

I'll never get tired of flipping the tessellation over and seeing the same pattern again - and then flipping it another 10 times so I can try to wrap my brain around the sorcery involved.

When you really look at this pattern you'll start to realize that there actually isn't any true hexagonal symmetry in it - there's always three things on one side and three on the other.

But just as you can use six equilateral triangles to make a hexagon, this tessellation appears hexagonal in nature because it's centered on a point where six triangular clusters come together.

Try it!

The best way to start folding more advanced designs is to try to reverse engineer the tessellations you see.

When you reverse engineer, look for patterns that you could use to make variations.

A key pattern in this tessellation is the clusters of four triangles.

You'll see that again and again in my upcoming posts too - it's a versatile and under-explored symmetry flavor that's available for tessellations using only equilateral triangle twists.

In this particular tessellation, the choices I made were that I wanted clusters of four triangle twists, the cluster would have a closed center and open corners, that neighboring clusters would alternate sides of the paper, and that clusters should be spaced as densely as possible.

Those four choices define the infinite tessellation, but not the specific piece of infinity that I folded.

For the specific choices I needed a crease pattern to calculate the grid rotation, the amount of grid needed for the repetitions I wanted, and whether I'd get a clean border.

If you're interested in folding this tessellation, you can get the crease pattern here.