Dancing Pyramids Origami Tessellation

A Whole Lot of 3

Sometimes it can be hard to tell what's 3 and what's 6.

Now this may sound confusing - how can you not know the difference between 3 and 6?

But when there's 6 things, with alternation between each one, is it 3 or is it 6?

In terms of rotational symmetry, the answer is 3.

In terms of twists in a loop, the answer is 6.

And to make matters worse, you can't look for 3-fold symmetries around the point in question to decide - they're all going to be 3-fold in either case!

You can, however, look for 2-fold rotational symmetries - those only show up around 6-fold centers.

And this pattern doesn't have any 2-fold rotational symmetry - no matter how much it looks like it in backlighting.

A Closer Look

Seen up close, it's hard to see how we might think that this tessellation has 6-fold rotational symmetry.

It's very clear that there are 3 triangles on each side per cluster, three trapezoids close to those clusters of triangles.

But are those trapezoids connecting directly to the triangles on the front?

Each trapezoid is really connecting to the paired trapezoid on the long edge, two trapezoids on short edges on either side, and a closed triangle twist on the back with the middle short edge.

If Woven Strips from last week was Hexagons and Triangles 6-fold with a split hexagon, Dancing Pyramids is Hexagons and Triangles 3-fold with a split hexagon.

Symmetry Rules

Whenever you have patterns like Dancing Pyramids where there are three positions of 3-fold rotational symmetry - no matter what is in the pattern - the three rotationally symmetric positions are always equidistant from each other and form 60° angles.

In other words, the locations of 3-fold rotational symmetries always form an equilateral triangle grid.

This is what it means for a tessellation to have fundamentally triangle nature.

For a tessellation centered on a grid intersection, this means that the two other 3-fold symmetry points are either both on grid intersections or both on grid triangles - not one grid triangle and one grid intersection.

So, which option did Dancing Pyramids go with?

Drop a comment below with your answer!

Symmetry Types

Every tessellation folded from a regular grid fits into one of six symmetry natures: hexagon, square, triangle, rectangle, rhombus, or stripe.

The symmetry nature defines the repeating tile of the tessellation - the portion of the pattern that contains everything you need to know to fold the whole pattern - and is closely related to the tiling.

The distinction is that a single tiling may have multiple possible symmetry natures - there are many ways to cut a hexagon or triangle out of a Hexagons and Triangles 6-fold tiling.

And just because a tiling is typically used with the simplest available symmetry nature doesn't mean that it's the only one available.

Many tilings have hidden depths, if only we dare to explore them.

And that's exactly what I've been exploring with my Tessellations by Tiles students - how to fold in 20 tilings and what hidden depths are available in each.

This Trapezoids and 6 Triangles tiling is one of the ones I explore with my students, along with three other tilings that use trapezoids.