It's easy to get sucked into the details of the exact placement of mountain and valley folds when you start looking at crease patterns for tessellations (and then think you need to precrease and collapse them!), but there's a simpler way to see these patterns.
When I look at a crease pattern, I look for the positions, relationships, contents, and spacing in the pattern.
Positions and Relationships
The positions and relationships that I'm referring to here are for the generalized form of the tessellation - the tiling.
So, I may have square positions in an alternating relationship or square positions in a mirrored cluster of four relationship (among other possibilities).
I may have equilateral triangle positions in alternating, mirrored clusters, or other relationships.
I may have a mix of shapes, like hexagons and triangles in 6-fold symmetry or hexagons, triangles, and rhombi in 6-fold symmetry.
There's a limited set of tilings used in simple tessellations, namely the square tiling, equilateral triangle tiling, hexagons and triangles 6-fold tiling, and a smattering of others involving rhombi and right triangles.
In fact, if a tessellation uses a different tiling I wouldn't call it a beginner tessellation.
The relationships are constrained by symmetry relationships, and therefore constrained to the 17 wallpaper groups.
The importation relationships for our purposes consist of mirror lines and rotational symmetry, either 2-, 3-, 4-, or 6-fold.
If the highest rotational symmetry is 2-fold, it's paired with mirror lines and up to 3 other distinct 2-fold rotationally symmetric points.
If the highest rotational symmetry is 3-fold, the pattern will have two other distinct locations with 3-fold rotational symmetry.
4-fold rotationally symmetric patterns will have two positions of 4-fold symmetry and one position of 2-fold symmetry.
6-fold rotationally symmetric patterns will have 3- and 2-fold rotationally symmetric positions that outline a hexagon around the 6-fold center.
So, what I'm really looking for in my first glance at a crease pattern is:
- What shapes of twists are used, in what arrangement?
- Where are the symmetry points and/or lines?
Looking at the crease pattern in skeleton form can give me answers to both of these questions.
The skeleton simplifies the crease pattern so I'm only looking at the twist type and arrangement - not the specifics of the pattern.
It's also easier to spot symmetries from the skeleton - whether a twist is in a position that allows its maximum possible rotational symmetry is immediately apparent.
For more details about how to draw and analyze a crease pattern skeleton, check out the video at the bottom of the post.
Here's a hint: twists can go in positions with rotational symmetry that are factors of the maximum rotational symmetry of the twist on its own.
So, a hexagon twist can go in positions of 6-fold, 3-fold, 2-fold, or no rotational symmetry but never in a position of 4-fold rotational symmetry.
Contents and Spacing
Where positions and relationships were focused on abstract, general properties of the crease pattern, contents and spacing are focused on the particulars of the crease pattern.
Here's where we ask questions like:
- Are all the twists of this shape the same type?
- What kind of twist is this - closed, open, hybrid, stacked, isoarea, other?
- Is the twist on the front or the back of the paper?
- How far apart are neighboring twists?
- Is this distance the same between each pair of twists?
- If it's not the same, how can I remember which spacing goes where?
- What's the shortest description that would tell me what to fold?
So, if I'm working on a pattern with hexagons and triangles in 6-fold arrangement I might look at the crease pattern and say:
"This tessellation has closed hexagon twists alternating with closed triangle twists, both on the same side of the paper, and with tube pleat spacing between everything."
That description would be enough for me to fold the whole thing - no crease pattern needed.
I analyze the pattern I just described and several more in the video below in more depth than I can write out here - check it out if you'd like to dive deeper!
The skeleton notation has been useful for my students in identifying what parts of the crease pattern to focus on, and I now include it by default in all of my crease patterns.
If you have any questions about reading crease patterns, either drop a comment below or send me an email and I'll get back to you shortly!