Solving Origami Tessellations:
Single Sheet Geometric Paper Art

This tessellation is a reverse engineer from a photo I had saved several years ago. I found it sitting in one of my google photo archives. It had been there a while and I had forgotten about it. I believe I tried to solve it once before when I'd originally saved it, but hadn't been successful.  So upon discovering it anew, I decided to try again. This time I had no trouble at all unravelling its secrets. It's amazing how much you'll learn in a year or two of steadily working with origami tessellations. Things that once seemed impossibly out of reach, now are almost second nature.  It's actually a flagstone on its back side, although the rhombus/hex flowers on the front side are really the star of the show in my opinion. The flagstone side is a hexagon of triangles and then large pyramids.  I don't know who the original designer is. Like I said, I saved the photo several years ago and did not note where I found it.  It looks like something that might be a Joel Co

 So, as I was working on figuring out someone else's rhombus based tessellation, I kind of inadvertently stumbled upon this idea. It's actually an interesting example of how the same configuration of shapes can yield very different results depending on which sides of the paper each are placed.  In this case there are small hex twists on the back side and there are rhombus twisting emanating off them from the other side.  There are also some triangle twists closely nestled into the hex twists.  And there are some pyramids between the triple intersections of the rhombuses.  This was a next to impossible fold. It was really hard. Just cause it's your own idea doesn't necessarily make it any easier to execute.  Given that it's such a difficult fold, I'm pretty pleased with the result.  Knowing its intricacies better now, a second attempt would no doubt be more perfect, but that's a project and a post for another day. 

  This tessellation is building upon the idea that I used in my Pathways design . It uses the same collapse method as that model, but in a different configuration.  The final model doesn't appear all that complicated, but there are many layers involved.  I don't know if this technique has been done before, but it was new to me.  It doesn't offer a lot of repetition on a 32 pleat grid, but it's an intriguing concept that could be applied to larger grids.  Personally, I just enjoy that it's a another way to manipulate the paper into a new variety of shapes.  There are not cuts, other than the initial hexagon . It's a single sheet of paper.  It's a little easier than some of the more complex tessellations, while still yielding a pretty intricate result. 

  A good flagstone tessellation is a glorious thing. Over the years I've folded many and enjoyed almost all of them.  I started out folding flagstone tessellations using the crease patterns generously offered on the internet by their respective creators. It taught me so much.  This idea may have, probably has, been done before, but it was new to me. I was thinking of a similar creation I'd folded before by Joel Copper and came up with this pattern.  It's a series of rhombuses and triangles arranged around small hexagonal twists on the reverse side.  Rhombuses and triangles are common shapes in so many tessellations. In this case, the difference is how those shapes are arranged.  Personally, I love a good flagstone tess. They just make so much sense to me.  Once you get the hang of it they're not terribly difficult.  And once you learn the structure of how they're created, it's not too hard to seek out your own patterns.  I do have my crease sketch. Just need to

 This tessellation just sort of came to me as I was thinking about possible new ways to integrate triangles and hexagons. I was inspired by a lot of the designs done by Arseniy K and was hoping to come up with one of my own.  This is the result. As I was mapping and folding, I was sure that the triangles up side (second picture) was going to be the front.  By the time I had finished, I was convinced the opposite side (first picture) is the front, because I like that pattern so much more.  I wish I had used sturdier paper so that I'd been able to get cleaner points on my triangles, but I do like the color of this paper. 

  This is an origami tessellation that I saw posted by multiple instagramers. I believe they creditted @arsenikoom as the original designer. I really liked the model and decided to fold it for myself.  At first, I was a little puzzled as to where to begin. Mainly because I wasn't able to see the sizes of the shapes from the pictures available.  At some point I realized I had already done a tessellation of my own that used large triangles in the same way and that gave me the size of the rhombuses. From my triangle tessellation I was also able to see that the connecting rectangles were inverses of the triangle points.   From there I mapped the rhombus shapes around the backside triangles/front rectangles and was able to see what belonged in the remaining spaces. 

  This is a design I came up with a few years ago. I originally posted it as the Space Invaders Tessellation .  I found the original model and decided I would give it another try to see if I could improve upon my first attempt.  While I'd love to get those hex twists on the backside nice and tight, it's next to impossible.  I do like those blade shapes and should try to see what other patterns can be done with them.  I think, if I remember correctly, I first saw that shape in a tessellation done by Byriah Loper. 

This tessellation was inspired by a reverse and fold I did of one of @gatheringfolds designs. In my post for that design I called it Tribal Tessellation .  I really liked how she used overlapping triangle and rhombus twists in that tessellation, so I wanted to see what I could come up with using that technique.  That's how I came up with this curved weave tessellation.

  This is a reverse and fold of a design by @gatheringfolds. I've been doing a bunch of hers lately. She's pretty active on instagram, so I come across them a lot. They're usually not hard for me to figure out. So when I'm in the mood for something less infuriating, I'll go for one of hers.  This particular pattern is pretty unusual. I haven't really seen anything too similar to it.  In actuality, it's just a series of rhombus and triangle twists arranged in a fresh new way.  In some places the twists overlap each other, which I thought was interesting.  The reverse side also generates a really cool back lit pattern. 

 Square grid origami tessellation are not something I do very often. They feel, at least to me, much simpler than their triangle grid counterparts. They are more limited in their ability to diversify. Still, they are fascinating in their own right.  Every once in a while, I'll come across a square grid idea that compels me. Every once in a while, I'll feel the need to refresh myself on those basics.  I'm sure in many ways that square grids help build a foundation for understanding more complex tessellations. Their mechanics certainly transfer some to a triangle grid, at least in my experience.  Browsing instagram, I was inspired to create this particular configuration. It's just repeating octagons and squares. Nothing fancy.  However, working this tessellation led me to some new ideas on the triangle grid that I had not previously considered. So it was successful in more than just the idea itself.  I'm still working on realizing the ideas it inspired. But it's c