I found a pdf crease pattern at origamitessellations.com that kinda caught my eye. Eric Gjerde and his book are what originally inspired me down the path of folded tessellations in the first place. If not for him and his book, I doubt that I ever would have discovered the genre that I now enjoy so very much.  The pdf and instruction were for offset triangle twists. An idea that had never occurred to me, but looked intriguing.  So I went ahead and folded it.  It felt very similar to other triangle and hex based twist tesses.  It was pretty easy as far as the actual folding was concerned.  The result was not far off from standard triangle and hex twist folds. But definitely worth doing. If only to maybe discover new possibilities in the grid. 

Here's an odd little tessellation that kinda just happened. I really don't recall anything in particular which inspired me to start down this road.  Basically, it's just rectangles folded across a square grid. The rectangles bisect the gird lines rather than following them.  It's just one of those interesting patterns that was hiding in the paper waiting to be found.  I consider this first pic the front.  But the other side is a pretty compelling pattern in its own right. It kinda reminds me of a tile floor or a stone patio.  Pretty straightforward to fold assuming small intersection folds are not a problem for you. I use a blunt point metal nail file to help with difficult and/or small creases.  Just take care to arrange your overlaps in an orderly pattern. They don't have to all flow in the same direction, but something orderly looks best.  

This is a square grid tessellation that I came up with while I was thinking about other people's square grid tesses.  The diamond shapes are folded from the standard Fujimoto clover/hydrangea square collapse.  The points or arrows work the same way.  From there, it's just fiddling with the connections and spacing to get an intriguing pattern.  It's not terribly different from other diamond shaped based tesses, but it's an interesting variation.  The way it lights up is particularly appealing in my opinion. Its exotic appearance belies its humble architecture. 

  I saw this tessellation on reddit that was just diamonds nestled together with no spaces between then. Being in the mood for something less mentally taxing, I decided to go for it. The folder had uploaded a really nice picture and it was easy to see how it had been executed.  The thing about square grid folds is that they are all pre-creasing. By the time you get to the actual folding and collapsing there's very little left to do.  Whether or not tha tis good or bad is personal opinion.  In my version I went with a lower density. My paper really wouldn't allow for anything more. I folded some of the corners differently to get two different sizes diamond shapes instead. But it's the same design. If I were to fold in the corners of my smaller diamonds I would have the tess that inspired this.  The gaps that appear are only because it's a backlit reverse view. Unlit there are no spaces between the squares. It's a not too complicated design that yields a really n...

 I found an image of a really interesting tessellation using google. It was a pretty complex flagstone with many different elements.  The center was the same as is shown here. But then they went with rhombus flowers surrounding it. I started out intending to replicate, but part way through decided to do a simpler variation.  It has some unusual petal shapes off the center open back hex. Then there are elongated rhombuses, which I've not encountered before. But as I was mapping out the repetitions, they showed up in the grid.  I didn't do a great job folding it and it doesn't really back light all that nicely. Even when your folding is a little sloppy, back light usually hides it. But in this case, not so much.  It's an interesting pattern though. And I do have a crease pattern for it. 

  I was messing around with square twists and weave patterns. Square grid tesses are not my strong suit, but they can be a nice change of pace. I know that Eric Gjerde did a very similar tessellation and this is just a riff on that. Just for the fun of it.  The Gjerde tess used just the larger offset square twists. In my case I just combined those with smaller square twists in an alternating pattern.  Not a major breakthrough, but interesting nonetheless.  What is nice about square grid tessellations is that they are usually pretty easy to fold. If you're in the mood for something less intense they are a nice little distraction.  Back side pic and front lit pic included below.

 In an attempt to recreate the tessellation of another, I misinterpreted the structure and created a minor variation on the idea. I've since realized the difference. But what I wound up with is something pretty cool in its own right.  From looking at the photo I thought it was hex twists on the back. So that is how I crafted my design.  Turns out, they needed to be offset hex twists. By the time I realized that I was too deep into it. I just went with what I had.  The result... not a replica of what I'd seen, but rather an interesting cousin to that original design.  New? Maybe. I don't know for sure.  Beautiful? Definitely!