My previous square grid triangles tessellation gave me an idea for another. It's not that different. It seemed way cooler in my head. I wanted them pointing at each other as well as going in opposite directions. Same basic principles. Just reverse a few things. I thought about leaving no gap between the flat sides and decided against it. But maybe that would've been more interesting.  Honestly, I didn't want to do all the extra folding that would've been involved. 

 Decided to do some square grid tessellation work. Just to keep up my understanding of it. Thought triangles might be interesting. I've casually wanted to do something with triangles on the square grid for a while now.  Used what I learned from this tessellation as a jumping off point since the triangles are really just half diamonds and the flat side is the same as a single side of a square.  It came together rather uneventfully.  It needs a lot of small diagonal folds in both directions. A little bit tedious in the prep work.  The actual collapse is pretty simple. However, it does overlap some on the backside. As you get closer to finishing the collapse you'll need to tuck certain folds under others. The specifics  will become apparent as you approach completion.  I do have a rough crease pattern that I haven't yet photographed. Will add it at a later date.  Update: Adding crease pattern....

 This tessellation was inspired by the way in which a small hexagon can be folded offset from the grid creating triangles very closely surrounding it. Like in the ninja stars tessellation . I figured why not see how it works out if I turn those triangles into rhombuses. Sounds logical, right? It worked out pretty nicely.  A very straightforward construction. Completely non-irritating to fold.  Still you wind up with a pretty picture for your not so hard work. The negative spaces are a larger version of the shape seen in the ringing bells tessellation . If there's a name for this shape, I don't know it. I guess it's a trapezoid variant of sorts.  I went with two pleats between where the rhombuses meet each other. You could go only one pleat. That's how I drew the crease pattern included below. I'm pretty sure you could even go no pleat and wind up with an even more compact version of the pseudo-trapezoid in the negative space. 

  This is an idea that I had a while back, but it seemed next to impossible to actually execute.  While it's not a very clean fold, it's satisfying to have done it. Slightly better paper might have yielded a somewhat neater finished product.  It's a very small, very densely structured design.  It's rhombus flowers, like the classic just rhombii flower flagstone . However, it's the tiny rhombuses natural to the grid.  Collapsed hexagons are at the center of each 'flower'. Small triangles negotiate the spaces in-between. Crease pattern is included below.  . 

So I came upon an interesting tessellation someone had created and I went about reverse engineering it. I thought from the single module I had come up with that I had it all figured, but I missed a few details which later revealed themselves as I went ahead and folded the larger incarnation.  Upon realizing, I returned to the original photo and figured out the steps I'd missed. A fold for another day.  My 'mistake' actually resulted in a pretty nice finished design. Pictured here first front, then back.  Central small hexagonal twists have double length rhombuses two pleats off of their axes. Repeat. Very large triangles form on the reverse side. They bisect three of the grid's tiny rhombuses.  It's an uncomplicated and relaxing fold. For what it's worth, those seem to be the sort that yield the most brilliant photos.  I have a very messy crease pattern photo. There are other ideas in the background, but the red lines still adequately illustrate the design....

  This is a Joel Cooper flagstone tessellation that I'd seen before, but for some reason, I had not really paid much attention to it. Until now.  It's his rhombus stars and open back hexes repeating. Fairly straight forward design once you're familiar with the structure of flagstone tessellations.  He does a video tutorial for it at his website. But video tutorials feel slow and frustrating to me.  He doesn't pre-crease beyond the standard grid. Which boggles my mind. My paper just doesn't behave that way. He also folds it in stages. One shape at a time. My paper doesn't do that either.  I always have to ease everything into place as a whole. End result, pretty much the same.  It was easy enough to map it out on a triangle grid and follow the crease pattern.  I used slightly better than crappy paper and it was a pretty fuss free fold.  Crease pattern is included below for those that want it. 

  In further exploring what shapes and configurations might work in an origami tessellation with a central six triangle twist core, I arrived at this result.  It's not a terribly elegant or clean tessellation. Although on paper and in theory it works, it's a bit fussy and messy to fold.  The first time I tried to do it I used flimsy paper and while I was able to complete it, it was sloppy and poorly constructed.  I tried again, with slightly better paper. It did work out nicer (as shown) but still not entirely as I had envisioned.  Really sturdy paper, I avoided, since I figured it would never allow all the overlapping and too close together folds required. But perhaps, patience and sturdy paper are actually what this tessellation requires. But I'm not up for fold it a third time to find out.  Crease pattern is at the bottom. 

  I had seen a pic of a tessellation someone had done that was the classic circle of flagstoned triangles surrounded by large blunt pyramids. This led me to wonder what other shapes might possibly be able to tessellate around that familiar circle of triangle twists.  That is how I arrived at this configuration.  It begins with the familiar flagstone circle of triangles emanating off of a small hexagonal twist on the reverse side. It's from there that I wondered how open back hexes would work as the next layer. It turns out that they work out quite nicely. Oddly offset parallelograms form on the back to reconcile the shapes on the front.  I had been fumbling around with ideas for flagstone open back hexes for a while and not really finding anything useful when this idea came quite suddenly. It all just coalesced pretty rapidly.  I used kraft paper thinking its sturdiness would be helpful, but it was actually really difficult to fold a true flagstone using such th...

This is a pretty simple design that I stumbled upon quite accidentally. I was messing around with more complicated folds and not really succeeding, when this idea just jumped out at me. So I went with it.  Collapsed hexagons in one direction and elongated rhombuses in the other. Both the rhombuses and the hexes are natural to the triangle grid.  Some offset triangles result in the spaces between.  It's a pretty simple design. Fairly easy to fold. Any old paper will do. I have included a basic crease pattern for those that want it.

 This is a variation on the open back hex twist and natural triangle tessellation . Instead of repeating the the hexes, they are half hexes, also referred to as trapezoids.  In order to repeat, you just use a mirror half hex off of each of the center hexes. Easy peasy.  This was my idea. Of course, it may have, probably, has been done before. But I've not encountered it.  It's not hard to fold. It is however, hard to keep track of the directions on the folds. Many of them are bidirectional.  So finishing a cohesive pattern kind of results in a lot of trying to keep track of which fold went which way and trying to determine what makes a sensible pattern to you. Circular or linear or something else altogether.  Using a 32 pleat grid further clouds the process since there are no full repetitions.  It's an interesting idea from which other interesting ideas might be gleaned. Make of that what you will. 

 I was going through some crease patterns that I'd saved from the internet many moons ago. I came across a few that seemed interesting. Little did I know how diabolical this one would turn out to be. Truly diabolical  I used good paper or else it would've been a complete loss.  Even the good paper took quite a beating in the process.  There were multiple times when I thought it simply was a theoretically accurate tessellation that wasn't actually foldable in real life.  However, I saw glimmers of it being possible as I stubbornly persisted in coaxing and cajoling the increasingly mangled paper into the shapes I desired.  Several hours later, it did bend to my will. I was stunned to have finally succeeded.  I don't remember where I found it. I don't know to whom it belongs. It was just sitting ide in my google photos when I rediscovered it. I almost wish I hadn't. But I'm glad I did.  I cannot stress this enough. A truly diabolical fold. Next to im...

 I found the crease pattern for this tessellation sitting in google photos. I'd saved it a long while ago and then forgotten about it. I don't remember where I found it. I don't know to whom it belongs or from what site it originated. I'd be happy to credit the designer if I could.  What I do know is that I was able to fold it. Which was no mean feat. This was truly one of those agonizing folds where you know it can be completed, but it's just really, really, really difficult to do so.  Maybe it was because I used pretty crappy paper. It looked like it would be simple. So I thought cheap paper would do. I was very wrong. And then wrong again and again.  Maybe it really is just one of those painstakingly tedious tessellation to fold that I both love and hate.  Perhaps, a little of both.  At any rate, it's a pretty neat result.  Left side is the crease pattern for the finished fold above.  Again. Not my crease pattern. Don't know whose it is. All cr...