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...