This is an insane flagstone tessellation that I came up with a little while back.  Originally, I only had the beginning of the idea. It was just recently that I was able to flesh it out to completion.  At the center is a triad of alternating rhombuses and triangles around a central triangle on the reverse side. The entire pattern is based on the fact that three sides emanate from the rhombuses, while the other three stem from the triangles.  Somehow, with a little persistence I was able to figure out how the two different patterns would coalesce.  It's a crazy intense fold full of what seems like endless shapes on both the front and the back.  At many points during the execution, it felt as if I would never finish. But finish I did.  The front is both many rhombuses and many triangles. The back is just an endless sea of triangles with a handful of rhombuses.  I do have a crease pattern that I need to upload. Will do that soon.  If you wish t...

This tessellation puzzled me for a while before I realized that what I thought would be hex twists were actually triangle collapses. Once I realized that, it was smooth sailing from there.  In his post he had the second photo as his front. But I liked the other side so much that I posted it first and used it to inspire the title.  Essentially, it's right triangles on one side around collapsed triangles on the other. As you repeat the pattern you get the large hex twists.  I gave it a title for the purpose of this post. His post didn't have a title for it.  It wasn't too hard to fold. It wasn't that easy either. But his ideas are always intriguing and unique. Had I'd used nicer paper the right triangles (second pic) probably would've nestled together more neatly, but the other side (first pic) came out really great.  There's also an unlit version of the first pic as well as a basic crease pattern below. 

This is a linear origami design using trapezoids and triangles. Instead of your typical triangle grid tessellation that repeats circularly, this one is all horizontal and vertical repetitions.  The top of each trapezoid tucks into the base of the one above it.  The triangles collapses serve as a means to make the trapezoids repeat horizontally. The triangles tuck into each other the same way the trapezoids do.  I really like this collapsed style of tessellating the paper. It allows for far more repetitions in smaller space than classic twists.  As far as a crease pattern, it's probably not necessary. Like I said before, it's just trapezoids and triangles. A very uncomplicated design with a lot of repetitions.  I did a very quick sketch before folding, but it was kind of mixed up with other ideas.  If I remember, I'll draw a quick, new one and post it here later. 

 This tessellation is pretty simple in structure and yet, yields an impressive finished product.  It's just open back hexagons with  rhombuses tucked into them.  Even though the shapes are quite spread out, it still has a wonderful result on a small 32 pleat triangle grid.  It's one of your less intensive origami tessellations. There is not a lot of pre-creasing.  Once you get the hang of the collapses, it's a breeze to complete.  It is, however, important to work the reverse side as you go. You can't just fold the front shapes. Working the back side is essential to getting everything nestled nicely into place.  A crease pattern isn't really necessary. The shapes are very apparent. I do have one though. It's included at the end of this post. 

 This tessellation comes from my idea for intermingling trapezoids and rhombuses. I just kind of started with the central pattern and worked my way outward.  Collapse style tessellations such as this one offer a plethora of new ways to configure the various shapes you might choose to incorporate.  You can also mix twists and collapses in a single tessellation. This one however, is all collapses.  Collapsed tessellations are pretty easy to map out on grid paper. The connections between the shapes are pretty straightforward.  They can be a little tricky to execute. Especially on the back side. They have a lot of folds in common with flagstones, although they are not quite as intense. They also tend to result in a lot of shapes to fold. Even on a 32 pleat grid, you can often wind up with many repetitions. So patience is a must. All the work is in the preparation. By the time you've finished all the pre-creasing, you'll feel like you've done so much. But at that poi...

 This tessellation began as an idea for trapezoids around a collapsed triangle. As I was working the paper, it became clear that additional trapezoids were needed.  I began working three around the triangle center, but quickly discovered that additional ones were needed. With the central six sorted out, it was just a matter of repeating and using the grid to follow the geometry for how it would all coalesce.  It's a pretty easy pattern to fold, so long as you're okay with doing small triangle collapses.  I haven't seen it before, so it may be new. The negative spaces are large pyramids.  I've included my crease pattern at the end for those who might be interested.  There's also an unlit closeup of the back side to help clarify how it's folded.  . 

This is a partial solve of a tessellation by Arseniy Koom. It's a rare square grid tess. Rare for both him and me.  I don't recall if he gave it a name.  I solved the central area of his design without even realizing that there was more to it. Later, after I noticed my oversight, I went on to solve the complete design. However, I've yet to fold it.  The center point for both tessellations is the start of a classic clover fold.  Moving outward, there are octagons and negative space squares.  His design used larger versions of the same shapes. I wanted to use smaller shapes so it would repeat more.  Afterward, when I looked back at his photo and saw the outermost layers of the design I realized why he had used the larger shapes. Since, some of the elements wouldn't really work in my smaller version. That is a fold for another day.  The crease pattern for this tessellation is included at the bottom of this post. 

Here's an interesting tessellation that I discovered recently. It combines several techniques in a different way. There are a lot (and I mean a lot) of triangle and rhombus twists in this one. They repeat in such a way that some small hex twists are created on the opposite side So on the back side you have open back hexagonal twists. Around those you have alternating rhombus and triangle twists.  This creates some cool negative space twisting stars that appear to be in woven cages. In order to make it tessellate flatly, I had to add many additional triangle twists.  Once those get situated that's where the small hex twists come into play on the other side.  This creates the caged flowers pattern. Which is rhombuses around the hexagons combined with a nice woven effect.  It looks like some of the rhombuses twists are on the front, but they are not.  The open back hexagsons, all the triangle twists, and all the rhombus twists are on the back.  The small he...

 This tessellation is designed by Madonna Yoder @gatheringfolds. She calls it 'Hazard Discs'.  It's neat because all the negative spaces are open back hexagonal twists even though they appear as two different shapes.  This is because they alternate between ones with equidistant triangle twists and ones with triangle twists that vary in distance by one pleat.  In the areas where three of the open back triangle twists converge there are a triangle collapses on the other side.  It's a really cool design that received unsolicited praise from casual passersby.  I myself, find it to be a really interesting configuration as well. Which is why I took the time to solve and fold it.  It's a really elegant arrangement of just a handful of fairly standard origami tessellation techniques.  The view of the reverse side is just as intriguing as the front.  My mapping sketch is included at the bottom of this post.