Here's an extremely difficult tessellation that I recently created. Even though it was my own concept, I still needed a second attempt to actually fully execute the proper folding of it.  The design began with a circular series of triangles. Their connections alternating between tucks and flagstone style collapses. See the crease pattern at the end of this post for further details. That circle of triangles led me to discover the incredibly hard to collapse center portion. It's a small hex twist with upside down right triangles. They collapse into the next repetition.  The photo of the reverse view shows pretty well how it works. It's small and cramped and has a lot of fiddly little folds.  Thick and sturdy paper is a necessity. As is quite a lot of patience.  It's advisable to work much more heavily on the back side. Getting those collapses to cooperate really is the key to successfully folding this  particular origami tessellation.  Crease pattern is...

 I saw the crease pattern for this tessellation on the instagram feed of gatheringfolds. No solving required. I just followed her diagram.  Sometimes, it's helpful to turn off the deductive/creative part of your brain and just follow the map that's already been plotted.  Often, when I'm stuck without a new idea of my own, doing something like this will sort of reset my brain. Let it rest a little, but still keep the electricity flowing.  It's usually right after that a new inspiration or two springs to mind.  This one is an elegant configuration of small triangle twists paired with open back triangle twists.  Her original the crease pattern . Enjoy. It's a good fold. 

 Here's a trapezoid based tessellation that I don't believe I've done before. It's composed from just trapezoids and triangles. No twists. Just tucks.  The trapezoid shape lends itself well to a variety of linear, non-twisting origami tessellations. It's just a matter of finding the various patterns that work.  I used some pretty flimsy paper, so it's a little wonky. But it's a cool pattern that isn't difficult to understand or execute.  It also makes some other possible patterns come to mind. Using trapezoids and rhombuses.  I did create a diagram, although I haven't yet taken a pic of it. Will upload at a later date. Though, the shapes are pretty obvious. Nothing tricky here at all.  Update: added diagram. 

Here's a crazy hard tessellation that I created. It really felt like I was flirting with the impossible to fold.  When I drew it out on the paper, it looked so simple and innocent. It's anything but.  It was only after I began my first attempt to fold it that I realized that it was much more difficult than I had anticipated.  My first attempt was only 80% successful. It did however, inform me on how to fully realize the finished design on my subsequent attempt.  My second attempt with tant paper was successful. Though at many points throughout the process, I was not sure it would be.  It's a pretty obvious pattern. Or at least it was to me. Alternating rhombuses and triangles tucked into one another rotating around large hexagons.  The real difficulty comes from the triad of triangles that are created by repeating the pattern. It's no small feat to get everything working together as it should.  I wanted all the triangles to be fully forward. As I strug...

 This tessellation is from a design by gatheringfolds. However, I only used the center portion of their design. I just really liked the center so much that I wanted to see how it would look in straight up repeats. It came out pretty cool.  Interesting thing about this design is that the axes of the small hex twists flow into one another rather than all in the same direction.  It was while solving her design that I came up with a different design using the same principle. It is featured in my " Idling Time Machines Tessellation ".  This tessellation is just me repeating the center point from her idea   It's open back hexagons with small hex twists and blunt triangles in the spaces that facilitate the repetitions.  But actually, I found it problematic trying to fold it from that perspective.  Folding it mostly from the back side (second pic) made a lot more sense to me  Your results may vary.  For further insight see my diagram below....

 This is another rhombus twist based tessellation that I created. It's composed entirely of rhombuses and nothing else other than the corners required to connect and repeat them.  It's an uncomplicated design that isn't at all difficult to fold.  I named it after the way the negative spaces create a cantilever pattern.  The back side is typical of this type of fold. It has a tightly constructed linear weave pattern.  Personal preference which side you like most. They're both pretty neat! Crease pattern is very straightforward and not at all hard to follow. There are no tricky twists or any near impossible to execute connections. I'm still waiting for when it won't be humid to tackle some trickier designs. So, it's just a really simple structure that yields, what I think, is a wonderful finished model.  I used a 32 pleat triangle grid and rectangular printer paper.  My crease pattern is included at the end of this post. 

While I was in the throes of trying to solve someone else's tessellation, I came up with this design in the process.  It's one of those tightly packed kinds of folds that is a pain to execute, but it's a pretty cool pattern.  That's what I love most about solving other designer's tessellations. It often leads to inspiration and new ideas.  This fold starts with open back hexagons. They are connected to small hex twists in a rather unorthodox manner. This creates the right triangles that are the main feature of the design.  It feels a little familiar, but I'm not sure if I've seen or done this exact fold before or just similar ones.  The reverse view shows the two kinds of hex twists used. But the true secret to folding this is kind of hidden in the structure. The twists of the small hexes tuck into each other. Where the folds of a normal small hex twist would just twist all in the same direction, in this case they twist into each other and overlap.  It's...

 This is a nifty little tessellation that starts with open back hex twists. There are some large natural triangles twisting off from it. When I connected the spaces between the triangle points I was able to fit some blunt pyramids in next. At that point, I was able to repeat the pattern  It's the other side though (first photo), that I feel produces some really interesting shapes and lines.  A lot of creating origami tessellations is just about following the shapes. Once you have a starting point, it's all about reconciling how the next set of shapes will work together.  Solving the tessellations of others is a little different. That's more about examining the structure and finding markers. Once you can identify a shape or two, you can gradually piece together the missing information Solving other artists' tessellations is a great way to hone your skills as well as find inspiration for your own designs.  While it would be difficult to solve this tessellation usi...

 This tessellation is about as basic as they get. Straight up rhombuses. One flowing seamlessly into the next. It's simple, but I just love the pattern.  When the paper is damp because the weather is wet, less complicated designs are a wonderful exercise in the fundamentals.  Someday, if it ever stops raining and being humid, I'll hopefully get back to some more complicated structures.  In the meantime, there are lots of interesting ideas to explore that are more rudimentary.  I like how in this tessellation the rhombuses almost appear to have movement. A little bit of an optical illusion effect.  The design is the same on both the front and the back. The two sides are mirror images of each other. Also known as mirror symmetry.  My diagram is included for reference. 

 Here's an interesting, almost linear flagstone tessellation that I created. It's almost both circular and linear at the same time. The shapes do rotate around hexes, but they also repeat in a kind of linear fashion.  It consists of trapezoids and triangles. Four trapezoids and two triangles per hexagonal twist. The repeats are petty standard. One iteration leads seamlessly into the next.  It features an interesting pattern on the reverse side. Mostly triangles, but also some linear connections between the trapezoids.  It's a little unusual compared to any of the flagstone tessellations I've done before. Similar, but different enough to make me think.  The back side of flagstones always look more interesting than the fronts to me.  Your mileage may differ.  It was a little difficult to fold, but nothing crazy.  For more insight see my crease pattern below. 

 Here's an interesting tessellation pattern that I recently found. It looks like there are small triangle twists, but it's actually rhombuses that are partially folded over.  When I drew it out on the grid paper, it looked pretty simple. When I went to fold it, it was kind of tricky.  I used standard printer paper, so a more substantial paper might facilitate the execution better. But I was more interested in a getting a nice back light.  The main circles of rhombuses twist around a large hexagon. However, the way that they twist and overlap kind of hides that fact. At the center you wiid up with just a small negative space hexagon.  On their outer edges the rhombuses are connected by small hexagonal twists.  When the pattern is repeated open back triangle twists take up the spaces between the small hexes, aligning with their axes so they can twist synchronously.  All of this is made much more obvious in the crease pattern that's included below.  ...

Searching for something simple, yet interesting, I came up with this pattern.  The weather lately, has not been conducive to intricate designs. So I was hungry for something new and intriguing that I could fold without the dampness interfering.  I came up with a rhombus with triangles twisting at all four axes. In order to repeat the pattern, I used the corners on the outside of the triangle twists to facilitate the next array.  The shapes overlap each other in some areas. It's folders choice how they land from top to bottom.  It's pretty easy to fold, as was my intention.  It has interesting layers and weaving on the reverse side.  It's just one of those wonderfully simple designs that looks much more tricky than it actually is.  Crease pattern included below. 

 This tessellation is curious in that I conceived of what most folders would consider the backside when creating it. It wasn't until after I'd finished folding it that I noticed the other side had more of a 'front look'.  What is the front and what is the back? Maybe both sides are fronts. Can such a thing be? My initial idea was simply triangle twists around open back hexagons. Those then flow into a triad of triangle twists around collapsed triangle.  What happens on the other side are rhombuses around the open back hex. The rhombuses also form triads around the collapsed triangles which on that side are up facing. This side has a similar look to a flagstone tessellation, but the gaps are a little wider because of how the central triangles collapse rather than twist.  It's an interesting variation on familiar techniques. It also boasts a lot of repetitions for a relatively small 32 pleat triangle grid.  It's a lot of shapes, but it isn't too difficult t...

 This tessellation by gatheringfolds had me stumped. She alluded to a fold she referred to as a closed hexagon twist. This was not something with which I was familiar. This was not something I'd ever discovered or encountered. It was totally new to me.  I've been doing origami tessellations for the better part of 5 years and I thought that I knew all the folds and all the tricks. I did not.  Watching the beginning of her video tutorial showed me the unusual fold which she had used in this particular tessellation. If I had not watched it, I'm not sure whether or not I would've ever figured out the concept.  So, my take away is that there's always more to learn.  Once I understood this new shape/fold, I was all good to finish the process of replicating the design.  It also occurred to me that it works in a similar fashion to what I call offset hex twists. Where in an alternating pattern, three of the shapes that emanate from it can exist on the front and the ...

 This is an interesting little tessellation that I found. Essentially, it's a hex twist and then there are some more overlapping hexes around it. It's at that point that you can create the triangle twists that overlay the initial folds.  It has a similar look to triangle twists around a hex, but it has some more layers.  I drew the crease pattern to express the finished fold in two different ways because I wasn't sure just one way would get the point across.  You can fold this in stages.  First the central hex. then the outer hexes. Which aren't actually hexes. They're double wide. But when you overlay them you wind up with what resembles hex twists.  From there you can do the triangle twists. They're not easy to do in such close quarters, but I managed okay with pretty cheap paper.  See below for the diagrams.