This is a tessellation that I came up with after having done a very similar one. That first one ( Controlled Chaos Origami Tessellation ) had exceedingly small rectangles, which made it nearly impossible to execute cleanly on both sides.  In this case, I made the rectangles bigger and it became infinitely more manageable.  I'm pretty sure that I've seen this one before. I think that I've seen it done by  Daniel Kwan as a crease pattern posted to his flickr account.  I was not using that crease pattern while conceiving of this, though I did have some recollection of having seen it in the past.  The front is all right triangles.  The back side is small offset hex twists, the aforementioned rectangles and small triangle twists.  Even with the larger rectangles, it's still a difficult to fold tessellation. The triangles and rectangles overlap each other.  My crease pattern diagram is at the end of this post. 

 This one has a bunch of shapes in it, all working together to form the whole.  When creating a tessellation, I usually just start with one shape that interests me at the moment.  From there, I choose whether it'll be a six axis or a three axis rotational repeat. In other words, will I try to repeat it around some form of triangle or some form of hexagon.  Once I have a starting point, I mess around with additional shapes which may help me to further create repetitions.  The last step is just to figure how to fill in any gaps that are created.  In the case of this tessellation, I started with natural triangle twists off of the six edged pyramids shown in the first photo.  At the point, I added an open back triangle twist to repeat the natural triangles. Rhombuses were the final puzzle piece needed.  On the reverse side I wound up with some rhombus based cubes and two different sizes of negative space triangles.  This tessellation is not difficult. I was able to execute it using ord

 This is tessellation is interesting because I actually messed up the pattern that I had created. It still looks really cool, but it is asymmetrical.  My original diagram called for all the repetitions to match each other.  This is also another difficult to execute to design. Regardless of whether you follow the pattern precisely or you accidentally. deviate from it as I did, it's pretty intense to fold.  My tant paper actually tore a little bit in one spot from all the stress.  The center is easy enough to get situated, but after that all bets are off.  It starts with an open back hexagon around which are situated interconnected rhombuses.  From there I just add a whole lot of triangles and more rhombuses. The pattern is meant to be three triangles and two rhombuses and then repeat the center.  I have included my crease pattern diagram at the end of this post. 

 This tessellation was pretty ambitious undertaking. When I plotted it out on some grid paper, I knew that it would be kind of crazy, but I loved the design too much to not try.  At the center point is an open back hexagon. From there I add three trapezoids and three triangle collapses.  Then I add another layer of trapezoids.  This in turn, creates the opportunity to add a whole bunch more small triangle collapses. Ten total per third edge.  It is ridiculously difficult to get all those triangles to work together while also making them behave nicely with the repeating trapezoids.  Obviously, it is doable, because I did it, but it's pretty tedious.  Well worth it though, for the end result.  I actually messed up the orientation for one of the triangles. Can you spot it? The nice thing about folds like this one is, once you get everything collapsed, it's not hard to re-orient the directions  of some of the folds if you need to do so.  When working with tessellations like this on

 Here's a fun little tessellation that utilizes just a few not too difficult techniques to create a moderately intricate pattern.  Rhombuses twist off from offset hexagons on their reverse.  Then on the outer edges of the rhombuses we add some open back hexagons.  At that point, the pattern can be easily repeated.  It's relatively simple compared to a lot of the tessellations that I've done in recent years.  It's a great exercise in some basic ideas that can lead to many other interesting configurations.  It also concisely demonstrates the close relationship between rhombuses and hexagons in geometric origami tessellations.  It would probably back light beautifully using thinner paper for those interested in folding it.  I used tant paper. The more pastel colors light up well, but the darker ones, unfortunately, do not.  Just another thing to keep in mind when crafting your tessellations.  Diagram below. 

 This is a tough one. Right triangles around an offset hex twist. When you go to repeat the pattern these very tiny rectangles are in the spaces between the points of the triangles. It's doable, but you probably need really thick paper in order to end up with a neat reverse side.  My tant paper worked for the front, but on the back I was only able to get some of the rectangles to land neatly.  In the diagram below they show up as micro rhombuses, but when you twist them they form very small rectangles.  Either kraft paper or elephant hide would be your best option for getting everything really neat on both sides of the paper.  As I was working with this pattern, it began to feel familiar. As if I might've seen it somewhere else. Perhaps on Aresniy K's instagram feed. I'm not certain. But I wasn't actively solving anyone's else's design when I came up with it. 

 Here's an unusual tessellation that uses a few different techniques.  It begins with a small hexagonal twist. Pretty typical.  Next, we have trapezoids spinning off from it. They are situated to twist the hex on bisected axes rather than the natural ones.  I call this an offset hex twist. Instead of using the grid lines to twist the hexagon, you create folds that bisect them and twist the hexagon that way.  The trapezoids have open back triangles in the spaces between them.  These open back triangle twists are not symmetrical three ways. They connect slightly differently to the next set.  There are also small triangle collapses tucked into the ends of the trapezoids. that connect to the next repetition.  All this is made more clear in the crease pattern provided at the end of this post.  Instead of having different shapes on the reverse side, this tessellation is a mirror image of itself on the back.