I've been sticking with the parallelogram shape that I learned from the gatheringfolds website. The photo I found there had four different variations on this shape. So this is another. I really like this one in particular. It's simple, elegant and yet impressive.  It's pretty straightforward once you get used to the underlying structure. The nature of the main shape requires hexes on the back to be offset from the triangle grid.  The parallelograms cross two pleats lengthwise and bisect one on the short sides. Small triangle twists on the rear unite the triads.  It's helpful to draw the pattern out on a piece of triangle grid printed paper beforehand. It really makes it easier to understand.  Large hex twists form as you work the repetitions. .  They occupy the area of the grid between two pleats wide and one pleat wide. I have a crease pattern that I drew, but I'm not sure it's appropriate to share as gatheringfolds is a for profit website.  

Fiddling around on google searching for crease patterns I came across an image from a site called gatheringfolds. It was a single image of four different folded tesses. None of which were crease patterns, by the way, google. Hint. Hint.  The image is copyrighted, so best not to post it here.. But I liked the designs.  When I clicked through to investigate I learned that the site sells their crease patterns and tesses.  I like figuring these things out on my own anyway. So I worked out one of them so far and was able to fold it also.  I have the crease pattern I drew, but since they are  charging for their designs and they have copyrights posted, I don't suppose it's right to publish that here.  If you like the design you'll either have to sherlock it on your own or pony up the money to purchase it from them.  It's an interesting pattern. It's pretty humble on the front without backlight. But when you light it up on the reverse side it really sparkles....

  For some reason I find it harder to wrap my mind around square grid tessellation crease patterns. Hexagonal/triangular ones just make so much more sense to me.  Still, just to add variety, sometimes I try to come up with one or fold someone else's.  In this case I found this crease pattern that Robin Scholz had uploaded for 'Roadkill Rose' by Philip Chapman-Bell.  Apparently Bell did a single molecule and then Scholz tessellated it.  I had to modify it slightly to fit my 32 pleat grid.  In hindsight, I would've liked to space the repeats even closer together. I had pretty weak paper and was still able to get a nice finished product.  It's an interesting design. 

 Here's a new origami flagstone tessellation that might be somewhat original. I've seen similar designs, but not this particular configuration.  It begins with a classic ring of triangle twists around a hexagonal twist on the back.  Then there are those odd polygons which alternate between following the grid and bisecting it.  I've included a photo of my crease pattern and a view of the backside below. 

  This is a literal repetition of the Robin Scholz amizade triangles around a hex twist on the rear.  I believe he's already mapped it and folded it.  But I thought I'd give it a try.  It's a circle of of six triangles twisted gingerly around a hexagonal twist on the back of the paper.  The design naturally repeats by flowing seamlessly one into the next.  No figuring out what goes in the gaps. There are none.  It's a tough fold. Especially when you don't have great paper.  Satisfying to complete though.  Hard to believe that it's just a single, uncut sheet of paper. Yet, somehow, it is.  Oh, those edges. The edges always suffer. I've usually lost my patience by the time I reach the edges. 

 I was messing around with a pattern by Byriah Loper and somehow went off on a tangent which gave me this result.  It's an odd pattern that is hidden in the triangle grid.  It was an accidental discovery.  Reminds me of an old Atari 2600 video game. 

 I saw a photo of a Robin Scholz tess that was alternating triangles and rhombuses around hex twists. I was going to suss it out, but whimsy struck and instead I took the pattern and found a variation on it.  It was a very nice result.  It's a great idea, which I had been playing around with in my head, but hadn't really fully explored. When I saw the photo of a finished model, it really cemented the notion and I just let it take me where it wanted to go.  These are the results.  The combo of rhombuses and triangles form larger triangles.  The negative spaces are oddly offset hex twists.

This is a hexagon of rhombuses and triangles that are nestled together along their matching fold angles. There is a concentric hex collapse at the center.  The tessellation repeats quite naturally. A larger grid and sturdier paper would really have done with model a great service.  We ran out of better quality paper at work, so my tessellations have suffered somewhat for it.                                            Still, the idea is solid. It would be really nice if someone with better paper and a bigger grid decided to fold it. 

This is hex collapses repeating. Kind of interesting. Nothing ground breaking here, but pretty cool still the same.  It's no doubt, been done before. It was new to me, though.  I was thinking about spread squashing the flaps open for a different look. That was the initial plan. Still, I like how it turned out before that.  I used hexes on the back in between the touching 'petals' to connect them flatly.  The photos that follow are front squashed unlit, rear lit and spread squashed lit. 

There's a neat technique that uses the fold angles of various shapes to have one flow into the other seamlessly. Unlike twist folds which just spin off the axis of another twist, these folds share the same fold angles.  It's similar, but a little different. It allows for slightly different arrangements and configurations of the shapes on the grid.  This type of fold relationship is also somewhat easier to execute. Each shape flows directly into the next.  Unlike twists, the patterns are more linear than round.  This one is rhombuses and open back hexes.  Instead of a typical circular pattern, they form rows because of the nature of their angular connections.  I've used this technique before. It offers some different structures while still using the standard shapes. 

I've done a bunch of tesses lately, but getting nice photos has been difficult. Either it's too late or it's cloudy or there's no time at all.  I'm not complaining, just saying.  This one is a riff on a tessellation in Eric Gjerde's famous origami book. That one is hex twists with open back triangle twists off of it.  In my variation I decided to see how it would look if the hexes and triangles were on opposite sides of the paper.  There was a lot of precreasing involved. Those triangular veins needed help folding. And it's a somewhat unnatural flat fold  But, not too crazy difficult and a pretty cool result.

Messing around with rhombuses yielded this interesting pattern. It utilizes reverse rabbit ear folds to create triangular points which connect the running rows of rhombus twists. The triangles form in the spaces between the rhombuses. Then there are mirror triangles to facilitate the the next row.  It kinda looks and folds like a square grid tessellation even though it's done using a triangle grid.  It's a fairly easy fold.  I do have a crease pattern. Just need to take a pic and upload it. Update: photo of crease pattern added below.

 I saw a tessellation by Byriah Loper on flickr that made me want to fold it for myself. After puzzling over it for not too terribly long I had the road map and was able to recreate it.  It wasn't too hard to fold. A little fussy.  It helps if you have very sturdy paper.  Congratulations Byriah on a wonderful creation.  I have a single module crease pattern as well.  I will add the pic of that at a later date.  Crease pattern as promised: 

  I've returned to rhombuses once again. They are probably my favorite origami tessellation shape.  This is small hex twists on the front with rhombus twists also on the front.  I stuck some triangle twists on the rear between the rhombuses to work out the flatness.  Rinse and repeat.  Some pyramid shapes naturally formed in the gaps. 

I was just playing around with patterns and came up with this.  I'm not really sure which is the front and which is the back. I guiess it's just personal preference.  Pretty easy to fold. Nothing really frustrating.  The second image seems to show the various shapes involved a little better than the first does.  It's a triad tessellation. So the pattern repeats in thirds and I just sorta filled in the gaps.