More fun with trapezoids in flagstone tessellations .  The idea for this was a triangle twist with three trapezoids spinning around it So even though the center is six triangles, that was just a way to make the shapes around the edges work.  Although, I would imagine that if you wanted to you could conceive of a design where the center point could repeat instead.  The six triangles with the trapezoids eminating out from it. That would be an interesting variation.  The back side is almost exclusively triangles except for the one center hexagon.  As far as flagstone tessellations go, this is one of the less difficult ones.  Crease pattern included.  Filed under: June 2025  Origami Tessellations

 This is one of your simpler ideas as far as tessellations go. A standard six sided shape with triangle collapses tucked into it. And repeat.  The main shape, besides the triangles. is a common one you might find it many origami tessellations. It consists of one cross grid crease, then one that follows the grid and continue until  you close the loop.  It's kind of a simplified version of the Bells tessellation I did several years ago using the same two shapes. In that case, I connected them using collapses, instead of tucks.  In that one there were only small triangles. In this one, there are small and open back triangles.  As you explore various shapes and folding techniques you'll find that many patterns can be recreated in similar, yet different ways. That's the amazing symmetry of the geometry.  Looking at the reverse side you can see the collapses of the small triangles as well as the larger ones. Because of the multitude of small triangle colla...

 This tessellation is a somewhat random idea. It's really only a pattern in the most liberal sense.  Basically, I started with rhombuses around open back hexagons.. Then I added small offset hexagons and repeated the two ideas as best as I could. Once you add a few offset hexes you can then repeat the larger hexagons. The pattern of rhombuses and triangles is rather unorthodox, but it does work.  It's anyone's guess how to arrange the moutnain and valley folds in order to create a consistent design.  It's make for a pretty picture though.  It's a lot of work, but it isn't very difficult.  My crease pattern is included below, but it's not all inclusive. I think I had to work some shapes out as I went along. 

 This is an odd one. I was just fiddling with small shapes I could nest around an open back hexagon. As I was trying to decide exactly how to repeat the pattern, I looked at the other side and decided I liked it when the spaces on the back were right triangles. So I went with that configuration. And here we are.  While the beginning idea for this tessellation was the narrow parallelograms around the open hex, the true inspiration point was when I looked at the other side to determine how to make it repeat. It was then that I saw that I could create negative space right triangles nested against the hexagon.  At that point, it was just map it and fold it and just like that another origami tessellation was born! I debated calling the back the front and the front the back. It's all subjective, I suppose. I like them both equally in their own way.  This is an intermediate level tessellation . Nothing too difficult. Nothing too easy either.  Crease pattern included....

This an odd little tessellation that I recently created . It was conceived of as a collapsed hexagon to which I then added a multitude of triangle twists.  It became quite messy with so many triangles overlapping each other everywhere.  However, as I was folding it, what I discovered was that the other side became an interesting set of unusual shapes in a flagstone style pattern.  It's bascially a small hex flower with oddly shaped petals. Then there are open back triangle twists to connect the repetitions.  I was so focussed on all the various triangle twists on the other side that I didn't even really consider what might be happening elsewhere.  I used thin printer paper thinking it would be easier to fold all the overlapping triangle twists. Had I taken a step back and seen waht was happening on the reverse side I mgiht've opted for a sturdier paper and gotten a neater finished product.  It's a somewhat difficult tessellation to fold because of how...

 This is a linear flagstone tessellation that I just recently created and folded. The idea for it came to me while I was thinking about some other tessellations that I've done that use trapezoids in an alternating directional pattern. And I thought to myself, what if I could do something similiar in flagstone style. And this is what I came up with.  Essentially, it's a cantilever pattern of trapezoids and hexagons.  Some might feel it's a difficult tessellation . While others might see it as only intermediate . I always find flagstones rather intuitive to fold. Sometimes it almost feels as if they sort of fold themselves.  But while the bulk of the folding process might flow rather smoothly, actually getting everything tucked close together can be a completely different situation.    If I had an iron, I would probably use that. But I don't have an iron, so I just use a heavy book and a dumbell. But not before I've spent far too much time pushing and cajoli...

This is a collapse based origami tessellation that I created pretty recently. It's all interlocked rhombuses and triangles in a back and forth repeating pattern.  This style of tessellation can be pretty difficult to do when using small shapes. Collapsing a multitude of small triangles requires patience and persistence.  On the other hand, these types of tessellations are fairly easy to conceptualize. Everything just tucks into everything else. There are no gaps to figure out how to reconcile. They pretty much take care of themselves.  Looking at the revesre side, I almost like it better. I do love the look of how those points converge in the y shape. And it shows the details of the work so much more clearly.  A crease pattern isn't really needed. Just fold all the shapes on the front tucked into each other. This tessellation holds no secrets.  Filed under: May 2025 Origami Tessellations

* This is a tessellation that was folded by 'Jessie'. We had a little e-mail exchange. They were interested in folding the ' hexagons and trapezoids tessellation ' designed by Peter Keller. I had previously posted my fold and solution for this tessellation back in 2022.  The crease pattern I had originally posted was pretty basic. So, they emailed me to ask for something a little more extensive.  I drew one and sent it to them the next day. And the day after that they emailed me again to show their completed tessellation (as shown above). It turned out quite well. Kudos to Jessie.  For better tightness on flagstone tessellations you might want to use sturdier paper or wait for a low humidity day. Otherwise, a really nice effort. Keep on folding! The  extended crease pattern is below. My original posting of Hexagons and Trapezoids fold and solution.  * photo used with permission from the creator. 

The inspiration for this origami tessellation was alternating rhombuses and triangles around six sided 'pyramid' shapes. As I was plotting out how to repeat it, I wound up with small hexagon twists on the other side.  When I went to fold it, I was thinking it was a pretty easy one. It is, provided you're familiar with how to nest six triangle twists around a small hex on the opposite side.  I hadn't done that particular fold in a long time, but it all came back to me pretty quickly. Even with thin printer paper, I had no problem executing it very nicely.  Had I realized before starting exactly how the design would fold (I had forgotten how this six triangle set looked in diagram form), I might've opted for thicker tant paper instead.  Which wouldn't have given me such a wonderful photo of the final product.  That's the dilemma I often encounter when beginning a new tessellation . Which type of paper should I use. There's no debating that thinner paper...

 At first glance this tessellation might not look very unusual, but it's somewhat different in subtle ways.  It begins pretty typically, with open back hexagons. But the rhombuses that twist off of them kind of go in the opposite direction than what you'd normally see. It's this deviation that creates the lines when I add the additional shape of open back triangle twists.  Personally, I find it to be an interesting variation. It's not hard to do. I'd say it's an intermediate level tessellation .  When you look at the opposite side you can see the hexagsons and triangles quite clearly. You can also see the long parallelograms that form in the remaining spaces.  It's not a very complex design, but it's an interesting one still the same.  My crease pattern is below. Filed under: May 2025 Origami Tessellations

I haven't posted an advanced tessellation in a while. So here is a new one. It's an unusual pattern that also features some very small folds. There are tiny rhombuses and compressed triangles with which to contend. It is an original origami tessellation that I just recently created. The main shape is a six sided polygon of alternating edges. One pleat bisecting the grid, then one pleat following it and so on until you close the loop.  One the other side are open back hexagons and some oblong six sided parallelogram-ish shapes.  When I folded it the small rhombuses weren't actually on the front at first. It was after I had completed it that I realized I could reverse some of the folds to bring them up to the top.  There are some very tight folds and also some very small ones, which is why I believe this to be an advanced tessellation .  Evenso, I actually used thin printer paper because I figured thick paper would never let me make some of the tighter/smaller fold...

 This is a wacky one. A just to see if I can pull it off kind of tessellation . For the most part, I was able to do so successfully. Could be better, but it's a tough one .  It's basically just layers upon layers of elongated hexagonal folds around open back hexagons.  I used thin paper because I figured I would never be able to layer everything using anything thicker.  When I first plotted it out using a small grid of only 16 pleats it seemed so easy and worked beautifully. However, as with many tessellations , adding more repetitions made it exponentially more difficult.  The crease pattern looks pretty innocent, but it's anything but.  As I was folding it, I found myself wondering if maybe medium thickness paper might've been a better choice.  Most of the folds are mountain folds with the exception of the open back hexagons and a set of convergence points. In the diagram the blue dots indicate where the paper should be tucked under. When I drew it I...