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Origami — Tessellations

Models folded and photographed by Michał Kosmulski. Tessellations designed by their respective authors (most are my own designs, though).
Click on images to enlarge them.

Two-in-one Flower Tessellation — whole

Two-in-one Flower Tessellation

There is only one kind of molecule in this origami tessellation, but depending on how you look, you can see two different kinds of flowers here, hence the name “two in one”. Each molecule requires an 8×8 grid.

The pattern is derived from four molecules of my Double Spearhead Tessellation slightly modified and combined into a single molecule.

I used this model to test Rhinoceros Hide paper for tessellations and I liked the result. The paper is a bit glossy which made taking a picture without any reflections hard, but in real life it looks beautiful. Folding was also pleasant and my feeling is this paper allowed me to fold finer details than Elephant Hide would at the same grid size and it felt considerably thinner. This contrasts with the data presented in Ilan Garibi’s review of this paper which suggests Rhino Hide should be thicker than Elephant Hide. But maybe other factors like softness make it easier to shape despite the thickness.

[ discuss main view on flickr ]
[ discuss close-up 1 on flickr ]
[ discuss close-up 2 on flickr ]

Two-in-one Flower Tessellation — close-up 1
Two-in-one Flower Tessellation — close-up 2
Cluster Tessellation

Cluster Tessellation

Cluster tessellation, designed and folded by Michał Kosmulski. There are several different ways of achieving a similar effect but with a slightly different pattern in the background (one can for example start from a four-sink-base). The variant shown here is the one with cleanest background, so that the clusters and not the background are exposed. It starts from what I call Clover Base, that is a level-1 Clover Folding by Fujimoto (just the central square, folded from 6×6 grid).

Folded from rare orange-colored Elephant Hide paper.

Crease pattern available here.

[ discuss on flickr ]

Square Pixel Tessellation

Square Pixel Tessellation

This is my recent design for folding arbitrary bitmap images as origami tessellations. Unlike using four-sink-base (see Saturn Tessellation), each molecule represents just a single pixel rather than a block of pixels, so all pixels look exactly the same.

The molecule is made from a 4×4 grid but the size reduction in folded molecule is just 1:2, compared to 1:3 with four-sink base which needs only a 3×3 grid. So, there is bit more precreasing but the resulting tess is larger, and also the collapse is much easier.

The molecule for visible pixels is similar to a square twist, but only the central square is visible. The CP is very simple so someone may have come up with it before, for example it seems Robson Renê Albuquerque got a very similar pattern at the back of his chessboard tessellation. The empty pixel molecule is just a square twist pulled back to the reverse side of the paper, so in the front just a clean pleat is visible. Anyway, even if the front molecule was already known, I haven’t seen a combination like this used for pixmap images, so I hope this brings something new to the bitmap origami genre.

This is just a quick demo of the technique — I plan to make something bigger and more photography-like using this method when time allows.

Crease pattern available here.

[ discuss on flickr ]

Sprout Tessellation — close-up

Sprout Tessellation

Sprout Tessellation, designed and folded by Michał Kosmulski. Derived from my Squares and Crosses Tess.

The first image is a close-up of my first fold.

Later, I refolded Sprout Tessellation for the German Origami Convention in Erkner. Asymmetry is inspired by an earlier workshop with Melina Hermsen (Yureiko).

Crease pattern available here.

[ discuss close-up on flickr ]
[ discuss refold with asymmetric ornaments on flickr ]

Sprout Tessellation — refold with some extra asymmetry around the margin
Squares and Crosses Tessellation

Squares and Crosses Tessellation

My recent design, derived from Ninja Star Tessellation. 6×6 square grid per molecule. This design can be further modified (pictures coming up soon).

Folded from Canford paper which is pretty thick (150 gsm) but with a slightly larger grid than usual and a bit of wet folding finish, it worked pretty well.

Crease pattern available here.

[ discuss on flickr ]

Ninja Star Tessellation

Ninja Star Tessellation

I came up with this design while playing with variants of my Propellers Tessellation but then realized that many others had already come up with this pattern before, so I used the same name, ninja star. Each molecule uses an 8×8 grid.

[ discuss on flickr ]

Stacked Propellers Tessellation molecule

Stacked Propellers Tessellation (single molecule)

This is a fractalized version of my Propellers Tessellation. Stacked Propellers Tessellation is folded from a 16×16 grid per unit in this case but you can go on and on stacking one propeller on top of another if you use a larger grid.

Elephant Hide paper painted with metallic acrylic paint.

I found a similar pattern designed by Ilan Garibi.

[ discuss on flickr ]

Propellers Tessellation — short blade variant

Propellers Tessellation (short blade variant)

This is the short-blade variant of my Propellers Tessellation. Molecule is 6×6 grid units but in this rendering an extra margin of one unit is added.

Image scanned rather than photographed.

[ discuss on flickr ]

Propellers Tessellation — long blade variant

Propellers Tessellation (long blade variant)

This tessellation is closely related to my Square Interlace Tessellation. Grid is 8×8 per molecule. Here you can see the variant with long propeller blades. I will publish pictures of short-blade variant shortly.

Designed and folded by me from a single sheet of black Khepera paper.

This image was scanned rather than photographed. For this particular tessellation, I found the scan much better than the photograph: lighting is very uniform and the scanner was able to reproduce the textured black color of the model much better than my photo camera could.

[ discuss on flickr ]

Twisted bird base tessellation — simple molecules, top-down view

Twisted bird base tessellation (simple molecule)

This origami model consists of the simplest molecules of my twisted bird base tessellation. Each molecule is basically a bird base with the central part raised and twisted for a nicer shape.

Collapse is a bit challenging because multiple points on the creased sheet of paper must meet at a single point in the collapsed pattern.

Apart from the simple molecule used here, many other variants are possible, some of which are shown in this model.

Crease pattern available here.

[ discuss top-down view on flickr ]
[ discuss angled view on flickr ]

Twisted bird base tessellation — simple molecules, angled view
Lily pond tessellation icture frame

Lily pond picture frame

This is a picture frame for 16:9 format, decorated with a variant of my lily pond tessellation. The lilies were modelled on top of four-sink bases instead of the sunk square twists used in the original tess since this allowed for a more symmetric frame.

The picture is attached by tucking its corners into the corners of four-sinks so that only a very small part is covered. This makes four-sink-base based molecules good building blocks for picture frames.

[ discuss on flickr ]

Sunshine tessellation

Sunshine tessellation

Sunshine tessellation, own design.

The molecule is 6×6 grid units, so normally 3×3 molecules would require an 18×18 grid. Since folding power-of-two grids is much more convenient than others, I skipped one grid unit at the edge from all but the central molecule which allowed me to use a 16×16 grid and resulted in a pinwheel-like shape for the finished tess instead of the conventional square one would normally end up with.

[ discuss on flickr ]

Square Interlace Tessellation

Square interlace tessellation

This is a more conventional way of folding my square interlace tessellation than the bracelet. Folded from a 32×32 grid, with each molecule using 6×6 grid units.

Crease pattern available here.

[ discuss version in blue Tant paper on flickr ]
[ discuss version in graphite Elephant Hide paper on flickr ]

Square Interlace Tessellation
Tessellated bracelet — square interlace tess

Tessellated bracelet (square interlace tess)

Bracelet featuring a new design of mine, the square interlace tessellation. Made from a 61×9 grid. Each molecule is six by six grid units and resembles two loops interlaced together to form a knot (or a pinwheel if you like). Wet-folded to get into final shape.

[ discuss on flickr ]

Twisted bird base tessellation — molecule examples

Twisted bird base tessellation — molecule shaping examples

I recently came up with the idea of the twisted bird base tessellation. As the name implies, it is based around the bird base which is folded with some extra paper and then twisted so that the flat part forms a square which enables tessellating. The central part pops out of the sheet and can be formed in many different ways. I plan to post a large version of the basic variant soon but in the meanwhile, here are some examples of how the molecules can be shaped.

These are obviously not all possibilities of shaping the molecules. Horizontally, the shaping of the four petals changes while the two vertical rows show two twisting directions of the central part.

Interestingly, each molecule in this example has three separate twists in three stacked layers, each with different properties:

  • the lowest layer is the twist which enables tessellating the molecules: like in the plain square twist, two neighboring twists must have opposite chirality
  • the middle layer is the stacking order of the four petals: this twist’s direction is independent from the twist below so you can have any twist direction that you like (many tessellations based around twists force a change in direction every other molecule); the middle variant is an exception since in order to thin the petals, layers have to be folded under the petal which is only possible if the rotation direction agrees with the twist that lies below
  • finally, the third, upper-most twist is the middle part of the base: this twist’s direction is again independent from the direction of the twist below

Unlike many others, this tessellation is not based around a square or hexagonal grid.

[ discuss on flickr ]

Lily pond tessellation

Lily pond tessellation

This tessellation is based on the sunk square twist pattern (visible in the second image which is the stage before shaping any of the molecules into flower shapes). It features four different water lily molecules. Certainly many more variants are possible.

Designed and folded by me. Grid was 40×40 which is a bit small for 25 cm paper so the details are folded less than perfectly. Each molecule is 6×6 grid units, with one unit of margin on each side.

[ discuss main view on flickr ]
[ discuss pure sunk square twist tess on flickr ]
[ discuss flower molecule 1 on flickr ]
[ discuss flower molecule 2 on flickr ]
[ discuss flower molecule 3 on flickr ]
[ discuss flower molecule 4 on flickr ]

Lily pond tessellation — pure sunk square twist tess
Lily pond tessellation — close-up of flower molecule 1
Lily pond tessellation — close-up of flower molecule 2
Lily pond tessellation — close-up of flower molecule 3
Lily pond tessellation — close-up of flower molecule 4
Saturn tessellation

Saturn tessellation

Saturn Tessellation, designed and folded by Michał Kosmulski from a single sheet of Elephant Hide paper.

This tessellation is based on a 128×92 grid. Image is created from a four-sink-base tessellation by folding up some of the squares’ corners. While I’ve seen a number of designs using this molecule, I haven’t so far seen it used as a way of rendering bitmaps in paper. Since each sink-base molecule makes up four pixels, the “on” pixels are not all identical which adds some interesting variation to the image. Certainly one could use a more complex fold to distinguish the on and off pixels but I went for simplicity.

Compared to the other approaches I’ve seen so far, such as EZ tessellation or variations on square twists, this design has the advantage of being flat. However, when one looks from the side, one can get a feeling of depth like in a woodcut or another printmaking technique: the pixels look as if eaten away from the smooth background.

Following the woodcut analogy, I tried to make a print using this tessellation (through paper in order to not discolor it) but it didn’t work out. However, I think that using origami tessellations for printing is an idea worth exploring.

One of the images shows the tessellation after folding the basic molecules but before folding any corners in order to create the image (so it’s basically a pure four-sink base tessellation at that phase).

Crease pattern available here.

[ discuss main view on flickr ]
[ discuss close-up on flickr ]
[ discuss clean slate before adding image on flickr ]

Saturn tessellation — close-up
Saturn tessellation — clean slate before adding image (pure four-sink-base tess)
Super-pineapple tessellation

Super-pineapple tessellation

This is my design which extends Ilan Garibi’s pineapple tessellation. In this version, the pineapple is larger: the central pleated part is two grid diagonals long instead of one and there are five instead of three radial pleats. Apart from the margins, each molecule is folded from an 11×11 grid, so the whole 4-molecule model with margins uses a 24×24 grid.

Collapsing is a two-stage process like in the original pineapple tessellation. Sheets with larger numbers of molecules will probably be quite challenging to collapse.

Crease pattern and basic instructions are available here and a phototutorial of the collapse was published in 13th Bulletin of Polish Origami Society.

[ discuss on flickr ]

Super-pineapple tessellation
Stars and Squares tessellation

Stars and Squares tessellation

Yet another fine example of me reinventing the wheel. After I designed this pattern, I found out it had been already published — in 1982 (!) — by no less than Shuzo Fujimoto. According to the description of this model, folded by Raymonde Bonnefille, the crease pattern can be found in Fujimoto’s “Invitation to Creative Playing with Origami”. Since I don’t have the book and could not find any name for this tessellation, I refer to it by the name I planned for my “original” design, Stars and Squares.

Folding is very simple: the squares are just Kawasaki Twists and the stars are the same twists with the radial angles of the squares bisected and tucked below each other (this process is called pursing — thanks for your remarks, Robin Scholz). Each molecule requires a 4×4 grid, so the whole model featured here uses a 24×16 grid.

Folded from Elephant Hide paper, inkjet-colored magenta.

[ discuss top view on flickr ]
[ discuss close-up on flickr ]

Stars and Squares tessellation — close-up
Bed of nails tessellation — version with longer nad sharper nails

Bed of nails tessellation — long nail variant

This variant of my Bed of Nails tessellation features longer nails which have additionally been sharpened at the tips compared to the basic version.

Based on a 32×50 grid (11×10 per molecule, including margins of 1 and 2 grid units (horizontal/vertical)).

Folded from a single sheet of metallic paper.

Crease Pattern was published in Techno Origami, book of 15th Origami Outdoor Meeting.

[ discuss on flickr ]

Bed of nails tessellation — best view of the nails

Bed of nails tessellation

Bed of nails tessellation by Michał Kosmulski.

I was inspired to design this tessellation after a comment from P. Colman regarding my Fenced tiling of Fujimoto’s Clover Folding. This tessellation is based on just the “leaves” of the clover, made two-sided and arranged into multiple rows.

A single molecule uses a 6×4 grid and one can vary the borders around them in both directions in order to control the nails’ density. This particular model is based on a 21×30 grid which results in 3×5 finished nails (7×6 grid for each, so margins are 1 and 2 grid units). One can vary the nails’ length or fold the edges of the nails inside, transforming them into triangular spikes. I also see some potential for creating pixel images, in a manner similar to EZ Tessellation, by bending some of the nails.

Folded from a single sheet of Elephant Hide paper, inkjet-printed for a more metallic look.

Crease Pattern was published in Techno Origami, book of 15th Origami Outdoor Meeting.

[ discuss best view of the nails on flickr ]
[ discuss front view on flickr ]
[ discuss back view on flickr ]

Bed of nails tessellation — front
Bed of nails tessellation — back (fish-like pattern)
Fenced tiling of Fujimoto’s Clover Folding — close-up

Fenced tiling of Clover Folding

I designed this tiling of Shuzo Fujimoto’s Clover Folding after I saw the tiling by Peter Budai and thought it would be better to make the borders between molecules with the nice petals that appear on an individual clover. These petals go straight up, creating a kind of fence around each molecule.

I believe this to be a new design but please comment in case you’ve seen it elsewhere before.

[ discuss close-up on flickr ]
[ discuss top-down view on flickr ]

Fenced tiling of Fujimoto’s Clover Folding — top-down view
Down the rabbit-hole tessellation

Down the rabbit-hole

This tessellation consists of concentric square twists of growing size. The medium is self-adhesive holographic foil glued onto tracing paper. The spiral is a fractal and with larger paper could be extended indefinitely with more and more levels.

Just after I designed this piece, I realized it was so simple that someone had probably invented it before. After a bit of searching on flickr, I realized I had just designed from scratch a tessellation almost identical to the Hill/Bump model by Matt Humberstone which I had seen on flickr and faved just a few days before. It’s disappointing when what you thought was an original design shows to have been designed by others a few years before. The only difference is that in my model the initial twist is based on a 2:1 slope and Matt’s starts with a 3:1 slope. As a consequence, my version should be folded from an odd-numbered grid (the one in the picture is from 25×25 grid including an extra-wide frame). My model also lacks the locks holding the model together in the corners because it was designed up-front to be viewed from the “tunnel” and not from the “hill” side. Since it was designed to be flat, there are small additional creases that prevent it from bulging upwards. Apart from these small differences, you can use the same crease pattern as for Matt’s version.

[ discuss on flickr ]

Hydrangea Tessellation picture frame

Hydrangea picture frame

This picture frame can hold a standard 15×10 cm photograph. It consists of four molecules of the Hydrangea Tessellation (designed by Shuzo Fujimoto) spaced apart in such way that a picture can be inserted into the space between them.

My sister-in-law Bogna gave me the idea of making a photo frame when I showed her the standard low-density Hydrangea tess.

The picture is held in place only at its corners by the petals on the inner edge of the frame. The picture’s edges are not covered by the frame at all, so almost all the area of the image is visible. Despite such small tabs, the picture sits inside the frame quite securely.

The size of the hydrangea flowers and the thickness of the frame relative to the size of the area used for the picture can be made arbitrarily large or small. This requires a bit of calculation to lay out the paper well (tip: the picture’s edge lies at 13/16 of the width of the Hydrangea molecule). Precise folding is required, especially if you want to make the frame narrow relative to picture size.

If you fold the frame a little bit too small and the picture bulges when you try to insert it, you can pierce the paper edge which is hidden under the petals on the inner edge of the frame with a knife. The picture’s corners will then be able to go through the slits instead of being pressed against the paper so that a larger picture can fit in.

[ discuss on flickr ]

See also:

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