geometry

WIP – a duo of interlinked Torusses

| wonko

For Xmas, my son Matt gave me a day a page Gary Larson desk calendar. Knowing me, I noticed the tear-off pages were square so, in a moment of madness I decided to fold something with ALL of it:

Exploring scale (and remembering the disaster in finished size that was my last attempt at a torus), I decided a PHIZZ based structure would be fun, with a torus being suitably visually interesting so set about folding units for them from the quartered pages. After 3 months I had enough (360+) to connect up the first torus, then began accumulating units for the planned second.

A longitudinal fold like this requires patience and persistence – making any part of the structure required LOTS of units, so you just gotta wait for the days to go by,

During July, I had enough units to complete the second Torus, deciding they needed to be interlinked, and began to work out HOW to connect up the units while being connected to another torus. Logistically this ramps up the difficulty exponentially – coupled with the fact that I initially connected the loop INSIDE OUT, requiring me to unpick it and invert it before re-joining it.

I am not sure if the inside diameter of the donut hole is wide enough to have 2 torusses passing through it – I _think_ it will be ok with a small amount of squishing, but I guess we will find out in 3 months.

The resultant dual torus chain is visually awesome, the geometry of 5,6 and 7-sided polygon meshes that comprise the curves are really interesting to look at, and it sort of self-supports, making display fun.

Folding over such an elongated period of time is exhausting. Keeping the model save and clean for that time so it all looks like it was made together is also a challenge I am currently working through.

Half way through the journey and i am still keen … we shall see how long that lasts.

1205: Blooming Bow

| wonko

Having international connections with folders continues to be a gift. Connecting via my Instagram account, I was approached by a new Kusudama designer Syed Faraz approached me regarding his new design “Blooming Bow”:

He asked if I could diagram it for him. I insisted on folding it first.

In folding it, I got a better appreciation for the steps (from a rough photo diagram sequence), and helped me refine the shape of the unit, capturing more accurately the sequence.

This kusudama is of the “curler” variety – it differs from others in that rather than a tab-pocket unit connection it relies on the interlinking of paper curls.

I have folded a couple of “curlers” and note that construction is generally more difficult – the units tenuously interlock until you “triangulate” and they stabilise. Getting the curl right is really important – the centre of the kusudama gets very dense as you add more 5-3 groups.

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1204: Basket Case

| wonko

Before Covid hit back in 2020, I had bought a slew of origami books, determined to work through them. One of them – “Origami Tessellations” by Eric Gjerde is one such volume that I barely scratched at the time. There is much richness to explore there, including this beauty:

This is “Basket Weave”, designed by Eric Gjerde. Folded from a 750x500mm rectangle of Kraft paper, on a fine triangle grid over the last few days.

The geometry here is mesmerising – the entire field used only 2 twists, alternating: Open Hexagon twists that rotate counter-clockwise and Closed Triangle twists rotating clockwise. The alternating rotations absorb the intersection conflicts but the folding is so dense (ie. these twists overlap) making the collapse an exercise in patience and perseverance indeed.

I struggled to find a regular rhythm when collapsing this tessellation – each molecule caused deep and awkward pleat overlaps and I could not devise wraps that would make them any less awkward. Working simultaneously on multiple molecules seemed easier as I did not bury so many axial creases while getting it to fold flat. The margins continued to be tangled.

I decided to pre-crease the open hexagons because I could derive the spacing accurately, reasoning that the triangle twists were the glue that set the hexagons in place – this I think saved me making some major blunders in collapsing, although I did notice a few times where the triangles twisted in the wrong direction making surrounding areas clash.

I am looking for a weave that I can fold VERY small, perhaps to make a lampshade of similar decorative inlay. I am not sure this is it as I am not sure i could physically fold this one much smaller without a lot more MUSH. This sheet ended up having pleats about 1cm wide – pleat width determines the size of the weave. My fat clumsy fingers would have struggled to navigate and collapse a smaller grid with this crease pattern.

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1203: Artist Book

| wonko

It is common for paper artists to “bind” their artworks into “artist books” – a broad display category that ranges from purely decorative through linear narrative forms, and everything in-between. Having recently made a linear design using cyanotype, inks and paint, it occurred to me that a book-like thing might be a fun way to display it.

Traditional “artist books”, in my observation at least, involve cutting, gluing, sometimes stitching and binding. I wondered if something could be achieved using FOLDS alone.

One “type” of artist book that I know of is a thing called a “Concertina Book”, sometimes including cut and folded “extrusions” which elevate parts of the page. I figured something like this should be possible using folding techniques, so took a scrap of paper and began a fold doodle with a simple fan fold.

Using a pair of pleats running across the fan fold gives me pleat overlaps that can then become “gussets” that then force layers up and off the resting surface in interesting ways. This makes “extrusions” that change the dimensionality of the shape.

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1202: Pineapple Tessellation

| wonko

At the beginning of the pandemic, I bought myself a copy of Ilan Garibi’s wonderful book “Origami Tessellations for Everyone”, but looking through recently I realised I had not folded many of the wonderful designs before, so set about remedying that:

This is a SINGLE molecule of the “pineapple” tessellation. After a simple set of diagonal pre-creases are imposed on a square grid, the first level collapse is really satisfying. You then have to perform a transformation (essentially turning part of the previously collapsed part inside to create the final structure.

You can then inside reverse the “scales” and you have a lovely form that resembles the body of a pineapple, kind of. The molecule tiles a number of ways – given it lies on the diagonal, you can either tile them in X or O formation – I chose to do a 4-molecule O form, just to see how difficult it was dealing with the interactions, but it turned out fairly easily.

By spacing the molecules correctly, and arranging them in an X you can create a rather lovely “Dish” that is dimensional, stands freely, has a satisfying volume and most importantly gives you free paper to shape pineapple “tops” to act as legs.

It was a fun fold, particularly if you accurately place the pre-creases, and get them in the right orientation (mountain/valley) before you attempt the first collapse. It is a terrific addition to the “what can I do with a square grid” pile.

I must explore more, Ilan’s work is well described, challenging but fun to fold.

1201: Corrugated Tubiform Trefoil Knot

| wonko

The internet (in this case Instagram) sometimes delivers to you by pure chance (or deliberate algorithm) inspiration that is timely:

A recent work (a square-tube based mobius strip) by Henk van der Vorst sparked a curiosity that led me to damaging a few A0 sheets of Kraft paper to explore a tubiform corrugation, and then work out something to do with it.

There is something interesting (for me, recently) in corrugations, and Henk’s work uses simple right-angle hinges, first documented by Paul Jackson, to use a large-scale fanfold without the tiresome necessity of reversing sections of the crease, and allowing you to curve that fanfold onto itself in an interesting way.

I discovered I could hinge on proportions of 6 and 3, making rectangular tubes that articulate and bend in very interesting (the kids would call it “satisfying”) ways.

I fashioned a bunch of different sizes to test the proportions and see just how small I could fold it reliably and accurately. On the large test folds I glued the seam – not sure why, but as I got smaller, the seam just seemed to keep itself shut and become invisible – especially when the tube was twisted.

A Trefoil knot is historically interesting – it is like a set of interwoven mobius strips, and originally was associated with the “Holy Trinity” : the Father, the Son, and the holy GOAT, or something similar. Renditions of it exist in historic engravings, statuary, heraldic depictions – even common images like the Girl Guide logo/symbol … thing.

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1200: Road To Nowhere

| wonko

I have this image in my head, of a petty little orange man, walking in circles because he has not realised he in on a flight of M.C. Escher’s stairs:

Oddly … this abstract concept is not that far from what the petty little orange man is actually doing (but, I do not really understand the lure of a golden ballroom), but I digress.

I first saw this model on John McKeever’s Flickr, and think it is a Fujimoto-style set of Escher steps, but the etymology of the model is less clear as it seems to be a variation on a clover-like tessellation, but is deliciously evil in it’s convoluted crease pattern.

I decided I had to try it, but really struggled to understand what the actual floop was going on with the crease pattern – it seemed like the prescribed creases could not co-exist. Naturally I turned to an old trick – I folded a maquette:

After a few days of twiddling with printer paper CP copies until they disintegrated, I finally found a collapse sequence that … somehow … sorted itself out by repeatedly bending back on itself. The real trick was working out which vertices go up and which go down – when you sort that out it is still counter-intuitive … until it isn’t.

I started with a 55cm square of Kraft, using a pencil I divided it into 12th, then trimmed 1 unit off 2 adjacent sides to reduce the grid to 11×11. I then used a stylus to place all the of the pre-creases, ensuring I oriented them mountain/valley as indicated. I was soooo chuffed at how CLEAN the pre-creases were, knowing how important it was to NOT mark some faces that would be solid squares in the final model.

I then had to walk away from it, as pleased as I was with the eventual success on maquettes, committing it to the actual fold is a step that made me oddly nervous.

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Torus

| wonko

When gifted a Larson a Day tear off calendar (thanks Matt), I was faced with a problem – each tear off day is a perfect square and there are 365 of them (for the year), and … I obsessively fold paper so naturally:

This is a 360 unit modular, based on Tom Hull’s Phizz unit – sort of origami lego.

The construction relies on inherent curvature of clusters of units. If you link 6 into a hexagon, the resultant shape is flat. Less than 6 units and the structure curves into a bump (ie. positive curvature), and groups of 7 or more negatively curve (like a saddle).

The basic structure is an inner strip of 6s, either side is a strip of 7s – this then forms the middle of the donut. A strip of 6s, then a strip of 5s to outcurve and then a strip of 6s to close – sounds more complicated than it is, but boy is it fiddly. Docking 3 phizz units together requires interleaving layers over a bend – when there is nothing else in the way it is simple, when there is lots surrounding it then it becomes very difficult, particularly when you cannot reach both sides of the join in the later stages of lacing it up.

The result is lovely, the geometry draws the eye. This used up what will be 1/4 of the total sheets torn off for the year – whether I keep going is up in the air at the moment – long term projects are fun so we shall see.

1199: Get Knotted

| wonko

As a paper folder, when someone tells you to “get knotted” … you have “options” – right?

I was playing around with offcuts – those inevitable slivers of paper you cleave off a sheet as you are squaring them and an idea struck.

I keep all my off-cuts, particularly those off beautiful papers – you never know when you might need some colour/texture. In the past I have added them to my paper pulp to add “thread-like” inclusions, and sandwiched them in-between translucent layers of wenzhou in paper mache sculptures etc.

I wanted to do something more “origami” oriented … so I tied a knot in a thin strip, and remembered that a flat knot resolves into a perfect (all things aligned and taut) pentagonal knot. If you string a few knots along the length then the strip does some pretty sculptural zig-zagging. I found I could decide the direction of the zig/zag by how I tied the knot, and that I could “graft” other strips on by simply knotting them there and hiding the extra end inside the graft knot.

I played around with Kraft paper strips to get my bearings, then added coloured accent strips of Hanji (purple, and green with acrylic ink spatter) and Kozo (red dry brushed with gold), knotted to intertwine like tendrils of an invasive weed. The original composition had a bunch more colours, but as I kept coming back to it, simpler seemed better so I gradually removed down to what you see here now. Initially I photographed it resting atop a sheet of my hand-made Kozo tissue because it looks classier like that. Should i ever decide to show/frame it I would prolly do the same. The geometry and composition is pleasing to me none the less.

It reminds me a little of the bold linework of Joan Miró, or the architectural geometry of Piet Mondrian, the fiddly intricate linework of Wassily Kandinsky, or the delicious geometry of Alexander Calder. We can all aspire to greatness I guess.

Origami “purists” will probably look down their noses at this because it is not folded from a square, contains multiple pieces and used some glue under each knot to anchor it to a sheet of olive Canson Mi-Teintes. That sort of folder snob can go get knotted 😛

1198: Shuriken Trunk

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I seem drawn to corrugations lately – there is something cathartic about folding such geometry, and this one, designed by Boice Wong is very satisfying to play with:

Although the CP and demo from Boice is based on a 24 gridded square, it is possible to expand the pattern infinitely on the long axis – I decided to try it as a 2:1 rectangle and found it fairly easy to fold accurately. The collapse, although a little more exhaustive, is none the less straightforward.

This corrugation is a self-sealing “tube-like” construction that folds back on itself – I think there is a more positive lock possible, but this works fine. The base structure is a crenellated plus (+) sign, that you then shape the arms using a series of inside reverse folds.

Once collapsed, and flexed a little, it becomes deliciously bendy – you can transform it in a variety of ways, twist it tightly and then it collapses back into a compact stack form – what fun.

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1194: 3 Gyroelongated Square Dipyramids

| wonko

Just trying to get my head and mouth around the name of this geometric modular provided the imperative to fold it:

Designed by Daniel Kwan, it is based around Francis Ow’s 60 degree unit, folded on a 4:1 rectangle that then has 30 degree crimps placed at thirds down the length of the paper, on opposite ends. The resulting units seem to spiral.

Units are joined in groups of 4, making a single solid descriptively called a “Gyroelongated square dipyramid” – “gyroelongated” meaning it is an extruded and twisted solid, “Dipyramid” because there is a regular square-based pyramid at each end of the solid.

Daniel illustrated they could be interwoven – 3 can be symmetrically interwined to make a visually startling whole.

The hardest part of this model was working out the symmetry of the intertwining. Merely seeing a finished one was not enough, you need to discover the scheme that, symmetrically, distributes spokes of each sold over and under, taking into account the twist, yet still meet at a pyramidal end WITHOUT overly distorting the units.

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1193: Ammonite

| wonko

There are many legendary folders out there and, thanks to the Interwebs, it is possible to connect with many of them via socials (and rare cases in the real world – wherever that is):

I am obsessed with the intricate sculptural pleat work of Goran Konjevod (@foldsome), and love playing in the space of densely pleated paper.

This piece, inspired by a piece from Goran, started as a 12:1 rectangle. With regular mountain divisions (1/2, 1/4…) until the creases were just over 1cm apart. I then successfully guestimated a tight and completely circular SPIRAL by pleating each mountain on the same angle, creating a lovely rosette.

Next, using a padded surface, on the reverse I scribed an irregular spiral track from centre out to edge.

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1191: Non-Euclidean Tttttsuru

| wonko

I have been playing with approximations of non-euclidean based geometric representations of “squares” – those shapes that have 90 degree corners. In curved space that geometry gets seriously weird, really quickly.

I operated on some standard Kami to create shapes that had 90 degree corners, but to my surprise, I managed a 2,3,5,6 and 8-sided “square”, depending on the size of the “plug” I grafted into the square.

I could then form “square bases” with these sheets – the preliminary fold is the base for many designs. Interestingly, the number of points a sheet has when put into a preliminary base is governed by the number of sides the sheet has.

Working on the 8-sided square, I then went about folding a traditional crane (Tsuru), and noticed I ended up with a surplus of appendages. With some re-arrangement I was able to return some of the classic vibe to the rear of the crane, but that resulted in 5 heads.

I have seen similar (like up to 3 I think) multi-headed cranes designed from conventional 4-sided squares, but the model efficiency is usually terrible because the point-splitting methods necessarily reduce the size of the final model exponentially.

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1190: The 5-Sided Square

| wonko

We are all very familiar with planar geometry – we see, for instance, a square or rectangle is a plain shape with all 4 corners being right angles (90 degrees). Curved space gets a LOT weirder:

It is possible, for instance, to construct a shape on a curved surface that has 5 (or even 6) corners, each having a right angle. Origami typically deals with sheets that start flat – a non-flat sheet affords fascinating properties.

After a conversation with Goran Konjevod (@foldsome), I wanted to try a technique he pioneered involving radially pleating such a non-Euclidean square.

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1188: Jeremy Shafer’s “Pyramid Tessellation”

| wonko

Few modern origamists are as prolific and inventive as Jeremy Shafer – he seems to be creating new models constantly, and most importantly, his designs are fun to fold:

This is his Pyramid Tessellation field – each molecule has pre-creases that have easy landmarks, meaning you could expand this field in any direction as far as you have patience for.

This version is a 4×4 field of 16 separate square-based pyramids – a lovely thing in itself but when you start playing with it it starts to do wonderfully weird things.

Using just the existing creases, the model flexes diagonally and also horizontally/vertically. When you flex it diagonally it turns in on itself and COLLAPSES down to a hexagonal stack – this initially broke my brain until I noticed the pre-creasing actually formed pyramidal faces that are equilateral triangles – the collapse then is just one state it can be arranged into.

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