geometry

Hedgehog Algorithm

| wonko

Eric Joisel was a genius – his sense of fun and play was amplified by his seemingly accidental discoveries. He typified himself as a “bad folder”, but no one breathed life into paper quite like him. When cleaning out my display case, I chanced upon an ancient and beleaguered fold of an “adult” hedgehog, based on Joisel’s iconic design, and decided to fold a replacement:

Looking through notes Eric himself left about the “Baby Hedgehog”, I discovered his “design” was really an algorithm – do it on a 16 grid and you get a baby, do it on a 32 grid and you get an adult – it is an algorithm because the method of raising a quill is repeated along a row. The method of adding another rank of quills is the same. Baby hedgehogs have 5 ranks of quills, an adult had 9. The leg formation, head and tail formation on both models is the same … the instructions are algorithmic.

I set about using some of a red roll I bought off Amazon (for something else, then changed my mind) – it is a scant 60cm square. It is red both sides, and takes folds quite well. You lay a 16 grid or alternate mountain/valley folds. then Diagonal grids alternating mountain and valley – this has the side-effect of, mostly, orienting all the creases you need for the base correctly. Nice.

The first rank of quills to set is the middle line. You lay as many pre-rank pleats as the model dictates before forming the first row of quills, then, it is a simple zig zag squash/collapse. Raising subsequent ranks all use the same method – popping points by liberating an internal pleat and reversing a point that is hugging the previous line of quills.

I remember being bamboozled by the collapse back when I first did it over a decade ago, but with experience you can see the process, anticipate what needs to mode and achieve each point without stressing the surrounding paper- very satisfying. Once all ranks of quills are raised, the side fans are box-pleated in half to form front/back legs. You then do something Joisel terms “True 3D”, where the body is curved via gentle stretching of the points – this subtly alters the inner gussets and locks the body in a domed shape – it takes time and a gentle tough not to distort and damage the paper – the effect is lovely however.

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1173: The Eye of Horus

| wonko

The masters of 2D side-on imagery, Ancient Egyptian Hieroglyphics and associated artefacts have always fascinated me:

This is Peter Bucan-Symons’ design, represented as a 2D colour change model (a genre that has had a recent surge of popularity), and folded from a 25cm square of Green/Gold Washi Deluxe.

This design is part of Peter’s forthcoming book “Folding Fantasy 3”, a collection of stunning complex original works, coming soon to a bookstore online.

I have not folded many 2D colour change models, but love the angular geometry, truly reminiscent of all artefact representations. The Eye of Horus is a left eye in a pair of Wedjat eyes. The right eye is the Eye of Ra, and often in modern illustration presented mirrored, curiously.

https://egyptplanners.com/eye-of-horus-vs-eye-of-ra/

Designing 2D colour change models is a specialised skill, manipulating and utilising raw edges (typically) in exotic and torturous ways.

https://www.livescience.com/archaeology/ancient-egyptians/what-is-the-ancient-egyptian-eye-of-horus-and-why-is-it-found-in-so-many-burials

This was a fun sequence, and would be easier on larger thinner paper. I found the pre-creasing folds on the Washi Deluxe disappeared, so it was harder than it would be on less textured paper.

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1173: Scarab Amulet

| wonko

I have always found ancient Egyptian symbology and art fascinating. Their attention to high graphic detail, the use of natural elements in depictions of deities, the use of gold and gemstones sublime. When I first saw the diagrams for Peter Bucan-Symons astonishing “Scarab Amulet”, I knew I needed to fold it.

From his forthcoming book “Folding Fantasy 3”, it is one of the many stand-out designs that are so terrifyingly complicated but so enticing. As part of his edit team, I also had the rare privilege of test-folding it before it has reached “the wild” as it were.

From a single 90cm square of Kraft (I keep a 90cm and a 60cm roll of Kraft in my stash all the time, they are my goto test-fold papers), via initial hex-pleat pre-creasing, we fold a many-lobed base that is then thinned, subdivided and refined to isolate, skillfully, all the necessary stickey-outey flaps in the right places.

Designing said bases is complex (at the moment beyond my feeble brain), but I like how PBS takes the time in this book to explain his methodologies, requirements and compromises for a whole bunch of the models in the book – a fantastic reference volume indeed.

Once all the flaps were isolated, the process of refining (thinning and shaping) them could begin. With so many required points, the layers really add up – managing that thickness is a real achievement. Thinner paper would help – indeed I am sure there are folders out there way more nimble fingered than me who could take it smaller, but you would need to really consider the paper choice here. Paper needs to be thin and tough (there is a LOT of sinking), it must also allow you to reverse folds cleanly (unlike a lot of laminated papers and foils that leave bunched up “kinks” when you reverse the direction of a fold).

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1171: Doomscrolling

| wonko

If you have been paying attention, you would know i am a member of PAQ – Papermakers and Artists of Queensland. in 2025 we are mounting an exhibition that explores contemporary interpretations of the scroll, entitled “On a Roll”. I decided that I wanted to mount a FOLDED scroll as one of my submissions, and envisaged a massive tessellation:

I needed a theme, and a style. For a theme, I decided to try and “tell” the progression of the first year of the recent Covid-19 pandemic … because I could see a sequence of “blossoming” outbreaks that progressively “break” regular society.

The style choice was more complex – I love the aesthetic of Lacquerware – the Chinese/Japanese technique of covering simple materials in coatings of red lacquer, texture and patterns. I also wanted to have hints of “Kintsugi” – the Japanese technique of fixing broken pottery using lacquer and gold.

I chose red/natural Kraft paper because the red reminded me of the lacquer aesthetic, and the natural grounds the work in a common/everyday material. I selectively also introduced gilded elements into the finished folded work – symbolising the “patching” of the broken world – I went for a really minimal touch here, arguing less is more. Read further….

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1170: Artichoke

| wonko

As a palette cleanser in-between gallery projects, I turned to my “must fold” pile and decided to have a crack at PRWorigami’s “Artichoke” Kusudama:

Vaguely resembling Xander Perrott’s “Conglomerate” in overall model morphology, the two designs are really different. “Artichoke” has deepish pockets and tabs, with friction locks whereas Xander’s has much more positive locking – multiple locks per unit.

The folding sequence relies a lot on alignment, making the risk of inaccuracies pretty high – certainly I got better at consistently folding angles as the unit production line progressed, but the points here rely on precision, getting pointy tidy points here is difficult.

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1169: Xander Perrott’s “Conglomerate”

| wonko

Multi-unit Kusudama folding is something I enjoy – the emergent geometry and intricate interlocking of units to make a whole is very satisfying. None more than “Conglomerate” designed by Xander Perrott:

When I saw his fold on insta I knew I wanted to fold it – I had not seen anything quite like it, so I reached out to Xander and he shared instructions – how wonderful is the internet?

Let’s break this down – the kusudama is composed of 30 units, each folded from a rectangle in the proportions of 1: sqrt(3). Each unit has a triangle grid imposed on it, with triangular gussets to allow the “facets” of the faces to become 3D.

The geometry of the unit is very pleasing to fold, it all feels really natural, and the tiny collapse of the unit to make it 3D also feels right.

Interlocking the units …. now that seems to have taken me an age – each unit cups around one lobe and inside another lobe, forming a many-layered, staggered icosahedrons – each locked into place at multiple anchor points. It took me a while to master the locking process, then re-learn it as the orientation turned this way and that, but this kusudama needs no glue, and when a unit is fully locked in it is really rigid and strong.

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1165: Kamillen – A Lovely Sunflower Ball

| wonko

I wish I remember where I found the “short” describing the unit folding for this lovely “sunflower ball”:

At the time, I practiced and learned the unit, modified it a little (removing the colour-change spiral) and then, a few weeks later decided to fold the whole thing. As I made progress in construction I realised I had NO IDEA whose design it was.

I turned to one of the many online origami communities I am a member of, and posted progress shots – fortunately it was identified as “Kamillen” designed by Irina Krivyakina.

I decided to hide the part colour change flap that causes a central spiral on each face, because I loved the shape of the geometry and thought the spiral detracted from it (my opinion only – I … have regrets).

Folded from 30 units, I decided to fold some of the 7.5cm square kami I have loads of … and … if I was to fold it again I would go bigger. The units have a lovely crenellated face, the tab-pocket mechanism is very fiddly but, when locked correctly, really positive.

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1164: Kimiro’s Ammonite

| wonko

With the development of origami design tools like Oriedta and Mu-Tsun Tsai’s astonishing “Box Pleating Studio”, origami design has in recent years gone through an explosion of complexity, innovation and artistry:

When I first saw Kimiro‘s new design for an Ammonite, I was astonished, then further amazed when he released the crease pattern – I knew I had to give it a try.

Ammonites are long extinct cephalopods (squiddy things) that lived in coiled shells – these shells are quite common fossils ranging from tiny (a few cm across) to massive (a couple of meters in diameter). Their closest living relative is the Nautilus I think. It is believed they, like the nautilus, could vary their buoyancy to control how deep they were in the ocean, and they probably had siphons that allowed them a little jet-propulsion.

Box pleating, by nature, results in blocky bases, the challenge is to make them feel rounded and organic. This model’s design has a good balance of fine detail and larger surface, and modelling and shaping it has taken me an age.

Initially I was sure the shell was fully closed 3D, but after posting some collapse and shaping progress photos on Origami Dan Discord, Kimiro himself popped into my DMs and showed me the back of his model – a half shell solved sooooo many problems. How amazing is it that in the age of the internet communication with the designer is actually possible. It continues to blow my mind how accessible origami legends are, and how helpful they are to noobs like me also.

I chose a lovely 45cm sheet of green/natural Damul matt Kraft paper (toying briefly with the idea of throwing shadow thai at it, but reasoned that I did not want a black shell or critter).

Laying in the grid was easy – finger pressure only (no bone-folded setting this time, as I wanted the creases to be as unobtrusive as possible), the CP is pretty simple, the shell spiral a lovely piece of geometry allowing a colour change.

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1163: Compleat II

| wonko

My second experiment in radial perpendicular pleating was based on a 30cm square of crispy Kraft paper, a regular 32 grid and a random number sequence:

I decided it would be interesting to see what happens when you use random numbers to control the collapse sequence for a micropleat corrugation based on a regular 32 grid of mountain folds. I typed “100 random numbers between 1 and 3” in google and blow me down but there was a website for that: https://numbergenerator.org/100randomnumbersbetween1and3

Each time you open that you get a new sequence. Mine was “3 2 3 1 2 3 2 2 2 1 2 2 1 2 2 3 3 3 2 1 1 3 2 3 1 1 1 3 3 1 1 3 1 1 2 1 2 3 3 1 2 3 3 1 3 1 2 1 3 2 2 1 2 3 3 1 1 1 1 1 1 1 2 1 1 3 1 2 1 1 1 2 1 2 1 1 1 2 2 1 3 3 3 2 3 1 2 1 1 3 1 3 1 3 2 3 1 3 1 2”.

Starting on the pleat just down from the centre line (in retrospect I wish I had started with the centre line, but… meh), I used the first 3 mountains and laid in micropleats (partial 128ths) across the sheet as uniformly as my fat clumsy, nerve damaged fingers would let me. I then rotated the paper 90 degrees clockwise, crossed out the first number and used the second number to determine how many cross-sheet micropleats to lay in. Rinse and repeat to the edge of the sheet.

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1162: Zen

| wonko

There are times when thoughts turn to the complexity of simplicity:

A few simple folds on an eccentric waterbomb base and you end up with a model that encourages deep contemplation, invoking a calm.

This is Pierre-Yves Gallard’s wonderful “Buddhist Monk”, folded simply (if not accurately) from a 15cm square of orange-white kami to simulate the otherwise vibrant saffron robes.

I am always delighted by how few folds it takes to evoke a human form – we seem innately able to recognise “people shaped” things, faces also.

A lovely exercise in restraint – this is my first fold, as I was discovering how to isolate the head and form the robes – I have no doubt subsequent attempts will be an improvement, but real life doesn’t give you a “re-do”, so here it is.

I posed it here with my Vietnamese soapstone dragon behind. The model is free-standing, it has a fold-back tab that makes it pretty stable. I like the the juxtaposition of the meditating monk playing against the hidden dragon in sharp focus, with the background blurring into obscurity.

I wonder if paper is folded in the forest and there is no one there to see it, does it remain as origami?

1159: Polly Verity Corrugation

| wonko

Polly Verity is a paper artist I have watched since she came on Instagram (@polyscene)

She has a singular style and a seemingly superhuman touch when it comes to teasing character from paper – her paper silhouettes are like nothing else.

Polly is drawn to the geometry of plain paper, expertly capturing the light and shade that corrugated paper naturally causes.

Today she posted a corrugation that LOOKS curved, but as I bathed in it’s posted beauty, I recognized it had to be a much simpler underlying box-pleated crease pattern, and I knew that I HAD to try it.

Peeling off a 2:1 rectangle from my 90cm Kraft roll (90x45cm resultant sheet), I lay in a longitudinal axis gutter (valley), then added a deeeeep zigzag. This afforded the laying in of standard box-pleat fill-in creases that I then alternated mountain/valley to make the sheet one giant accordion pleat that folds up and on itself (resulting is a stored size that is tiny – 23x2cm).

The joy of this corrugation is that when it opens up, curves emerge from the straight lines – like by magic.

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1155: Dasa Star

| wonko

Inspired by a friend and fellow folder (should out to @aboy021), I decided to throw a 60cm square at Alessandro Beber’s “Dasa Star”:

Carving a hexagon and laying in basic axial creases, initially the paper is collapsed into a tato (envelope) and then re-folded into a tato to form a pinwheel structure as the base.

Then, in a process reminiscent of the algorithmic fractal sequence of Shuzo Fujimoto’s “Hydrangea”, we go through processes of teasing paper until it is no longer free, then flipping over and feeding more paper through the middle structure in a “paper pump”, then flipping over and teasing again.

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1153: Tanteidan “Cubes”

| wonko

At any moment, I have a half-dozen Tanteidan magazines, from my subscription to the Japanese Origami Society, on my chair-side table yet to be filed:

Flicking through them, it is impossible not to be intrigued by the challenges, fold tidbits, crease patterns and full diagram sequences.

Sadly, I contrast it to British Origami Society Magazine – I used to subscribe but let that membership lapse when I got to the situation that I folded nothing from them for 4 issues in a row.

These models are modular “cubes” designed by Jun Maekawa – delicious geometric puzzles with radically different design methodologies. The “Cat Ear” cube is a 6-module cube that is related to the “Business Card” cube I have made into a huge Menger sponge.

Paper friction and tabs and pockets offer structural strength, properly interleaving the tabs make it a really stable piece of geometry.

The “Zig Zag” cube is different – it uses 2 pieces of paper total – each “half” of the cube is folded complete (internal and external surfaces) from a sheet of Duo paper. Ingenious in design, fiendish the first time you try to link the 2 halves, delightful together and apart.

I love these little gadgets – every Tanteidan has at least 2, I have barely scratched the surface of them.

Short-Beaked Echidna

| wonko

There are only a few origami figures I MUST have in my collection – Steven Casey’s “Echidna” is one of these:

This adorable little monotreme is covered in one of my favourite square-grid tessellations, but skillfully crafted to allow all the other body bits to be where they need to.

I bought the British Origami Society booklet describing how to fold this treasure as soon as I knew it existed, and have folded it a few times now. Some sequences are nightmare fuel – this one is just so enjoyable to fold.

I recently received a shipment of paper from Origami-shop.com and in it was a 65cm 11 colour pack of the NEW Shadow Thai paper. I last bought it in 40cm square form but it was THICK so to my delight this version is thinner and takes complex folds really nicely. I chose this fur-like colour because it most closely matched the quill and hair colour of an echidna.

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1150: Star Katrina

| wonko

As a closet botanist, I am interested in floral geometry – many flowers are based on pentagons:

This is “Star Katrina”, a beautiful kusudama designed by Xander Perrott. Folded from 30 x 2:root 3 rectangles cleaved from squares of Tuttle Indigo dye duo paper over the last couple of days.

The unit is based on a tight triangle grid – fairly easy to fold accurately and the locking mechanism is so positive that this kusudama is held together via paper tension and friction only (no glue, truly, none).

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