tessellation

1138: The Power of Many

| wonko

I love the geometric world of Tessellations, and have folded many. It is doubly satisfying when you design that tessellation molecule and how it tiles yourself.

This is a hex-point tessellation, and is based on a mathematical algorithm discovered by Aurélien Vermont ( @auregamiiii ) and described in a paper written by them as part of their study in Engineering. The algorithm describes a geometric construction method that lest you raise a n-finned spike from a flat surface and have the surface “heal” around it.

It does so by placing strategic dart pleats that seamlessly absorb the excess paper caused by the spike in a controlled and very flexible way. You can raise a spike at the intersection of a collection of creases (2 or more intersections). The folding gets progressively more fiddly the smaller the spike and the larger the number of intersecting lines.

I chose to derive a hex-spike, that is a 6-crease intersection spike molecule, based on a regular hexagon. Once I had derived all the creases necessary to allow one spike to be raised, I test folded it (just to check – theory and practice are sometimes at odds – some paper designs for origami seem to ignore the thickness of the paper which then breaks the symmetry or distorts the shape) and all was good.

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1084: Square Spaceness

| wonko

I have a huge pile of “must get around to folding this” models and “Square Spaceness” designed by Alessandra Lamio is one of this legion:

1084: Square Spaceness - plan view

Take a square, divide it into a 16×16 grid, lay in strategic mountain and valleys and you get this almost Escher-like tessellation molecule (meaning you _could_ put multiples of these if you had a more expansive grid with some tweaks and a bit of smush).

Charged with the confidence Advent of Tess gave me, I knew it was time to give this a whirl. There are many long slight diagonal valleys that make up the bulk of the geometry for the inward sloping spirals, and the corner widget is ingenious as a lock, and adjusting the outside pleats lets it sit flat – love it.

1084: Square Spaceness - diamond view
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1068: Russian Lilac Module

| wonko

I stumbled across the instructions for a glorious checkerboard kusudama designed by Andrey Ermakov, an insanely talented designer from Russia:

Russian Lilac unit designed by Andrey Ermakov
Russian Lilac unit

I decided to try and make ONE module – an exhausting process that starts with a HEXAGON initially divided into a 16 grid, then you dance through moves that flash and hide the reverse colour of the paper until you get this lovely pattern. This took me in excess of 2 hours!!! For ONE unit!!!!! You then crenelate and interweave them to make a spikey ball, tucking in tips to complete the tessellated surfaces.

Russian Lilac shaped to allow others to interlock.

Had I no life, and a LOT of paper, I would consider making all 30(!?!?!?!) of these things necessary to make the most complex spikey ball there is – a beauty that is not within my reach (for now) due to time pressures.

It is a timely reminder that astonishing and beautiful things come from Russia; ugly political and military action does not diminish this fact.

1058: Twister A

| wonko

Clocking on for another round of procrastigami, I decided to give the first of the “twister” series a go:

Twister A by Ilan Garibi - 2x2 molecules
Twister A – 2×2 molecules

This is “Twister A”, designed by Ilan Garibi, a lovely dimensional fold with a final twist to finish it off.

I have folded a few square twists, this one perches a twist on top of the intersection of opposing ridges, contains remarkably few folds on top of the base square grid.

Twister A by Ilan Garibi - molecule
Twister A Single Molecule

The basic molecule tiles awkwardly – because of the directionality (it forms in a clockwise direction) of the molecule, you have to reverse adjacent molecules if you want them to line up.

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1057: In ‘da ‘Hood

| wonko

Exploring Ilan Garibi’s lovely book “Origami Tessellations for Everybody”, the next “family” of folds starts off with “Childhood” and then evolves into more of the same:

childhood evolved
Childhood-Evolved (4×4 molecules)

This is almost a corrugation, as there are nearly no layers overlaying others – the surface treatment is deliciously dimensional, and the distortions are caused by paper tension and torsion of the underlying square-twists.

Childhood Molecule
“Childhood” molecule

I started with standard cotton-based photocopy paper (which for me is a LOT like thin Elephant Hide) and laid in a square grid. Both childhood and childhood-evolved use off divisions. I folded a regular division (halves or thirds), then halved until I was close to the required grid sizes, then sliced off unneeded units before laying in the wedge-shaped mountain creases.

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1056: Red Flower Tessellation

| wonko

Exploring the rich and intense world of tessellations, I decided it was time to try the second family of folds described in Ilan Garibi’s wonderful book “Origami Tessellations for Everyone”:

Redflower Tessellation Scale

This is ‘Red Flower’, the base fold of which there re many variations, but the base molecule is based on a square grid and (for single molecule at least) simple to pre-crease and collapse.

Redflower Tessellation Molecule

When you scale up, accuracy shows itself as important – slight errors mean that the internal collapses twist the whole sheet out of shape.

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1031: Cubes Tessellation

| wonko

Looking through my Origami Library, I realised I had bought “Origami Tessellations for Everyone” by Ilan Garibi back as the pandemic hit early last year, and realised I had yet to fold anything from it at all:

Cubes by Ilan Garibi
A field of cubes

Early last year was crazy times – bushfires, floods and then lockdown from Covid-19, this book got buried in my reading pile so it is time to begin the journey of exploring tessellations more formally.

Starting at the beginning, with the “Cubes Family”, this is “Cubes”, a deceptively simple tessellation of twisted cubes. I present the “molecule” – that is the tileable unit:

Cubes by Ilan Garibi Molecule
Cubes “molecule”
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Their Lives In Our Hands

| wonko

Much has been made in the media about the current bush fire situation in Australia. Truth is the scale of devastation is impossible to grasp, in terms of sheer acreage of scorched earth, number of homes lost, lives lost and livelihoods ruined. When we add the effects on environment, habitat and wildlife (flora and fauna), the effects of the 2019/2020 summer will have long-reaching and potentially permanent ramifications:

their lives in our hands

I want to say that our leaders are on top of this, but have never had confidence in politicians, and am not convinced any can see past getting re-elected to make the hard decisions necessary for our continued existence. Indeed, when our PM chooses to go on holiday during the worst of it, when he and his colleagues continue to deny climate change, despite the overwhelming scientific evidence. They display a vandalistic attitude to environmental policy, and offer reckless abandon to fossil fuels and non-sustainability.

Their lives in our hands. “They” are our children, their children, the animals and plants that make up the biosphere in which we live. The “they” are US.

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967: Spike the Echidna

| wonko

Australia is the home of many unique animals – few come odder than monotremes, mammals that lay eggs – an echidna is one such critter.

spiny norman

I had seen folds of Steven Casey’s Echidna but struggled to find a source of diagrams – only by drilling down in Pinterest did I find some copyright infringer’s scanned pages of the diagrams (sorry, I would have purchased them could I find a publication that had them) and knew I had to have a go at it.

Central to the success of this model is the lovely crop of spines – these are treated scales (much like those that adorn Satoshi Kamiya’s Ryu Jin 2.1+), a lovely “preliminary base” tessellation that I had already mastered. the rest of the model is making the surrounding paper do the work of all the other stickey-outey bits of the animal.

spiny norman views

I particularly love the snout and head, so simple but so nice. It has 4 feet, each with toes – just genius.

You fold it, the resultant shape before you collapse it into it’s end 3D shape looks a lot like a pelt – not sure National Parks and Wildlife would appreciate the notion of an Echidna Pelt, but it then becomes round and plumptious and locks together ingeniously into an adorable spikey ball full of character.

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966: Green Turtle

| wonko

As part of an origami challenge on Fakebook, we were challenged to fold the Green turtle from Origami Pro #4, designed by Jang Yong Ik :

green Turtle designed by Jang Yong Ik

I split off a square from a 70cm roll of black/natural Ikea Kraft and started the diagonal pleat pre-creasing.

Over a period of a couple of weeks, fitting it into life, the universe, and everything, I folded this intense model, really determined to enjoy the process.

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964: Tetsuya Gotani’s Ankylosaurus

| wonko

Being most of the way though the proof-reading of a new book by Tetsuya Gotani, I decided to test-fold his Ankylosaurus:

Tetsuya Gotani's Ankylosaurus

The final model, folded with 50cm printed Ikea Kraft paper, is a freaky cow-like critter covered in spikes with a club-like tail and a cow-like head.

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943: Dragon Heart Tessellation

| wonko

Researching tessellations, I stumbled across a paper, written by Helena Verrill (Queens University, Kingston, Canada) that generally introduced the concept and looked at a number of common tiling patterns, but the first CP is one I had not seen before:

dragon heart

I did a small tester and loved (fluked) the collapse, and decided to scale up to a full A3 sheet, starting with a square grid. Then nested adjacent squares are layed in on diagonals to provide odd inverse hinges.

I am quite happy with this, and if more ambitious, I would fold it much smaller on a larger sheet – it would make amazing dragon skin.

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942: Triangle Waterbomb Tessellation

| wonko

One of the interesting things about being associated with “Pinterest” is that their algorithms continually look for stuff it thinks will interest you. Given I only browse Origami, I get some interesting leads. I saw a triangle-based tessellation/corrugation and did a little digging:

triangle waterbomb tessellation

Seems Ron Resch, in the early 1970’s, was heavily into paper-based corrugation and this design emerged around then. The basis of this fold is 2 triangle grids, one at twice the scale of the other, offset at 30 degrees to the other. It took me a couple of failed attempts to get the crease layout to work but in retrospect is is much simpler than it seems.

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941: Home is where the Hive is…

| wonko

Over the last few years I have played with origami tessellations – the theory of a repeatable pattern that interacts with other repeats (molecules) is fascinating and a real testament to the accuracy of the pre-folding. As part of another project, I have been exploring triangle grids, and a devilishly tricky to collapse hex-cell tessellation by Robert Lang he calls “Honeycomb”.

Robert Lang’s Honeycomb Tessellation

After folding this a number of times, and then schematicizing the molecule, I noticed that “cells” were deep and, due to the nature of the collapsed layers inside I did not think they were very tidy nor kept their shape nicely. All to often, in origami design, paper thickness is disregarded in the theoretical collapse – in this case hiding away most of the paper in canyons between cells deforms them in ugly ways.

Original Lang molecule (right) and my shallow modified one (left) – same size paper

I started playing with the corner mechanism, and discovered I could halve the height of the cell wall, making the tuck much less bulky and doubling the size of the resultant folded field on the same bit of paper. Additionally it held itself together nicely with edges that are easy to stabilise. With a little practice (I am sure my work colleagues thought me obsessed, given the number of times I folded this tessellated field) I was ready to scale up … well, down in truth as I folded a “tiny” triangle grid on my target mustard leather-grain paper and then set the corner widgets before collapse only to then realise that folding this small was a real challenge with my nerve damaged, fat clumsy fingers.

CP of molecule (red=Mountain, blue=Valley) Thick lines are visible edges, thin are hidden
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937: Diamond Bracelet

| wonko

I hate throwing things out – having cut the biggest 2×1 rectangle possible from an A2 sheet, I was left with a lovely thin strip of scrap:

I was playing with a corrugation, with the view to design a millipede, but stumbled across a molecule that I then tessellated along the length. Continue reading →