I had to teach myself a bit of Python recently, so decided to do it by implementing this thing (it’s uploaded here as a text file). It takes a nice intuitive mark-down for phylogenetic trees (below) and spits out a Newick file (-f nwk), or a Wikipedia clade format (-f wp). It’s not amazing (hey, it was my first attempt at Python in anger) and I know there’s a bunch of repetitious stuff in the functions that could be lifted out, but whatever. Maybe you’ll find a use for it.


Concrete jungle

The Barbican is a sprawling concrete brutality as apt to divide opinion as Marmite or skinny jeans. Built in the 60s and 70s as a housing estate and cultural centre on a vast bomb-site in the City of London, it does not sound like a promising place to go botanising, but after the fire-weed was concreted over, other plants moved in…

Erigeron karvinskianus [CC-BY-SA-4.0 Steve Cook]

Fleabane (Erigeron karvinskianus) growing on a bastion that formed part of the original London Wall

Aside from the fleabane gussying up the scheduled monuments, and the friendly rivalry of the residents’ window-boxes, the Barbican also houses a large conservatory containing more than 2000 species of plant. It is open to the public most Sundays.

Barbican glasshouse [CC-BY-SA-4.0 Steve Cook]

Barbican Conservatory: main glasshouse

A highlight of the conservatory complex is the arid-zone glasshouse, which crams a surprising variety of cactuses and other succulents into quite a small space. Like many botanerds, my sickness started in childhood, when I broke out in a rash of succulents and carnivorous plants – as I enter my dotage, nostalgia grips me and I am drawn back to these plants like a homeopath to idiocy.

Barbican arid zone glasshouse [CC-BY-SA-4.0 Steve Cook]

Barbican arid zone glasshouse

Is this a botanic garden? Can it be included in my baggings? As far as I’m concerned, the key difference between a garden and a botanic garden is whether the plants have merely been plonked there to be gawped at, or whether the plants are properly labelled with Linnaean binomials and can thus be ticked off a list as God intended:

Lophophora williamsii [CC-BY-SA-4.0 Steve Cook]

Labels: check. Lophophora williamsii

The plants are labelled; Flora be praised; this is a botanic garden.

Anizoganthos [CC-BY-SA-4.0 Steve Cook]

Kangaroo paws (Anizoganthos) – very on-trend.


When we went to the Berlin Botanischer Garten in 2005, we saw some pretty cool stuff,  including a flowering Welwitschia and some fantastic tree ferns. However, like every other botanical garden I’ve ever visited, mosses , hornworts and liverworts were effectively ignored, or – worse – treated as dreary also-rans at the start of one of those ‘evolution houses’ so beloved of curators without the wit to find something accurate to do with their cycads.

Conocephalum conicum [CC-BY-SA-4.0 Steve Cook]

Most large glasshouses will have a crop of the liverworts Conocephalum conicum (above) or Marchantia polymorpha in them somewhere. Of course, glasshouse curators all think these are weeds; however, I like to think of them as a middle-finger to every so-called botanist who thinks plants are only plants if they’re wafting their genitals at some poor insect or other.

The main problem with displaying mosses is that they’re mostly quite small, so you need to be up close to appreciate them. Berlin have solved this by creating a reverse sunken-garden: you walk down some steps into a basin, around the edge of which are arrayed the mosses. It’s a lovely idea, spoilt only a little by the netting they have put over the planted area at ground level to stop people from trudging over them in their pursuit of angiosperms. For my long-suffering companion this was too much: “It looks like the backyard of an abandoned bungalow”. Tsk. Sticks and stones… are all just substrates for them to grow on.

Racomitrum canescens [CC-BY-SA-4.0 Steve Cook]

Racomitrum canescens, the hoary fringe-moss

Although most mosses are quite small, a few conduct water sufficiently efficiently up their stems to grow much larger. Some of the mosses in the order Polytrichales can grow to almost a metre tall, but even the smaller ones from this order, like the smoothcap below, are still rather chunky and striking:

Atrichum undulatum [CC-BY-SA-4.0 Steve Cook]

Atrichum undulatum, the common smoothcap

The leafy green bit of a moss that mostly attracts your attention is haploid – that is, it contains only one copy of each of its chromosomes in each of its cells. This is unlike most of the cells in your body (and those of most plants), which are diploid, with two copies of each chromosome. The only part of a moss that is diploid are the little stalked capsules that you can see growing out of the leafy mass in the photo above. Those diploid capsules make spores through the specialised cell division process that reduces diploid cells to haploid cells. This process – meiosis – is the same one that occurs in your ovaries or testicles. The result is a mature capsule full of haploid spores, which drift away on the wind and eventually germinate into the leafy green body of the moss. It’s the leafy green part that then makes the eggs and sperm. The sperm swim to another moss plant, fertilise its eggs, and the diploid cell from that fertilisation then grows up to make the capsule.

This life-cycle is inside-out compared to the way humans reproduce. The only cells in your body that are haploid are your eggs and sperm, and they never have an independent existence. So that whole section of a moss’s life – from spore to leafy plant to eggs and sperm – is condensed into just the eggs and sperm of a human life-cycle. If we reproduced like mosses or liverworts, we would all start as a kind-of unfertilised ‘egg’ that grew directly into a big mass of haploid cells. That blob would then sprout vaginas and penises all over its surface. The penises would fertilise nearby vaginas, and then a diploid baby would grow directly out of each one. That baby would have a gigantically long neck, and when it hit puberty, its head would explode in a shower of eggs to restart the cycle.

Biology is simply disgusting.

Hypnum cupressiforme [CC-BY-SA-4.0 Steve Cook]

Hypnum cupressiforme, the cypress-leaved plaitmoss

At the Moosgarten in Berlin there is a vending machine where you can hire a small magnifying glass to  appreciate the details of this pornographic cycle.

Tortula ruralis [CC-BY-SA-4.0 Steve Cook]

Tortula ruralis, star-moss

Tragically, there was one promised – but missing – bit of phylodiversity. The moss mugshot guide promised a hornwort, but there was no Anthoceros or Phaeoceros to be seen. These plants aren’t technically mosses (or liverworts), but something slightly more closely related to ferns and flowering plants (probably). My search continues.

Another overlooked part of the vegetable empire that Berlin showcases brilliantly are the spikemosses and clubmosses. Despite the name, these are not mosses at all, but are sister to the huge group of plants that contains all the ferns, conifers and flowering plants. Clubmosses are no longer a major player in the world’s ecosystems,  but during the Carboniferous (300 million years ago) their relatives were a dominant part of the forests that formed the coal we seem so keen on burning.

Sigillaria sp. [CC-BY-SA-4.0 Steve Cook]

Sigillaria sp. – the trunk of a giant clubmoss: this is at the new main hall of the Natural History Museum in London, rather than in Berlin

The true clubmosses include Lycopodium, whose spores once found use as fingerprint powder, but which is now mostly used (if at all) as a pyrotechnic flash-powder in magic acts.

Lycopodium hippuris [CC-BY-SA-4.0 Steve Cook]

Lycopodium hippuris

Their relatives the spikemosses may be a little more familiar: the genus Selaginella includes one of those resurrection plants you can occasionally buy whilst exiting through the gift shop, and Krauss’s spikemoss, a plant from the Canary Islands that is becoming an option for outdoor ground-cover in the UK because of climate change: it’s hardy down to 5°C (and has the RHS Award of Garden Merit, lah-dee-dah) but I’ve managed to overwinter it in a sheltered spot in London despite it getting covered in snow this year.

I get to appreciate spikemosses in my garden only because burning the remains of their fossilised relatives is hastening us on the way to extinction. There’s probably some lesson to be learnt here.

Selaginella umbrosa [CC-BY-SA-4.0 Steve Cook]

Selaginella umbrosa

As you can see above, the leaves of spike-mosses look rather fern-like, but what you are probably identifying as leaves are actually whole boughs of tiny scale-leaves attached to a much-branched stem. The resemblance to ferns is not coincidental though: the leaves of ferns (and of conifers and flowering plants) are probably derived from a much-branched stem – but with fused webbing rather than individual overlapping scales.

Some of the spikemosses are really extremely pretty: this sumptuous specimen seems to be a William Morris.

Selaginella erythropus [CC-BY-SA-4.0] Steve Cook

Selaginella erythropus

Botanerd highlights (as if more were needed)

This picture isn’t going to win any awards because the plant  was trapped in a locked-off area of the glasshouse so I had to take the photo from a looong way away, but – drumroll, please! – here is an actual Amborella trichopoda, in the flesh, as it were. This plant is in the sister group of all the other flowering plants – indeed, is the sister group, as there’s only this one species in it. If you don’t think this is the most exciting thing you can experience on a holiday to Berlin in early September, then I pity you.

Amborella trichopoda [CC-BY-SA-4.0 Steve Cook]

Amborella trichopoda. Don’t bother clicking – this is about the maximum size the resolution can manage!

Here be dragons

Previously on Bagging Botanic Gardens: Maspalomas et al.Wisley, Brussels, Down House, Much Else Besides.

The minute I found out there was a botanic garden in the north of Gran Canaria as well, a return visit became inevitable.

The Jardín Botánico Canario “Viera y Clavijo” is in the outskirts of Las Palmas, and is much larger than its Maspaloman sibling. As well as showcasing a Canary Island pine woodland, it also has a impressive path carved into the side of the cliff that overlooks the site. This affords splendid views across the gardens – assuming you can stomach them.

Jardin Canario [CC-BY-SA-4.0 Steve Cook]

Jardín Canario: view across the gardens from the aptly named “steep stairs” of the cliff-side path

The cliff-side path is planted with lots of native flora, including the endemic maple-leaved mallow (Lavatera acerifolia):

Lavatera acerifolia [CC-BY-SA-4.0 Steve Cook]

Lavatera acerifolia

haresfoot ferns (Davallia canariensis):

Davallia canariensis growing epiphytically on a palm [CC-BY-SA-4.0 Steve Cook]

Davallia canariensis growing epiphytically on a palm

and especially, a whole weyr of dragon-trees (Dracaena draco):

Dracaena draco pathway [CC-BY-SA-4.0 Steve Cook]

Dracaena draco – dragon-tree pathway

Entry is free!

Botanerd highlights: the cactus and succulent garden is fantastic. It is large, and has many mature specimens of cactuses, Euphorbia, Pachypodium, and even a Madagascan Alluaudia.  Some of the specimens are vast: this Pachycerus weberi is about 8 m high, and many of the Euphorbia are similarly imposing.

Pachycereus weberi [CC-BY-SA-4.0 Steve Cook]

Long-suffering husband with Pachycereus weberi cactus for scale

Hairy kneecap

Our microscopy practical always turns up something new, which is the main reason I enjoy it. This year it was a ciliate called Euplotes patella. This kneecap-shaped critter is a single-celled organism masquerading as a tiny animal. The appendages that look like long hairs or legs are bundles of extra-long cilia called cirri, which it uses to scuttle about like a microscopic woodlouse. The cirri around the front of the cell (top of the image below) are also used to sweep food into a region of its cell membrane that acts like a mouth. Euplotes will eat bacteria, yeast, algae, pretty much anything it can fit into its mouth really: much like me at Christmas.

As Euplotes is quite fast-moving and camera-shy, it was tricky to get a good image. Fortunately, one of my students got a good picture, and kindly agreed to let me license it under Creative Commons: thank you Yikai!

Euplotes patella [CC-BY-SA-4.0 Yikai Feng]

Euplotes patella [CC-BY-SA-4.0 Yikai Feng]

Stems all the way down

In the days when potted plants were more than just disposable land-filler, my Nan would try to force her wizened poinsettia to flower by sticking it into a bin-liner with some apples every night from October until December. Sum total of flowers she ever saw: zero. Ho hum. But how many poinsettia flowers can you see in the picture below?

Euphorbia pulcherrima [PD, USDA Scott Bauer]

Poinsettia (Euphorbia pulcherrima)

If you’re wondering whether you can include the ones that are partly off-screen, you are asking the wrong question. There are many hundreds of flowers here, not six or ten. The red leafy-looking things you might have been cruelly mislead into thinking were petals are actually brightly coloured leaves. The flowers themselves are found in the tiny yellow-green blobs in the middle…

Euphorbia pulcherimma cyathia [CC-BY-SA-2.5 André Karwath]

Poinsettia cyathia [CC-BY-SA-2.5 André Karwath]

…but they are not – as it turns out – the blobs themselves. Each one of those blobs is itself a bunch of tiny petal-less flowers: one petal-less female flower, and a bunch of tiny petal-less male flowers consisting of just one anther each. This structure – a cyathium – is found in all the species of Euphorbia, the genus to which the poinsettia and 2000 other species of spurge belong.

Cyathium [CC-BY-SA-3.0 Ies, Aelwyn]

Cyathium of a different species of Euphorbia, but showing the single female flower, and here two males flowers [CC-BY-SA-3.0 Ies, Aelwyn]

In one of the close relatives of the poinsettia – the crown-of-thorns – two coloured leaves surround just a single cyathium, making it a flower-squared: a bunch of tiny flowers masquerading as a single bloom:

Euphorbia_milli [CC-BY-SA-3.0 Ezra Katz]

Eight crown-of-thorns cyathia [CC-BY-SA-3.0 Ezra Katz] – as the author says on the upload page on Wikipedia “a close up of Euphorbia milli cyathia (not of the flowers!)”

This means that the poinsettia ‘flower’ is really a flower-cubed: it is a set of tiny petal-less flowers grouped together into a cyathium, and then a bouquet of cyathia are grouped together and surrounded by coloured leaves. It is a flower reinvented – twice!

The cyathium is one of many re-invented wheels in the business of making flowers. All the 33 thousand species in the daisy family have flowers-squared, as do all of the arum lilies. The flowering structures of grasses are flowers-squared or cubed according to taste.

Silybum marianum [CC-BY-SA-4.0 Steve Cook]

Thistles (like this Silybum marianum) and all the other daisy family have flower-heads that are composed of many very small flowers clustered together and – in this case – surrounded by spiky rather than ‘pretty’ leaves.

Deeper down the rabbit-hole…

When I said the ‘red leafy-looking things’ on a poinsettia weren’t really petals, I wasn’t lying, but from an evolutionary point of view, that’s all a petal actually is. Petals are a ring of leaves that became useful for attracting pollinators in some long-dead ancestor of the flowering plants.

Something similar applies to the male anthers and female carpels at the centre of a flower: these are also modified leaves, wrapped around the tissue that makes the pollen and egg-cells. So even a single real flower can be seen as a cluster of smaller male and female ‘florets’ (botanists call these sporangia), themselves made of rolled leaves, and surrounded by yet more (prettily coloured) leaves. A flower is a leaf-squared, so a poinsettia ‘flower’ is a leaf-to-the-fourth-power.

Liriodendron tulipifera [CC-BY-SA-.4.0 Steve Cook]

Tulip-tree magnolia (Liriodendron tulipifera) – some pretty leaves wrapped round some sexy leaves

Oh, but we’re not done yet.

What is a leaf? Well, if you hold up the leaf of most land plants to the light, you’ll see a lot of veins. Depending on what you’re looking at, these may be arranged in neat parallel lines, or in a tree-like branching network.

Cotinus coggygria [CC-BY-SA-4.0 Steve Cook]

A tree within a tree (Cotinus coggygria)

The latter kind gives you a clue as to what a leaf really is. A leaf is a much-branched but flattened stem with tissue stretched out across it like the webbing between the toes of a duck.

So if a leaf is a stem-squared, then a poinsettia ‘flower’ is a stem-to-the-fifth-power:

  • Start with a stem
  • Group a bunch of stems together with some webbing to make a leaf
  • Group a bunch of reproductive leaves together with some pretty leaves to make a flower
  • Group a bunch of simplified flowers into a cyathium
  • Group a bunch of cyathia together with some pretty leaves to make a poinsettia

Plants: what you get when you take one good idea and run with it for 500 million years.

More botanical baggings

Previously on Bagging Botanic Gardens: Wisley, Brussels, Down House, Much Else Besides.

Eden Project

An old clay pit in Cornwall doesn’t sound like the most promising place to find a tropical forest, but the alleged proximity of the entrance to Magrathea might go some way to explaining it. The Eden Project’s bubble-wrap domes are now such a familiar part of the Cornish tourist board’s marketing that you’re not really expecting them to be as impressive as they actually are in the flesh:

Eden Project [CC-BY-SA-4.0 Steve Cook]

The domes are made of plastic. The left side houses a tropical biome; the right Mediterranean.

Inside the tropical biome there is a tree-top walkway that takes you up a stairway suspended in mid-air to a platform suspended in the same. This platform sways from side to side at the exact resonant frequency of the human bowel. There was probably a view. I don’t really recall.

Eden walkway [CC-BY-SA-4.0 Steve Cook]

Stairway to heaven. One way or the other.

As it was the depths of winter when we visited, there was not all that much happening outside in the Cornish biome, but the clean air and high humidity supports a lush growth of mosses and lichens on every available surface.

Eden lichens [CC-BY-SA-4.0 Steve Cook]

Tickets are a breath-taking £25+ per person, and I should imagine pre-booking is sensible in the summer months.

Botanerd highlights: my memory is terrorised, fragmentary. Some palms. Definitely some rubber trees. Jewelled crabs?

Westonbirt Arboretum

On the way back from Cornwall, we dropped in at the national arboretum in Westonbirt, which is run by the Forestry Commission. We’d been there once before for a wedding, but hadn’t had much time to explore the grounds. As previously noted, it was the depths of winter, so quite a lot of the trees were dormant, and my photo-roll is a bit scant. However, there are several areas planted heavily with conifers, so I spent a happy couple of hours running about trying to find fallen cones to add to my collection. I would pay good money for the cone of a Taiwania cryptomerioides (seriously – call me), but – in keeping with most of specimens of this species in the UK – Westonbirt’s coffin tree was far too young to have any. Oh well.

Taiwania cryptomeroides [CC-BY-SA-4.0 Steve Cook]

Taiwania cryptomerioides, the coffin tree

Tickets are £15.

Botanerd highlights: trees. Lots and lots of trees. If we’d timed it better, I would have the pretty pictures to wax lyrical about the 79 ‘champion’ trees they have, which are the biggest of their species you can find in the UK. But it was too bloody cold, so I shan’t.


In pre-emptive defence, Las Canarias = winter sun ∩ not-imprisoning-the-gays ∩ relative affordability.

Eschewing the fleshpits of the Yumbo Centre, we walked for mile after sweaty mile under the very sun we had paid to enjoy, to the Parque Botánico de Maspalomas. Only to find that Epiphany is still a thing on Gran Canaria, and so a glimpse through some locked gates was all we were likely to get:

Maspalomas botanic gardens [CC-BY-SA-4.0 Steve Cook]

Jesus says “no”

Never one to let an unlikely man-god get in my way, we tried again on the morning of our flight home. The gardens are all outdoors, and showcase sub-tropical plants, including a number of the Canary Islands’ wide range of endemics. The Canary Island spurge – which looks like a cactus but isn’t – is everywhere on Gran Canaria, and looks absolutely gorgeous in the sunshine. The photo below is not at high enough resolution for you to see that is it also covered in spider webs. Hundreds and hundreds of them, each with an attendant angry arachnid. The stuff of nightmares. Yours obviously, not mine. My nightmares are wholly taken up with falling to my death from great heights.

Euphorbia canariensis [CC-BY-SA-4.0 Steve Cook]

Canary Island spurge (Euphorbia canariensis)

Entry is free! In addition to the botanic gardens, the Paseo los Gatos Rabiosos (officially, the Paseo Costa Canaria, and Calle las Dunas) have some pretty nice planting along them, including a lot of Norfolk Island pines (actually monkey-puzzles), none of which – tragically – had cones within scrumping distance. The latter street also takes you to the edge of the amazing dune system and nature reserve behind the beach.

Araucaria heterophylla [CC-BY-SA-4.0 Steve Cook]

Araucaria heterophylla, the Norfolk* Island Pine**. *Nowhere near Norfolk. **Not a pine.

Botanerd highlights: along with the native spurge, the Madagascan traveller’s ‘palm’ Ravenala is the Maspalomas go-to for generic corporate planting. However, the one at the botanic gardens was considerably more majestic than most of the hotel frontage offerings.

Ravenala madagascarensis [CC-BY-SA-4.0 Steve Cook]

If you’re trying to work out how Ravenala madagascarensis ends up looking like this, think of it as a neatly flattened banana plant growing on top of a palm-tree trunk and see if that helps.

Wakehurst Place

We timed this final UK visit a bit better. Wakehurst is the country outpost of Kew Gardens, and home to the Millennium Seed Bank, which aims to preserve the seeds of 25% of the world’s seed plants by 2020.

Fritillaria meleagris [CC-BY-SA-4.0 Steve Cook]

Snake-head fritillaries (Fritillaria meleagris)

Being a contrary dick, I went hunting for plants that don’t bear seeds to photograph. Here is Cthulhu escaping from R’lyeh.

Osmunda regalis [CC-BY-SA-4.0 Steve Cook]

Royal ferns (Osmunda regalis) – crosiers grasping for your soul

And some mosses: diploidy is for wusses.

Wakehurst mosses [CC-BY-SA-4.0 Steve Cook]

Mosses. I really need to put my money where my mouth is and go on a bryophyte identification course.

Tickets are £12.50, but factor in an unexpected £10 for the car-park too, unless you fancy a 6 mile walk (or a once-every-two-hours bus) from from Haywards Heath railway station.

Botanerd highlights: the huge stand of royal ferns above are reason enough alone to visit, but Wakehurst’s pinetum also has some interesting species: Keteleeria evelynianawhich I’ve seen nowhere else before, and a lot of small firs which have beautiful immature cones with brightly coloured bract scales at this time of year.

Abies squamata [CC-BY-SA-4.0 Steve Cook]

The flaky fir (Abies squamata). Lovely cones. Dreadful branding.

Modular origami

A few months ago, I went to a creative origami lunchtime session organised by some lovely people at $WORK. I’d done origami a bit when I was younger, but mostly just frogs and cranes, which have since helped me while away the hours when invigilating exams. However, at this lunchtime session I was shown how to make modular origami. This involves making lots of (generally quite simple) origami parts, and then slotting them together to make larger structures.

Sonobe origami assembly [CC-BY-SA-3.0 Steve Cook]

I went home with a simple 12-unit Sonobe ball that afternoon and was very pleased with myself.

Cumulated octadedron (sonobe) [CC-BY-SA-3.0 Steve Cook]

Entry-level modular origami: the 12-unit Sonobe ball. Mathematically, it’s a cumulated octahedron; practically, it’s 12 sheets of square paper and about 1 hour of your time.

Things have rather escalated from there.

Assorted modular origami [CC-BY-SA-3.0 Steve Cook]

Between running some of these lunchtime sessions myself now, and being asked on several occasions on Twitter about how I make the pretty things I keep tweeting, I thought it’d be useful to put together a quick guide (or a link-farm, at least)..

Sonobe units are very easy to fold, quite forgiving, and can be used to make a cube (6 units), a cumulated octahedron (12 units), a cumulated icosahedron (30), and a kind-of truncated icosahedron (90, basically a spiky football). They’re a pretty good introduction to the general principles:

Assorted (sonobe) [CC-BY-SA-3.0 Steve Cook]

Sonobe family: 90, 30, 12, 6 and 3 units. The 3-unit one is a trigonal bipyramid but barely counts! These have all been made with the slightly modified unit mentioned below. The 90 units one is the biggest Sonobe that’s really worth making IMHO: about 3 hours’ work

The 30-unit ball has the symmetries of an icosahedron (or dodecahedron). Once you’ve learnt how to construct that object in Sonobe modules, you’ve essentially learnt how to construct any 30-unit modular origami ball: they mostly involve slotting 30 edge-units into groups of three to form the 12 pentagonal faces of a dodecahedron (or equivalently/alternatively, slotting them into groups of five to form the 20 triangular faces of an icosahedron – the difference is mostly one of perspective).

Cumulated icosahedron (sonobe) [CC-BY-SA-3.0 Steve Cook]

Cumulated icosahedron made of Sonobe units: 30 sheets of paper, and – if you’ve got the hang of the 12-unit ball – still only 1 hour of your time

There are lots of variations on the Sonobe unit you can (re)invent, by adding back-folds that expose the other side of the paper, or that make the tabs narrower than the pockets, giving a more intricate look.

Cumulated icosahedron (sonobe variant) [CC-BY-SA-3.0 Steve Cook]

Cumulated icosahedron made of slightly modified Sonobe units

Although the 90-unit structure is quite stable, the next one up (270 units) tends to sag under its own weight over time, but by that point it felt like a right of passage to make one.

270 (sonobe colourchange) [CC-BY-SA-3.0 Steve Cook]

9 hours construction, plus some planning. This uses duo paper, which is coloured on both sides, and a modified Sonobe unit that has a reverse fold to expose the other side of the paper in each module.

The Sonobe units can also be assembled inside-out to make inwardly cumulated polyhedra…

Cumulated postive negative icosahedra (sonobe) [CC-BY-SA-3.0 Steve Cook]

Getting the last few units in on the inverted ball (left) is tricky.

…and they can also be assembled in pairs and then assembled into a spiked pentakis dodecahedron...

Pentakis dodecahedron (sonobe colourchange pairs) [CC-BY-SA-3.0 Steve Cook]

Pentakis dodecahedron, with a reverse-fold Sonobe unit that shows the other side of the paper.

…and other structures.

Rhombic triacontahedron (sonobe colourchange pairs) [CC-BY-SA-3.0 Steve Cook]

The site above describes this as a rhombic triacontahedron, but I’m pretty sure it isn’t. I’m not sure what it actually is though. Has both colour-change and the units are assembled ‘inside out’ to make it inwardly cumulated.

The next unit I tried was the Penultimate edge unit (attributed to Robert Neal), which can be used to make a wireframe dodecahedron, as demonstrated by Matt Parker, the stand-up mathematician. Other variations of this subunit can be used to make pretty much any other wireframe polyhedron.

Dodecahedron (penultimate) [CC-BY-SA-3.0 Steve Cook]

Dodecahedron. I was trying to use up the boring coloured paper on this one, but I quite liked the result in the end!

Thomas Hull’s PhiZZ edge unit makes similar wireframe structures, but the modules fit together more tightly and the resulting structures are much more robust than you get with the penultimate modules.

Truncated icosahedron (phizz) [CC-BY-SA-3.0 Steve Cook]

Truncated icosahedron – this is basically the shape of a football (12 pentagons, surrounded by hexagons) and of some viral capsids too.

You can also make colour-change variants using the technique shown in Lewis Simon’s decoration boxes.

Dodecahedron (phizz colourchange) [CC-BY-SA-3.0 Steve Cook]

Dodecahedron made from PHiZZ units with a colour-change.

For structures based on dodecahedra/icosahedra and made from edge-units, you can always get away with using just three colours and never have two of the same colour pieces touching. This is because you can draw a Hamiltonian circuit on a dodecahedron: that is a path from vertex to vertex that only visits each vertex once, and which comes back to where it started. You can represent this in 2D on a Schlegel diagram.

Hamiltonian circuit through a dodecahedron [CC BY-SA 3.0 Tomruen/Steve Cook]

Hamiltonian circuit through the Schlegel diagram of a dodecahedron [CC BY-SA 3.0 original by Tomruen, modified by Steve Cook]. The red and purple edges form the Hamiltonian circuit; the grey edges are what is left over. You’ll notice that every vertex has one of each of the three coloured edges. The diagram is a projection of a dodecahedron: imagine taking a wireframe of the dodecahedron and shining a torch through it: the Schlegel diagram is the 2D shadow this 3D polyhedron casts on the wall. It’s fairly easy to work out which edge in the 2D diagram correspond to which edge in the thing you’re building.

If you colour alternate edges of the Hamiltonian circuit in two of your chosen colours, and the rest of the edges in the third, then you’ll avoid having any colour-clashes. I only learnt this after I started making these structures, so not all of them have this optimal colouring! The same 3-colour rule is true for the other Platonic solids, and also for the truncated icosahedron.

Francesco Mancini’s star-holes kusudama uses a similar module to the PHiZZ, but with a little back-bend that gives a nice 3D star effect. This one is a dodecahedron-shaped (30 units), but a 90-unit truncated icosahedron should also be possible.

Star holes dodecahedron [CC-BY-SA-3.0 Steve Cook]

Star-holes dodecahedron.

UPDATE: yes, it is possible 🙂

Star-holes truncated icosahedron [CC-BY-SA-3.0 Steve Cook]

Star-holes truncated icosahedron

Lewis Simon and Bennett Arnstein’s triangle edge unit can be used to make very nice patchtwork tetrahedra, octahedra and icosahedra.

Icosahedron (triangle edge) [CC-BY-SA-3.0 Steve Cook]


They’re a bit fiddly to put together but are very robust once constructed. A similar patchwork effect for the dodecahedron can be achieved with M. Mukhopadhyay’s umbrella module; Sonobe units can be used to make analogous Battenberg-cake style cubes.

Assorted (triangle edge sonobe umbrella) [CC-BY-SA-3.0 Steve Cook]

Battenberg-cake Platonic solids. The dodecahedron is made from umbrella units; the cube from Sonobe. The tetrahedron, octahedron and icosahedron are all made from triangle edge modules.

The simple isosceles triangle unit (attributed variously to M. Mukhopadhyay, Jeannine Mosely and Roberto Morassi) can be used to make small and great stellated dodecahedra.

Small and great stellated dodecahedron (isosceles) [CC-BY-SA-3.0 Steve Cook]

Great (left) and small (right) stellated dodecahedra.

The small stellated dodocahedron is particularly pleasing and makes a fairly robust decoration if made of foil-backed paper.

Small stellated dodecahedron (isosceles) [CC-BY-SA-3.0 Steve Cook]

Xmas decs

The great stellated dodecahedron can be made from the same subunit, but is tricker to construct because a tab has to curl around into a pocket that is partly inside the next tab round. I used needle-nosed forceps to construct this, and I’m still not terribly happy with the result.

The opposite is true for Paolo Bascetta’s star module, which makes a great great stellated dodecahedron, but a rather *eh* small stellation. This module needs duo paper (i.e. paper that is coloured on both sides) for best effect.

Small and great stellated dodecahedron (star) [CC-BY-SA-3.0 Steve Cook]

Great (left) and small (right) stellated dodecahedra.

Dave Mitchell’s Electra module can be used to make a icosidodecahedron: it’s unusual in that each module corresponds to one vertex of the structure: the edge units described up to this point combine together to make each vertex.

Icosidodecahedron (electra) [CC-BY-SA-3.0 Steve Cook]

Icosidodecahedron made from Electra modules

I’m not that happy with my Void kusudama (Tadashi Mori): I should have used duo paper, but it was really tricky to put together. Maybe one day. It’s one of the few structures here that is back to the original octahedral/cubic 12-unit structure. I’m not sure the 30-unit version would be stable.

Octahedral void [CC-BY-SA-3.0 Steve Cook]

Octahedral void

UPDATE: Yeah, I don’t think the 30-unit version is do-able. I think the units are too wide to actually fit into an icosahedron: I couldn’t even manage it with glue, so I don’t think it’s just a stability issue. However, I did do a better 12-unit version, with duo paper and a little reverse fold on the outer edge to expose the second colour properly, which I’m quite pleased with:

Octahedral void (modified) [CC-BY-SA-4.0 Steve Cook]

Octahedral void (modified)

Tomoko Fusè’s little turtle modules are extremely flexible: they can be used to make pretty much any polyhedron that is made of regular polygons. However, because the flaps are only one paper layer thick, they don’t fit together terribly tightly, so I’ve only found them robust enough to make smaller structures without the help of glue. However, with glue, I’ve made a rhombicosidodecahedron, which is cool because it is built of pentagons, triangles and squares (all of the polygons found in the Platonic solids)…

Rhombicosidodecahedron (little turtle) [CC-BY-SA-3.0 Steve Cook]

The impossible-to-spell rhombicosidodecahedron.

…and also a pair of snub-cubes, which are even more interesting as the snub-cube has two non-superimposable mirror images, like hands, amino acids and amphetamines.

Snubcubes (little turtle) [CC-BY-SA-3.0 Steve Cook]

Snubcubes: left- and right-handed enatiomorphs.

I found Maria Sinayskaya Etna kusudama in Meenakshi Mukerji’s Exquisite Modular Origami book. It’s a really pretty model, and robust once it’s assembled, but it can be a bit fally-aparty during construction: I used very small clothes pegs to hold it together as I was making it.

Etna kusudama [CC-BY-SA-3.0 Steve Cook]

Etna kusudama.

Meenakshi Mukerji’s compound of five octahedra (inspired by Dennis Walker) is also a bit fally-aparty, but I like it as – unlike many of these models – it genuinely is the polyhedron so-named, rather than something where you have to squint at the holes in the wire-frame and imagine faces there.

Compound of five octahedra [CC-BY-SA-3.0 Steve Cook]

Compound of five octahedra. You can easily see the yellow octahedron here: the sixth spike is underneath the model; the other four colours are similarly interlaced.

The five intersecting tetrahedra are actually a lot easier to make than they look. Francis Ow’s 6-degree modules themselves are easy to fold, and the vertices are a lot more robust than they might appear. The most difficult bit is getting the modules interlinked in the right way. I’ve managed it twice, but only whilst staring at the YouTube video and performing assorted “purple = green” gymnastics in my head.

Compound of five tetrahedra [CC-BY-SA-3.0 Steve Cook]

Compound of five tetrahedra – party piece.

Michał Kosmulski’s page has lots of lovely illustrations, instructions and inspirations. I found Tung Ken Lam’s blintz icosadodecahedron (also credited as Francesco Mancini’s  UVWXYZ intersecting planes model) there. It has the same symmetry as the Electra icosadodecahedron above, but you can see the six intersecting pentagons more clearly. Both have the same underlying structure as the Hoberman sphere – that expanding/contracting plastic stick model thing beloved of science fairs.

UVWXYZ intersecting plane icosadodecahedron [CC-BY-SA-4.0 Steve Cook]

UVWXYZ intersecting plane icosadodecahedron

This last one is a bit of a cheat as (in theory, and mostly in practice too) the structures above are held together by nothing more than friction. Valentina Gonchar’s revealed flower star kusudama has to be glued, which is kind-of cheating, but I couldn’t resist as it is two structures in one:

Flower star [CC-BY-SA-3.0 Steve Cook]

Revealed flower star – shut (left) and popped open (right).

Things I’d still like to do:

  • Build a much larger PhiZZ ball (270 units): this would be useful for demonstrating the structures of viral capsids. UPDATE: Done!

270 PHiZZ knolled parts [CC-BY-SA-3.0 Steve Cook]


270 PHiZZ [CC-BY-SA-3.0 Steve Cook]


  • I’ve not yet found a good great dodecahedron model: they exist on Pintrest, but I’ve yet to find any instructions for one. UPDATE: Done! (Couldn’t for the life of me work out how to do 3-colouring, but module is from Saku B, recommended by Nick in the comments below)

Great dodecahedron

  • I have lost wherever it was I found the instructions for this inwardly cumulated rhombic triacontahedron: I’d quite like to rediscover them so I can credit the inventor! UPDATE: this isn’t where I originally saw it, but AresMares by Gewre has a video tutorial, and a kind commenter has let me know the designer is Silvana Betti Mamino – thank you!
Rhombic triacontahedron [CC-BY-SA-3.0 Steve Cook]

Rhombic triacontahedron of unknown source.

  • Invent my own module 🙂

Organism of the week #31 – Tardigrades

Tardigrades make me squee. These little relatives of the arthropods and velvet-worms are found in the water around mosses, and they are quite easy to find if you have a cheap microscope and a little patience. Like spiders, they have eight legs, but unlike the legs of a spider, they’re plump and stumpy, and end in the little ‘fingers’ you can just about make out in the photos below. Ignoring the excess of legs, it’s easy to see why they’re sometimes called water-bears.

One of my first-year undergrads spotted this one in a sample of moss I pulled out of the down-pipe from my bathroom. The fact that my skin flakes and spittle contributed in some small way to this microteddy’s food-chain makes me feel about as paternal as it is possible for me to feel.

Tardigrades [CC-BY-SA-3.0 Steve Cook]

Organism of the week #30 – Sticky situation

All science is either physics or stamp-collecting.

This rather mean-spirited dismissal of chemistry and biology as “stamp-collecting” is attributed to Ernest Rutherford, the physicist usually (not wholly fairly) credited with discovering the atomic nucleus and the proton.

Shortly after Rutherford’s death in 1937, particle physicists discovered the muon, pi mesons, kaons, the electron neutrino, the anti-proton, the lambda baryon, xi cascades, and sigma baryons. It took physicists the thirty years following Rutherford’s death to make sense of this veritable album of subatomic stamps.

Nature has a sense of irony, but its comic timing needs work.

There’s nothing wrong with stamp-collecting. Science very often begins with stamp-collecting, because it’s only once you have enough stamps that you can start reliably identifying patterns in the stamps, and – from there – finding the interesting exceptions and edge-cases:

Baryon supermultiplet [CC-BY-SA-3.0 Studzinski.daniel]

Subatomic particle stamp-album [CC-BY-SA-3.0 Studzinski.daniel]

I have been an on-off collector of carnivorous plants since I was very little. Most of them attract insects, kill them, digest them to tasty soup, and then absorb that soup. The soup contains useful nutrients that are missing from the soil in which they are rooted, so this helps them grow and set more seed. But there are many plants that tick some of these carnivorous boxes, but not all of them. Roridula is one I’ve blogged about before: it subcontracts out the digestion part of the process to an assassin bug. Another plant that walks the line is a kind of passionflower:

Passiflora foetida bud [CC-BY-3.0 Alex Lomas]

Passiflora foetida bud

The charmingly-named stinking passionflower bears sticky hairs on tentacle-like growths around its flower buds. There is some evidence to suggest that these help protect the flower bud from hungry insects while it develops. Similar sticky hairs are also thought to protect the flower buds of a number of other plants.

The stinking passionflower kills insects, and even appears to digest them, but it doesn’t benefit from the nutrients this releases. However, it does benefit from not having its flowers damaged by herbivorous insects. This presumably means it sets more seed, so the ultimate effect – more baby plants – is the same as for ‘true’ carnivory.

Passiflora foetida flower [CC-BY-3.0 Alex Lomas]

Passiflora foetida flower. The reason these plants are called passionflowers is because the various parts of the flower are supposed to look like hammers, nails and a crown of thorns – items associated with the Passion (crucifixion) of Jesus – rather than the earthly passions you might have been considering

Is this plant carnivorous or not? Well, whether you ultimately choose to paste this stamp into the carnivorous plant album or not is very much less interesting than the reasons you have for making that decision. I’m just glad that someone discovered this particular stamp and took the care to stick it somewhere for us to study. Here’s to collectors and taxonomists, the unsung heroes of biology.

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