Roman Dodecahedrons Part II

This is a continuation of part I on Roman Dodecahedrons.

Games on a Dodecahedron:

So I’ll go on to explain why I think it’s a string puzzle, but first I want to mention what my criteria for an acceptable answer is. To my mind, these are the observed things that need explanation,

  • The dodecahedra are (fairly) widely produced, over a wide geographical area
  • They are fairly expensive to make, both the bronze cost and fabrication time would make them more than just a disposable item.

So if it’s a toy, or puzzle, it must be one that is sufficiently engaging. Any puzzle which is too easy or too hard is not a good candidate, since it wouldn’t explain the observed popularity of the devices. And considering that the owner likely spent a reasonable sum of money on purchasing their own, then it’s probable that it’s a device they’d wish to use again and again. (Otherwise, you’d just wait till your friend got bored and borrow theirs)

So, my test for success with evaluating possible game ideas, is basically whether it’s the sort of toy that gets played with only on Christmas morning, or afterwards as well.

Back to the dodecahedron itself. The first thing that occurred to me is that the dodecahedron’s vertices form a graph, and that you can wind the string to traverse the graph.

https://en.wikipedia.org/wiki/Hamiltonian_path#/media/File:Hamiltonian_path_3d.svg

Variations on the game include:

  • Visit all cities exactly once, no crossing over allowed. (Equivalent to a Hamiltonian Path)
  • Visit all cities once, but return to where you started (Equivalent to a Hamiltonian Cycle)

Example of a Hamiltonian Cycle

And I play-tested them and found it nicely challenging, although once you figure out the algorithm it becomes a little more straightforward to solve, and you can just reuse the same solution after a while.

What suddenly makes it more interesting is a two-player variant. Player A makes the first five moves, then hands it to player B, who has to complete it. The randomness of the start shakes it up nicely, since you can’t just use your remembered solution, but instead have to think it through each time.

A multiplayer option is also made possible by the string, the first player can make the first five moves, finishing by making a knot around the final nodule. Player B then can attempt the puzzle, (perhaps while being timed), and when finished they restore it to the original state just by suspending it by the string and letting it unwrap until it hits the knot. Then player C can proceed, etc. In this way you could have a lineup of people all given the same puzzle, and see who is the fastest.

(I could imagine this being used as an impromptu skill test, or as a way to see how people deal with unusual situations. It’s not exactly the world’s best management metric, but compared to contemporary tests of character, such as apparently killing people that didn’t understand obscure riddles, it’s practically objective!).

And having played around with it, I can say it’s quite a satisfying object. Both the game itself is reasonably challenging, and the act of winding it around the posts is satisfying and somewhat meditative.

Hard Mode – Double Eulerian Walk:

I’m sure there’s lots of games you can play with the dodecahedron, but one I had fun working out was this. Use every road exactly twice, and end up back at home. (I don’t know if there’s an official math term for it, but for the sake of argument I’m going to call it the Double Eulerian Walk).

This gives a very pleasing pattern with the string, and is a much harder game than the 2 player Hamiltonian Cycle.

‘Double Eulerian Walk’ on a Roman Dodecahedron

At this point I was now convinced that the Roman dodecahedron is a string toy, and that gameplay involved one or more variations on the graph traversals. I started looking around on the web to see if anyone had previously had the same idea. And I was delighted to find this article here by David Singmaster:

https://www.comap.com/product/samples/UMAP_37_4.pdf

Which makes the connection between the Roman dodecahedra and a game actually invented by Sir William Rowan Hamilton himself in 1857:

http://puzzlemuseum.com/month/picm02/200207icosian.htm

(Adorably, there’s not just a tabletop version, but he also invented the travel version as well).

More info in a paper here.

 

Making the dodecahedron:

I went through several designs. First I started with an acrylic sphere I had laying around, and tried marking out the vertices on it evenly. That wasn’t too easy, so I worked out the diameter as a circle that would just enclose the pentagonal faces, and then lasercut a wooden template of the same size. In my mind this was a simple matter of doing a geometrical construction in the style of Euclid, but on a sphere. That didn’t turn out as easy as I expected either, and a lot of the vertices ended up kind of squashed. Sigh.

I’m amazed how much the ancient Greeks figured out, when they didn’t have any lasercutters at all.

I then decided to try printing another set of vertices from my lasercut icosahedron/Dymaxion maps, but that would have taken several hours, so while the printer was running I kept on working on designs for the lasercutter.

I tried a cable tied verted model, A snap-together model, a flat model, and one other one I didn’t bother to get a photo of.

Roman dodecahedron prototypes v01

The bag of rejected prototypes

And finally settled on this version, which most closely resembles its Roman ancestors.

 

Files up here for anyone that wants to make their own:

https://www.thingiverse.com/thing:4463142

 

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Roman Dodecahedrons Part I

I recently came across these fascinating old Roman artefacts:

https://en.wikipedia.org/wiki/Roman_dodecahedron

Untitled

Several hundred of them have been found, all around Europe, and what the purpose of them is is considered a mystery.

Some things to note about the dodecahedra:

  • They all appear to have ‘nodules’ on the vertices.
  • The hole sizes are not consistent
  • The size of the objects aren’t consistent

I decided to make my own model, and after playing with it for a while, I believe I’ve (re-re-)discovered what they were used for.

But first, I want to go over some ideas that are mentioned in the literature.

About the ‘rangefinder’ idea:

Several people have proposed that the dodecahedra were rangefinders, used for positioning something at a preset distance. Often this idea is accompanied by a diagram showing how to look through the holes at a distant Roman army staff as a reference, and position yourself accordingly. The implication being that up and down the front lines commanders would use these to ensure they were spread out appropriately.

I really, really have to object to that. That explanation just doesn’t work, and here’s why: If you were giving someone a rangefinder, to be used in the heat of (or just prior to) battle, you’d want to make it completely idiot proof.

What I mean is, when designing let’s say, ‘tactical devices’, they need to be able to be used in a hurry and when the operator is completely distracted. The correct way to use the device should be absolutely obvious, and any incorrect ways to use it must be so obviously wrong that they it becomes unlikely to be attempted. The dodecahedron fails on all these points. There are 12 different ways to look through it, only six of which are correct, and five of those six will still be the wrong setting for what you actually want to do.

Furthermore, there doesn’t appear to be any engravings or text to indicate what the different holes mean. If it were a tactical rangefinder, you’d expect something like [“5 stadia”, “2 stadia”, etc] to be really, really clearly marked on it, so you didn’t get confused or make a mistake in the heat of battle. Yes, there are slightly different concentric circle designs on the faces, but that is a really subtle code, (or more likely, an aesthetic flourish on the part of the artist) and not the sort of thing you’d want to rely on deciphering properly when stressed, such as when you knew there was going to be a lot of stabbing and bleeding happening in the near future..

And I want to point out that the Romans weren’t exactly strangers to combat. They were really good at thinking of the practicalities of warfare, had a ton of experience, and made a habit of equipping their soldiers with sensible and useful things. I don’t see them handing out a device like this for tactical use.

About the surveying tool idea:

It’s also suggested that the dodecahedra were surveying tools, used for working out angles between things, and laying down plans. I don’t really find this compelling, for two reasons:

  • One: if you wanted to make a protractor or similar device, you could do so much more directly. It would also allow you to measure any angle, or see how much error was in your measured angle, rather than only measure certain pre-set values. And it would use substantially less metal than the dodecahedron.
  • Two: the Romans were kinda fond of Cartesian designs. Check out their blueprints for how to set up an army camp. It’s rows and columns and 90 degree angles all the way. Guess what you can’t find on the dodecahedrons, no matter how you turn it? Exactly. There are no right angles. To try and use the angles present on the dodecahedron as a surveying tool would be irrational (heh).

The ‘Nodules’ are super important:

All the photos I’ve seen of Roman dodecahedrons show them having ‘nodules’ at the vertices.

They appear to be brazed onto the dodecahedrons at a later stage after the body was made. In other words, it added a step to manufacturing, and hence increased the cost and complexity of the device.

When a device is made across a large area and by many people, I don’t think it’s conceivable that any feature would be so consistently reproduced, unless it was an intrinsic and necessary part of the device’s function. In other words, people are both lazy and full of their own ideas. If the nodules were ‘vestigial’, then sooner or later someone would just make a dodecahedron without them, and save money and time.

I might be wrong on this, but nodules are present on all the pictures of dodecahedrons I’ve seen, and as far as I’m aware there weren’t any varieties of Roman dodecahedrons that were constructed without them present. There is, however, at least one Roman icosahedron that has the nodules present, but omits the holes in the face!

 

It’s not for grip:

I’ve seen proposals that the nodules were meant as aids to grip the dodecahedron, perhaps while using gloves. I find this a bit hard to believe, for the simple reason that dodecahedrons are constructed exclusively of pairs of flat, opposing sides. In other words no matter which angle you pick it up from, your fingers will intuitively find a pair of places to ‘pincer’ which perfectly balance the forces. (Compare that with trying to pick up something like a pyramid shape, which actually would be tricky, since you’d be forced to grip one pointy vertex, and one flat face.)

Even wearing gloves, it’s not hard at all to manipulate a dodecahedron. Certainly not to the point where you’d order the craftsman to carefully solder twenty balls to it, rather than just one ‘lollipop’ style handle, or some other method for holding it.

[Screw it. I just went and tested adding a lollipop style handle to my dodecahedron model. With a handle, you can easily hold it and swivel it to look through six of the twelve holes in a matter of seconds, without blocking the view with your hand. For a viewfinder, a handle is far more convenient than manipulating a spherical object directly, or adding 20 balls]

The nodules are not precisely calibrated:

Assuming it’s a surveying, astronomical or calendar device, then if you wanted to measure an angle in relation to a flat surface (such as putting it on a flat table and measuring the Sun’s position above the horizon), you’d need to make sure that the nodules it rested on were calibrated to the angle you wanted to measure. Casting or brazing bronze is not going to give you ‘Astronomical’ levels of precision right off the bat, there would need to be a calibration step involved.

In other words, when making the dodecahedron, you’d:

  • First build the rough shape of the device by casting and/or brazing the parts together,
  • Then measure the angle it made, and how much error was present,
  • Then carefully file down the parts of the feet contacting the floor, stopping to measure it occasionally, until it was perfectly tuned.

The photos of dodecahedrons I’ve seen don’t show any sign of having been filed down or ‘tuned’. Filing would have been necessary, rather than just bending the nodules apart, since bending one nodule affects the calibration of not one, but three other holes.

Both the Romans and Greeks were no strangers to precision craftsmanship, and could easily have made devices much more precise than these dodecahedra with little effort. (I mean c’mon! The Antikythera mechanism is actually older than these toys).

Observation: All the nodules are undercut, which is perfect for gripping string:

What is super interesting about the nodules is that they all have a spherical or an undercut shape.

To my mind, that immediately suggests that they were used in combination with string or cord of some sort, since if they were present just for added grip that wouldn’t be needed (and indeed, would just make it more likely to get snagged on things)

Also, this notion of the ‘undercut’ nodule being a functional requirement is consistent

with all the dodecahedra I’ve seen. Even these super unusual ones that has triangular holes:

http://www.la-detection.com/dp/message-75691.htm

Or these miniscule gold dodecahedra from Thailand and Burma:

https://journals.openedition.org/archeosciences/2072

Although these gold polyhedra are positively tiny (and possibly just intended to be decorative versions of the larger ones), the shape of the nodules is still clearly undercut, and it’s easy to imagine winding a small thread around them.

Back to the Roman dodecahedrons. Looking closer, although string or twine definitely seems to be involved, it doesn’t appear to be a primarily ‘practical’ object. In particular:

  • Using it as a ‘Knitting Nancy’ https://en.wikipedia.org/wiki/Spool_knitting doesn’t seem like a good explanation. Each hole has 5 pegs, so you’d get the same result no matter which orientation you used. It’d be far easier to carry something that wasn’t so bulky and redundant if you wanted to knit that way.
  • Using it as a yarn holder, or ‘NItty Noddy’ doesn’t make sense either. https://en.wikipedia.org/wiki/Niddy_noddy
    • Your wool would be trapped on there,
    • there’s more than one way to wind it, which is confusing,
    • And it doesn’t actually hold much string anyway.

There would be far better ways to do the same task without involving platonic solids.

  • You could maybe make an argument that it’s a drop spindle, https://en.wikipedia.org/wiki/Spindle_(textiles) but you can do that for pretty much any object. So I don’t consider that sufficient to explain why it was recreated so prolifically and with the same conserved feature set (dodecahedral shape, nodules on every vertex).

As a next step, I started thinking about the kinds of games you could play with string and a dodecahedron.

Continued in part II.

[Edit 2020/06/17 – Added link to dodecahedra with triangular holes]

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Homemade Pomodoro timer

Like most people, I’m stuck at home right now. Trapped, in an environment which is under my full control, and I’m surrounded by my books and technical resources, and free to do whatever I choose, whenever I choose. Which, it turns out, is not quite the recipe for unbridled productivity I had imagined only a few weeks ago.

(Although, hey, I fully acknowledge that merely being bored or distracted right now is a lovely problem to be able to have)

Anyway, I wanted something to help me focus on tasks and avoid distractions. I remember reading good things about the Pomodoro technique a few years ago, and I thought it might be a good thing to get into again.

There’s some online sites you can use as a timer, but I wanted something a bit more tangible.

I had an LED ring laying around, and I actually had the case designed and cut earlier for another project, so really all I had to do is reprogram the arduino.

The user interface is about as minimalistic as it gets; as soon as it’s plugged in, it starts counting down from 25 minutes. If you want to stop it, you unplug it.  If you want to reset it, you unplug it and plug it back in again. Voila!

The ring shows the amount of time remaining, starting in green:

Pomodoro ring green v01.JPG

Then the last 5 minutes are displayed as yellow:

Pomodoro ring yellow v01.JPG

And once the time is up the whole ring shows up as red:

Pomodoro ring red v01.JPG

The ring is from here:

https://www.jaycar.com.au/duinotech-rgb-led-circular-board/p/XC4385

And files here for anyone that wants to make their own:

https://www.thingiverse.com/thing:4284881

 

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Homemade masks w/ wire nosepieces

So, masks have been in the news a lot recently for some strange reason. Probably most interesting; in the Czech Republic masks were made mandatory, but as there wasn’t enough existing masks in the supply chain, people had to create their own. Their president is crediting the widespread use masks as one of the main reasons they’re doing so well right now.

There are a number of organizations online that have provided instructions on how to make your own mask, and, in areas which are much more severely affected, some hospitals have been requesting people produce homemade masks for them to use. Thankfully, there’s groups of volunteers around the world which have been stepping up and making them en masse. (This whole situation is something that we likely would have considered ludicrously implausible just 6 months ago)

Luckily, here in Australia the supply for medical professionals seems to be holding out, and I haven’t heard of any hospitals which need the public to make stuff for them.

But masks as a whole seem like a great idea, and they are certainly easy to make. So, my friend Kate and I decided to stitch up a bunch of masks for ourselves and others.

Making the mask:

We used the pattern and instructions from here:

https://freesewing.org/blog/facemask-frenzy

and made the following modification so that you get a much better fit:

First, grab some 1.6 mm diameter armature wire . (This is the sort of thing they use inside clay puppets so they can hold position while being filmed for stop-motion animation.)

We cut 190 mm of wire, and put it in inside the mask, at the bridge of the nose, with a stitch underneath to keep it held in place.

After assembly, here’s what it looks like:

Finished mask - crop v01.JPG

Stylish fabric optional

The fit is very comfortable. As a test, I wore my one for several hours straight while we were cutting and stitching the others, and I had no issues.

Also, the armature wire in the nose is a godsend. When I put the mask on, I can press down and ensure shape the wire so the top part of the mask sits tight against my face. Most importantly, no matter how hard I exhale, I can’t fog my glasses. I don’t think I’ve ever had a mask that I could easily wear glasses with before. (This probably just means I’ve been wearing dust masks incorrectly for years, but hey)

Some tips I’d recommend if you make your own:

  • If you’re using ribbons and not elastic to hold it on, then use fairly thick ribbons, (~10-20 mm?). The thin ones we made (~6mm) have a tendency to be hard to tie in a bow, and tricky to undo if stuck. (I ended up with one double-knotted behind my head and needed another pair of hands to rescue me, which kind of defeats the point of social distancing with a mask)
  • Use two different colours of fabric, with a boring fabric inside, so it’s obvious how to put it on.
    • Some sources say to use non-absorbent material such as (polyester or poly-cotton blend) for the outside of the mask, and absorbent material (such as pure cotton) for the inside. Others don’t specify, or just use the same material for all layers. If you have a choice of materials maybe do non-absorbent on the outside.
  • Make more masks than you think you’ll need, and that way you can give (a washed) one away or have spares if needed.

 

Do they work?

There’s the question of whether homemade masks work. I’ve seen some people say that masks for uninfected people are a bad idea, and other say they’re great, and everyone should be wearing them. My inner [socratic dialog/shower thoughts/ shoulder angels discussion] ran something like this;

A: Now I’m confused. Hmm… What do we think?

B: Well, the argument is that while a mask might intercept an incoming droplet from someone else, in doing do it then traps the virus right next to your mouth, where you’ll just breathe it in later

A: That logic seems weird to me. I think surely more would still be stopped than make it through? I mean, how can the steady state be worse than the no-mask case?

B: Hmm… good point. But then there’s the effect that the mask stays warm with exhaled breath, so is that giving trapped particles a boost? Like a mini incubator?

A: But what about touching your face? We’ve all learned recently just how often everyone does that. A mask completely stops you touching your mouth and nose. And makes you more aware if you try and touch your eyes.

B: So in a sense, a mask makes your hand washing more effective?

A: I’d wager so. But what about that ‘incubator’ idea? Is it a real effect? And if so, does it cancel out the benefits?

B: Hmm… How the hell would we test that?

And I just got tied up in knots trying to imagine stuff I don’t have enough experience to predict well. But thankfully we don’t have to mentally simulate everything from first principles to find the likely answer. For example, this study:

Click to access radonovich2019N95masks.pdf

did a randomised controlled trial of just under three thousand doctors that interacted with patients with influenza. The doctors were issued at random either an:

  • N95 mask (roughly equivlant to a ‘P2’)
    • These are specially designed to stop aerosolized particles, have a material with guaranteed effectiveness stopping particles of a certain size & are tested rigorously
    • Are a pain to fit properly. You can be ruled out from wearing particular sized masks because of your head shape, or if you have a beard.
    • Depending on your area, may periodically require special procedures for a Fit Test, involving wearing the mask while an aerosolized chemical is sprayed directly at your face. If you can’t smell it at all, while vigorously breathing in and out, (you can’t cheat by holding your breath, they make you read out loud long sections from a book), then your mask fits.
  • Surgical mask
    • These aren’t remotely air tight, they’re designed to be comfortable and easy to use
    • These have no special requirements relevant to stopping viruses. (at least for the couple of models I looked at). The only specs and standards I saw were: 
      • BFE > 98%  – this only applies to bacterial filtration ability, not relevant to much smaller viruses like coronavirus
      • EAN14683: Type II – this specifies that :
        • It has a low differential pressure. I.e. it’s easy to breathe through, and
        • specifically not required to have any splash resistance, and
        • specifically not required to have any sub-micron particulate filtering ability
    • The particular surgical masks used in the study were better than ordinary cloth masks, however.
      • They had fluid resistance ratings of 160 mmHg, indicating it needs approx 1/5th of an atmosphere pressure difference to force liquid through,
      • They had particulate filtering ability at 0.1um of 98% (this is about the size of coronavirus particles)

So the study describes a comparison between basically a ‘gold standard’ mask and a quite good mask, but which is not guaranteed to be airtight , tested by people wearing them whenever they were:

routinely positioned within 6 feet (1.83m) of patients”.

And they found no significant difference in the number of doctors contracting influenza.

Of course, the influenza-A & B from the study obviously isn’t the same as coronavirus, so perhaps it might turn out that there’s a true difference in PPE effectiveness. But, I mean, they’re pretty similar. Both are small, airborne, viruses made of RNA, and if significant amounts could traverse an improperly fitted mask so easily, then we would probably have seen that reflected in the study, which we didn’t.

The next study I looked at was this one: https://bmjopen.bmj.com/content/5/4/e006577 (Also note that the authors of the study made a recent response in light of coronavirus )

The study follows 1607 healthcare workers in Hanoi who used either:

  • locally produced cloth masks,
  • locally produced surgical masks,
  • or their normal procedures (which would likely have some form of surgical mask).

The study covers a four week period, and covers seventy four wards of varying types (including emergency & infectious/respiratory disease wards) specifically selected because they were high-risk.

Looking at the results of the study, at first glance the Relative Risk level of 13 to 1 seems terrible.  i.e. wearing a cloth mask is 13 times more risky than wearing a surgical mask, as far as Influenza Like Illnesses goes.

But when you look at the actual outcomes, the numbers don’t look so scary:

  • Surgical mask: 580 people, 1 got Influenza-Like-Illnesses
  • Cloth masks:  569 people, 13 got Influenza-Like-Illnesses

[Edit: Screw it, I decided the picture from the journal article didn’t convey it well enough, so here’s my own plot instead. Edit2: I removed the ‘control arm’ section as it was confusing and not relevant as it was their old procedure. The numbers here just show surgical masks vs cloth masks]

infographic pic v01.png

Yes, technically the cloth mask is 13x worse, but at this stage you probably don’t give a shit.

When you consider that these numbers are specifically selected from front line healthcare workers, in high-risk areas, and still indicate that you can wearing cloth masks for a month and still only have a 7.6% chance of catching a CRI, or 2.3% chance of ILI, that would seem to indicate cloth masks are still pretty fuckin’ awesome. 

Don’t get me wrong, I’m sure surgical masks are better, (and they should be mandated in any countries that haven’t yet), and if you’re a healthcare worker you should absolutely use them if you have access. But it looks like we’re talking about fairly subtle differences in safety here. If it were a car, it’d be the difference between having airbags with extra side-impact cushions, and just regular airbags. Whatever you have, it’s far better than nothing. 

Now pretty much none of the extreme scenarios in either of those studies apply to me, or anyone likely to get a mask from me.  I’m not the worst-case of someone spending all day next to a contagious patient, I’m just some schmuck making a quick trip to the shops. Or getting drive-through. I just want a little more protection when I have some short interactions with others, and that’s that.

I can then carefully take off the mask (without touching the outside), stick it in a plastic bag and boil it when I get home.  And I have a couple of masks ready so I’m still covered in case I have another errand later.

So as far as effectiveness of homemade masks goes, it looks cautiously encouraging. But at any rate, I’m working on the assumption that:

it’s a mask, not a magic wand:

  • Wearing it does not grant me magical powers.  I will not assume that I am in any way immune to infection because of my stylish facewear
  • I’m still going to keep social distancing, and not do any extra errands which I wouldn’t have done anyway.
  • I’m going to be careful taking it off, making sure I don’t touch the outside
  • I’m still going to wash my hands with soap and water, or hand sanitizer, as normal

However:

  • If I do unknowingly have the virus, I’ve probably made it less likely for others to get it off me. Win!
  • If I do run across someone that unknowingly has the virus, I have probably made it a bit harder for them to infect me. Win!

 

If you’ve got a sewing machine, why not make your own? If nothing else, it’s an excuse to use up those fat-quarters of unmatched fabric you’ve had laying around for years…

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Waves at Bessel-on-Sea

This is a lasercut version of the Bessel Functions, as a handy desk ornament:

Bessel mono crop v03.JPG

The helical diffraction theory, (and hence the Bessel functions) were the major key to solving the structure of DNA.

In 1952 (well before the DNA structure was solved) Francis Crick & Bill Cochran wrote a paper explaining how the expected form of X-ray diffraction from a helix is the sum of various Bessel functions:
https://onlinelibrary.wiley.com/doi/10.1107/S0365110X52001635

Quick bit of background. When you’re using X-rays & film to find the structure of something, what you get when the film is developed isn’t a picture of the structure. Instead, it’s (more or less) the Fourier Transform of the structure.

We can simulate this in python like so. Let’s say we have a simple helix, (which we’ll assume is smoothly continuous, and not made up of any yucky atoms). The pattern we get looks like this:

CrickCochran_ContinuousHelix.PNG

A continuous helix (left) has a diffraction pattern like a big X (right)

and then if we take a photo of a helix which is made up of a discrete atoms, we see a pattern like this: :

CrickCochran_30AtomHelix.PNG

A discontinuous helix (left) has a diffraction pattern which is a series of diamonds (right)

The way the maths works out is something like this; the ‘dotty helix‘ can be thought of as the (piece-wise) multiplication of two functions:

  • H – a helix with constant radius
  • K – a function for the ‘planes’, which is zero everywhere except at a plane every ‘p’ units

Cochran crick maths - real space v01.png

and the result in the ‘Reciprocal Space’ (i.e. what the X-ray picture will look like) can be neatly expressed as the convolution of the [Fourier transform of H] with [the Fourier transform of K].

Cochran crick maths - reciprocal space v01.png

In other words, the big ‘X’ is “stamped” on the image every where the red planes are. The result looks like a series of diamonds.

Let’s make a larger diagram. If we sketch out the expected pattern for a continuous helix, we’ll see an x-shaped pattern, roughly like:

cochran crick sketch - platonic helix v01.jpg

And if the helix is made up of discrete units (atoms or rungs), then we’ll see the above pattern ‘stamped out’ multiple times on the image.

For example, if we have a helix which has 10 layer lines per twist (like real DNA),  we’d expect to see a pattern like this:

cochran crick sketch - 10 layer repeate helix v01.jpg

Expected diffraction pattern for a discontinuous helix which has one twist every 10 rungs

That’s an amazingly good match for this (terrible quality) photo of the real thing :

Photo_51_x-ray_diffraction_image.jpg

Source here

You can see most of the characteristic features. The double diamond (4+ diamonds, really). Note that they meet up on the 10th line, indicating that every 10 rungs the helix makes one turn.

There’s a whole bunch more cool stuff covered in the Cochran/Crick paper, like:

  • They explicitly consider cases where the number of rungs per turn isn’t a neat integer
  • They do worked examples to show how it explains features in the Pauling’s recently discovered alpha helix
  • They propose practical methods for analog computing via paper charts and movable masks in order for people to be able to quickly synthesize patterns for arbitrarily complex helicices in the future.

 

Side note: the mathematician Alexander Stokes had also worked out the helical diffraction theory at around the same time, but didn’t bother to publish it. He famously did the work on the train on the way home, and presented it to the lab the next morning. You can see the lovely sketch he did here:

http://dnaandsocialresponsibility.blogspot.com/2011/03/short-and-simple-ish-guide-to-x-ray.html

Which Wilkins was so impressed with, that he stuck it on the lab notice board, with the name “Waves at Bessel-on-sea”.

It was after seeing Stoke’s picture, that I decided I wanted to make my own copy of Bessel-on-Sea for my coffee table:

Bessel crop v02.JPG

Files here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3872197

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The Crystallographer’s Watch

Finished product v01.JPG

Here’s a project I made almost accidentally on the way to a later design. I’ve wanted to make my own watch for a while now. It’d allow me to pick and choose all the features I really want, and it’s a fun exercise in design to try and figure out which features work smoothly, that I’d appreciate having everyday, and which features are more ‘fads’ that I can do without.

I have a metal CNC machine, so carving the watch body from solid metal is doable (if somewhat fiddly and time consuming). And it’s really cheap to design and make your own circuit boards these days, so the electronics are fairly easy.

But it occurred to me that this is still a multi-stage process, which plenty of opportunity to loose energy or procrastinate. If I wouldn’t get that reinforcing emotional feedback/reward until WATCH_CASE_DESIGN + MILLING + ELECTRONICS + SOFTWARE are all done, that’s a very long chain with plenty of ways it can fail.

So, as a way to break the the project into chunks, I figured I’d start with the circuit board only.
I bought a large men’s watch 2nd hand watch on gumtree and pulled out the guts, this left me with a big empty enclosure I can fill with my custom electronics.
Unmodified watch v01.JPG

I measured up the internal space I can use, and I lasercut a couple of ‘dummy’ cylinders of the same size:

Internal case dimensions v01.JPGdummy cylinder v01.JPG

The idea is that as long as whatever electronics I come up with are smaller than the dummy cylinders, I’ll have no surprises when it comes to assembly.

At that point I realised that the empty watch was essentially a wrist mounted display case.

The other day I’d been playing around with small ball bearings, to make a ‘bubble raft’ style display like those popularized by Sir Lawrence Bragg.

I figured that with a bit of fiddling, I could make a watch mounted version I could take anywhere. So I laser cut another plug, and some circular rings hold off the wood from the glass, which allowed the balls to move freely.

Ball bearing insert v02.JPG
It took a bit of tweaking to ensure the balls didn’t have enough space to ‘double pack’ when tilted. Brett and I had to have several rounds of taking it apart, sanding the ring down carefully, then reassembling before it worked nicely.

There’s a lot of interesting structure in the raft. You can see how the balls pack in regular order at a local scale, but don’t line up on a global scale.

Raft coloured v01.JPG

Grain boundaries and sphere packing

(Also note the red areas with square packing, everywhere else seems to be the more efficient hexagonal packing).

Every time you look at your wrist you’ll see a different pattern. Sometimes regular, sometimes chaotic. And by tapping and jiggling, you can often ‘anneal’ the structure into a lower energy state. Here’s one pattern that’s been annealed a bit.

Raft coloured v02.JPG

The watch annealed into a much more regular shape

(Note the lovely grain boundary, and two large grains which have steadfastly refused to merge together).

The semi-randomness of the pattern is quite appealing. The eye has no problems picking up the detail, and you can often see grain boundaries more easily than the individual balls. And with a quick flick, you can get a whole new arrangement. Sort of a wrist mounted I-Ching.

I’ve been wearing it for two days now, and it’s rather soothing. In fact it’s an anti-watch.
(Since a regular watch tells you the time and makes you stressed. This tells you absolutely nothing, but makes you calmer)

 

 

 

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The Dichroic Confuse-O-Scope

This is the ‘Confuse-O-Scope’, a device which allows you to enjoy all the fun of having mis-aligned RGB in real life! Guaranteed to cause both confusion and irritation in anyone that’s worked in TV, printing or theatre lighting!

Example view 01.JPG

If only I could make a telescope that messed with Kerning.

It uses the same dichroic beamsplitter cubes as my previous project, but arranges 3 in a row:

case inside 02.JPG

With the result that; while any colour light from the world can get to your eye, the red, green and blue colours all travel via different paths. And because of parallax effects the view of each will be slightly different:

example view 02.JPG

You can also flex the frame a bit and change the RGB alignment, making it overlap or separate.

Here’s what the view looks like from the other side:

image split v01.JPG

Files here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3591363

case overview v02.JPG

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Dichroic Moving Sculpture

This is a quick project I made to explore dichroic filters. I just love the colours that can be produced, and wanted a way to display it easily.

In the last few years, these dichroic cubes have appeared on eBay. They’re used inside projectors to combine red, green & blue colour channels into a full image. And, apparently making them isn’t a perfect process, so a bunch of defective ones regularly end up for sale.

IMG_2368.JPG

I used a couple of stepper motors to allow the cubes to be rotated at a slow yet precise speed, and used the same unwise technique from before to simplify wiring. I then used some high power LEDS to provide illumination:

Side note: The ‘unwise technique’ still seems to be paying off surprisingly well. Everything you see in the video is running straight off the arduino, via USB with no other power supply.

To collimate the light from the LED I used some lenses from eBay jeweller’s loupes (which were so terrible that their main value is for parts), and made a lasercut holder for the lens. The light assemblies are mounted on thin brass shims to allow bending and positioning them by hand:

box behind v01.JPG

Files up here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3591151

 

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A Planetary/Harmonic Hybrid Gearbox

I recently saw this amazing idea from Darren Schwenke on Hackaday.io:

https://hackaday.io/project/164732-mprt-modified-planetary-robotics-transmission

Which is (so far as I know) a brand new type of gearbox, inspired by a well known concept called a  “Harmonic-Drive“. Harmonic drives have been around for years and were used whenever light weight or small size was required (on the moon rover, for example). They work via the deformation of a flexible ‘strain wave gear’ to enforce the meshing between two gears with nearly matching numbers of teeth. This allows very small reduction ratios, in a compact space:

240px-HarmonicDriveAni.gif

Image from Wikimedia Commons, here

The red strain wave gear is where output goes. The downside is that a strong and really flexible gear like that is hard to make, and also difficult to couple the output from easily. Most designs I’ve seen require custom spring steel ‘cups’ which are precision manufactured via electro discharge machining or similar. (Although there are several awesome 3D printed versions out there. )

Darren’s idea with the MPRT gearbox is to take the basic concept of the harmonic drive, but remove the complicated strain wave gear and instead substitute ordinary planetary gears, which then do the same job of enforcing the meshing of the two outer rings at several key points:

Gear sketch v01.jpeg

The final gear ratio for this is around 66:1 reduction, which is amazing for a single stage:

IMG_2322.JPG

I liked his design, but didn’t want to wait a long time to 3D print it, so I drew up this one for my laser. I created the gears just using the involute generator in inkscape, then added all the bits and bobs to hold it on the motor and mount reliably. I laser cut the pieces out of bamboo ply, and screwed them together with M3 fasteners.

A NEMA 17 stepper motor sits underneath and drives the whole assembly:

IMG_2323.JPG

Also, I’d like to mention one of my favourite construction techniques, using M3 nylon standoffs as thumbscrews. I’ve had to put various bits of the gearbox together and pull them apart half a dozen times while prototyping, and being able to fasten bits securely by hand is a huge time saver.

I used a few dollops of furniture wax, which seemed to make it run smoother:

IMG_2328.JPG

The torque is large but not ridiculous, and I can make it stall by hand if I really try, but overall it’s still remarkable for a single stage gearbox. (Also, not many moving devices have sliding wood-on-wood surfaces, so if you made this out literally almost any other material you’d likely have better results. )

Files up here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3576090

 

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Unwisely driving 17 stepper motors from a bare arduino

This is a quick and dirty way to get a whole bunch (up to 17) small stepper motors working off a single arduino, with no extra circuitry whatsoever.

Why would you want to do that? Maybe you want to make a wall display, a clock or some other interactive object, and you can’t afford thousands of dollars for motors, and hundreds of hours spent wiring up boards for ‘proper’ drivers.

(Side note, the back story to this is for a while now I’ve been wanting to make a big-ish display simulating vector fields, and I was scratching my head trying to come up with a way to do it that wasn’t ridiculously expensive. I ended up finding one that was not only cheap, but lazy too!)

Anyway, here’s the finished display:

finished display.JPG

The main ingredient is the 28BYJ-48 stepper motors, which is a geared motor which is dirt cheap. They’re mass produced and apparently designed for air conditioning louvres?

Ordinarily when you use a stepper motor you need a dedicated constant-current driver to avoid damage. The essence of this project is how to avoid the cost and complexity of a driver.

The cost of the parts were:

  • $3 each for the driver and motor pair , from Little Bird Electronics.  (If you really wanted to, you could get them even cheaper in bulk, and also by not including the driver)
  • $10 for the arduino, from eBay.

I used 17 motors, so it was $61 AUD all up. Which is ridiculously cheap for something with almost twenty channels of precise motion control.

Here’s what it looks like inside:

A smarter person would have numbered them so they were correct as viewed from the front. Next time, Gadget, next time.

Also, the wiring is also about as simple as you can imagine, I basically just jammed the motor wires into the arduino’s digital pins and slammed the box shut before they could fall out again:

Science

Here’s how to do it yourself. But, before we begin:

If you try this, you might break your arduino.

If you try this, you might break your arduino.

If you try this, you might break your arduino.

And also,

If you try this, you might break your arduino.

Everybody clear?

This approach works*, but relies on several things which might not occur all the time. Don’t assume you can get away with this in other designs.

*Actually, I’ve really no idea if this will work long term, all I can say is that my one has been running for several hours now, and the arduino hasn’t yet caught fire, appeared broken, or visibly lost steps on the motor. Win!

Trick 1: The arduino digital outputs have a non-zero resistance

This is the reason you often see people getting away with plugging LEDs in to the arduino pins directly, without a current limiting resistor.

(For years I thought it the ATMEGA chip had actual current limiting circuitry, but turns out it’s just the internal resistance or something? At any rate, don’t assume you can abuse other chips in the same way. )

I don’t know for sure this is required, but I’ve found in the past the ATMEGA/arduino is way more tolerant than microcontrollers for badly connected loads, so I’ll assume it’s relevant.

Trick 2: We convert the motor from unipolar, to bipolar,

(This has the nice side effect of doubling the resistance of the motor, further reducing it to the point where the arduino chip can drive it without circuitry).

The motor from the factory has a coil arrangement we want to change from this, to this:

We do this by:

  • opening up the back cover,
  • cutting off the centre tap ( red wire)
  • Dremelling out the circuit board to disconnect the two pairs of coils from each other

Trick 3: The 28BYJ-48 stepper motor is crap. And that’s good news for you!

Or, to be more precise, the motor has (before the gearbox) only 32 steps per rev, or 11.25 degree step size.

Why this is relevant is that we want to be able to power down each motor’s coils between movement, so that the arduino is only powering a single motor at a time. But we also want the motor to not lose steps the next time it’s powered up again.

A big 400 step NEMA17 motor (such as you might find in a good 3D printer) has 0.9 degree step size. If you power on and off a big 400 step motor repeatedly, it’ll jiggle slightly. If it jiggles more than 0.45 degrees, then when it’s started it’ll be dragged to the next notch in the rotor, and hence the wrong  location. This will happen most when under mechanical load, or the influence of belt tension, etc. So ordinarily, turning motors off translates into lost steps, and poor position accuracy. Hence for a 3D printer, they typically leave the motors powered up, or under a reduced current whenever they need it to hold position correctly.

Because the 28BY-48 motor has a huge 11 degree step size, (and it’s behind a gearbox) it’s really unlikely any mechanical jiggling is going to move it far enough to be a whole step away from where it should be. So the next time it’s powered up, it’ll be pulled back to the exact location it was before!

And that’s it. With code to carefully avoid running more than one motor at once, it can be scaled up to as many motors as you like, and you only need to stop when you run out of arduino pins.

I think am going to enjoy making displays with this technique, and it’s quite satisfying to watch the dials spin around in person.

 

Files and code here for anyone that wants to make their own:

https://www.thingiverse.com/thing:3544304

Have fun, but don’t blame me if you damage stuff by trying this.

Edit: I’m still playing around, but it seems like you can get loss-free movement of at least 6,  8, 10+ motors at a time. Damn, this works way better than I have any right to expect. 

 

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