## Spiral Music Visualization using Teensy or Python

Here’s a project I finished a couple of years ago, but never got around to posting.

It’s a way to visualize music differently to the standard notation. The notes are wrapped in a spiral fashion, with one rotation per octave. (e.g. all ‘C’ notes are at 12 O’clock)

There are a lot of benefits to seeing the music this way:

• You can see an entire orchestra “Cooperating” to make a chord, without having to read 6 sets of sheet music at the same time.
• It makes it extremely easy to see transpositions (they’re just rotations)
• Melodic inversion is just a mirror flip
• Notes played stay visible for a time as a ‘histogram’. This makes it easy to see the Key signature
• Different instruments (midi channels) are different colours. Can see contributions of each instrument to the whole

Here’s the github repo for the hardware version:

https://github.com/mechatronicsguy/SpiralMusic_Teensy

And just recently I decided to make a software only version, so I didn’t have to drag out the hardware each time. Here’s the github repo:

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## Origami Maths – Face Graph of a Hypercube

Here’s a quick little project I never got around to writing up. A while ago I was doing a dive into group/category theory and playing around with graphs of various polytopes. I wanted to make a model of the faces of a hypercube.

I tried a couple of different models, using pipecleaners & straws, then another version with lasercut wood struts and cable ties. Eventually I settled on this method, just using string and lasercut cardboard:

Of particular interest, is that all the vertices look the same, in terms of having the same connectivity. (Not too surprising, but hey).

## Examination of the Hypothesis:

The hypothesis I put forward in the last post I think has great merit. In particular:

• This hypothesis explains the consistent presence on the dodecahedra of their two most defining features; the nodules and the varying holes, and furthermore, my hypothesis has the enormous advantage of requiring both parts as essential and optimal parts of the design.
• The nodules are optimal for string winding, being reliably ‘undercut’. Other proposed functions of the nodules (such as hand grips) are significantly less plausible.
• The varying diameter circular holes are optimal for cone mounting. No other hole shape (pentagon, etc) is as easily mounted. Consider If the device served some other function, with the nodules being functional, but the circular holes being an optional aesthetic feature, then (given the sheer number of artefacts found) we would expect to find at least one craftsman having decided to omit them, resulting in either:
• a dodecahedron without holes,
• or perhaps a skeletal ‘cage’ dodecahedrons with no faces altogether,
• or other variants with non-circular cutouts (such as pentagonal holes).

To my knowledge, none of these hypothetical variants have been found.

• The literature has, shall we say, a superabundance of hypotheses on what the dodecahedrons were for, but most ideas strongly focus on explaining either the holes or the nodules. I haven’t seen any that provide compelling explanations for both features at the same time. To my knowledge, this is the first hypothesis to do so.
• This hypothesis explains the reason for differing hole sizes, since they are needed to fit a tapering cone. Furthermore, we require the differing holes to occur not just randomly, but in rough pairs. Since inserting the cone to rest on the bottom face requires the upper face to be large enough for the cone to pass through. If the holes in the dodecahedron were just used for measuring diameter (e.g. as a pipe gauge), there would be no reason to expect that opposite holes should be similar in size.
• This hypothesis explains the almost total lack of standardization encountered in the dodecahedrons. They don’t need to be any particular size, weight, or absolute diameters, and can be made as small or large the customer would prefer and materials to hand suit.
• This hypothesis explains the reason why the instrument is both specific enough to require many different sized holes, yet relaxed enough for us to encounter dodecahedrons which have no obvious markings on the faces to delineate them. Since this information would be carried on the accompanying cone, it is redundant to carve on the dodecahedron itself. While identifying markings on the face may be present, choosing to carve them on the faces would be an aesthetic rather than functional decision.
• Contrast that with other proposed uses of the holes (such as pipe gauge, or rangefinder), where not only would the absolute size be important, but correct identification of the which face to use would be absolutely vital to the successful operation. It is simply not plausible that a plumber or craftsman verifying a part would be required to rely on either memory or sight to identify which of the 12 different hole sizes a given face corresponded to, when that information could easily be carved on the faces permanently.
• This hypothesis also has the potential to explain why ‘Roman’ dodecahedrons have not actually been found in Rome, but instead are found in Gallo-Roman areas. Since the Romans had their own calendar of a more irregular nature, it’s not as neat a fit, and becomes a ‘harder sell’ to adopt.
• If an open and free device that one might sell, it might simply have flopped and not found Roman interest, since 30 is a not-significant number to them as far as calendars go.
• Alternately the calendar may have been a cultural secret. Either because if knowledge that those-people-who-are-in-our-empire had still not adopted the official Julian™ calendar, they might send some soldiers out to give them a talking to, or alternatively because it was thought to involve secret cosmic and nature knowledge that was not for sharing.
• It also explains the existence of a lesser known Roman Icosahedron, since both dodecahedrons and icosahedrons have 30 edges, both can be used as calendars. Furthermore, it correctly predicts that a Roman Icosahedron will not feature the same varying hole patterns as the dodecahedrons, since the 20 faces are not useful for counting months.

## Falsifiable tests:

Any good hypothesis should have at least some conceivable observation which would allow it to be proven wrong. So I’ll stick my neck out and suggest a few ways this idea could be disproved:

(Disproving should ideally require finding a couple of examples, btw. Since perhaps one faulty dodecahedron may be a workshop manufacturing blooper, or made as a practical joke (similar to teapots with no hole in the spout), but consistently finding examples of dodecahedra which are not functional as calendars would blow this idea out of the water.)

1. The nodules must be undercut to retain string. Finding dodecahedra with nodules that are tapering spikes (instead of ‘wasp-waisted’ balls, or at the very least cylindrical pegs) would disprove the usage of string, since it would too easily slip off.
2. The holes, should be circular and varying in diameter. (Either in a linear fashion or in a looping fashion for an Analemma display). Finding dodecahedra with identically sized holes would disprove the usage of a cone. Only if the dodecahedral faces contain unique identifying marks, (which would allow identifying the month in the absence of the cone), could we relax this requirement.
3. Opposite holes must not have widely dissimilar sizes. If the cone is longer than the dodecahedron, as seems most likely for a vivid display, then putting a small hole directly opposite a large one would prohibit the large one ever being used, since the cone could not pass through the dodecahedron far enough to engage the large hole.
1. Note that the requirement for opposite faces to be in rough correspondence, but not necessarily adjacent in the sequence potentially allows falsifying any other hypothesis which might require alternate faces to be adjacent in size.
2. For each pair of holes the inter-hole spacing and the difference in diameter allow us to calculate the minimum taper on a cone that fit both without fouling, and hence calculate the maximum possible cone length. As a rule of thumb, I would suggest any implied cone length of less than, say, 120% of  the dodecahedron face spacing would disprove the calendar hypothesis, as this would not produce an obvious enough vertical travel to show the month clearly. (This is an aesthetic judgement on my part, but I do want to stick my neck out and make it falsifiable. I would be surprised if any were below 150% in practice).

For example, assume we have a dodecahedron of height 50mm, with a linearly arrangement of holes of diameter [10, 12.5, 15, 17.5, 20, 22.5, 25, 27.5, 30, 32.5, 35, & 37.5mm]. How should we place the faces?

1. If we placed the 10mm and 30mm faces opposite, the cone must taper from 30mm to 10mm within 50mm, which would imply the longest possible cone length was only 68.75mm, barely larger than the dodecahedron itself. This does not make an impressive display.
2. If we instead place the 10mm and the 17.5mm holes opposite, then the tapering implies a maximum cone length of 183.3mm, which is much more visible a change throughout the year, with the dodecahedron moving more than 3x its height.
3. Naturally, if we place holes in adjacent size order, we have the most freedom. With 10mm and 12.5mm holes opposite each other, then the tapering implies a maximum cone length of 550mm. We are free to shape our wooden cone to any value smaller than this, of course, and may choose a smaller number if that suits our scepter or walking stick of choice.

4. Any Roman Icosahedrons (with 20 sides instead of the dodecahedron’s 12) should definitely not have faces with the same differing hole pattern that the Roman Dodecahedons do. This is because that, whilst a dodecahedron can be re-positioned on its 12 different faces to show the month, using the 20 faces of the icosahedron in the same way would not perform any useful task for calendar keeping. Rather than using the icosahedral faces directly, we may instead use the 12 vertices (nodules). This may be accomplished by, e.g. by putting a ring, ribbon or other marker on one nodule to show the current month. For this reason we may expect that the icosahedrons will not have unique identifying marks to delineate the faces, but, if present, will have markings useful for delineating the vertices instead. However unique marks are not strictly required either for dodecahedrons or icosahedrons.

## Further Predictions:

The following are predictions which are verifiable, but not directly falsifiable:

• If an analemma scale is used, then then the obvious choice of information to mark on the faces would be to show the months containing equinoxes or solstices (furthermore, equinoxes are more likely to be marked than solstices, since the largest and smallest holes likely correspond to the months containing solstices)
• If all or most of the holes have markings, and they are not just month numbers, then the next most useful piece of information to include would be an indicator of whether the given month was 29 or 30 days long.

These suggest a subsequent line of research, verifying that the tests above hold for the available data.

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## Roman Dodecahedrons Part III

I was pretty happy with my previous hypothesis on the dodecahedron, the string idea fits so well with the observed shape of the nodules, and the device is very satisfying to use and play with.

However one thing that it didn’t explain at all was the differing hole sizes on the faces. I’d initially written it off as either the blacksmith conserving metal (bronze is expensive), adding artistic flair, or maybe being used for a string game variant (e.g. threading through holes worth different values).

However, hundreds of dodecahedrons have been found, and differing hole size is a consistently observed feature. Surely if differing holes were an ‘optional’, rather than  functional, then at least some of the builders would have come up with different versions. But no, an array of different sized holes is a too strongly conserved a feature for them to be non-functional.

Let me cut to the chase, and I’ll explain how I got there. Basically, here’s the idea:

Roman Dodecahedron, and a contemporary device

So, I’d previously considered whether the differing holes implied mounting on a cone, and, since that would allow up to 12 different mounting heights, that it might possibly imply it was a calendar style device. I’d discounted it however, as the Romans used the Republican/Julian calendars which, although it does contain 12 months (usually), has month lengths which are very irregular, and I didn’t see a calendar-dodecahedron as an appealing labour saving device, or ‘desk toy’, given that.

However recently I was watching a Clickspring video, when I heard him mention the Egyptian Civil Calendar: https://en.wikipedia.org/wiki/Egyptian_calendar

Which neatly divided the year into 12 months of exactly 30 days, and then had an intercalary month of 5 days to make it fit neatly with the solar year. Simple and elegant, and I’d imagine it made planning business and trade very straightforward:

Compare that to the nightmare of the calendar of the Roman Republic (with months lasting either 22, 23, 27, 28, 29, or 31 days, but never 30):

Or even the slightly more logical Julian calendar:

Watching Clickspring’s explanation and hearing of the simplicity of the Egyptian calendar made me suddenly realise that while the Romans obviously used their own calendar, the outlying portions of their empire may not have. I did a brief bit of searching, and found this Gaulish calendar for example:

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

Which has 12 months, all consisting of 29 or 30 days. (As well as a couple of further tweaks, to ensure solar accuracy on a longer cycle. For example, the month ‘Equos’, which normally lasts 30 days, drops to either 28 or 29 days on two out of every five years. As well there are two 30 day intercalary “bonus” months each of which only occurs once every 5 years). While that all might sound complicated on paper, in a diagram it actually looks much simpler:

The point being, that while for an ancient Roman a calendar is a wildly varying thing (with months varying between 23 and 31 days), whereas for an ancient Gaul, it’s a far more regular object, almost always involving 12 months of 29 or 30 days. That makes it very amenable to representing with the dodecahedron.

I’ll lay out my hypothesis for how the device functioned, and in a separate post I’ll explain the strengths & weaknesses of the idea, as well as proposing some specific predictions & falsifiable tests that I’d wager will be supported under closer examination of the artefacts.

I took a spare  broom handle, and spent some minutes on a belt sander until it was approximately conical. I’m sure using a wood lathe (even one made from saplings) would get a far more regular outcome:

I then lasercut 12 washers, with differing inner diameters, and installed them on one of my previous dodecahedrons, taking care that opposite sides should not have very different diameters, as it would cause fouling:

There is then the consideration of how the hole sizes should be distributed to make a good calendar. To my mind there are only two simple choices. The first is a simple linear sequence, with the hole sizes either smoothly increasing or decreasing throughout the year, e.g.:

The second way would be to arrange the holes such that the position on the cone shows some aspect of the season. (e.g. length of the days, or height of the sun at zenith being the two most obvious properties). In such as case the arrangement would look something like this, which I dub the “Analemma sequence”.

If you wanted to get super fancy, you could even incorporate this as part of a sundial, with, for example, an external gnomon casting a shadow at noon to the dodecahedron at the appropriate height. This is a strictly unnecessary embellishment though, and as far as I know nothing in the dodecahedrons implies that it was done.

I’ll go with a simple linear sequence of hole sizes for my model, but if we find dodecahedrons with definite non-linear arrangement of hole diameters, then analemma spacing (either symbolic of day length, or a literal measurement as with a gnomon arrangement) is the obvious hypothesis to consider.

Then after assembling the model, I spent a while considering what the natural way to mark the days would be. In other words, how might you wind the string so that the number of days represented is most visible with the minimal of effort. (Absolutely no pointing or counting on fingers should be needed to interpret the result).

If the dodecahedron is mounted vertically through a pair of holes, as seems most likely, then a series of layers in the edges immediately become obvious. The top and bottom faces have 5 edges each, then there are 5 verticals leading down the equator, which is exactly 10 edges around.

And if we start from the bottom, this way of winding is most obvious, as it allows quickly counting in groups of 5:

I think starting from the bottom is most likely, as unused string will hang down out of the way. (Whereas if wound from the top it will hang down and be potentially confused with a legitimate edge to count). Also, by attaching the string to the base of the cone, after being looped around to show the appropriate number of days, the remaining string can then be tied back around the base of the cone, firmly securing the dodecahedron in place. That would make the whole arrangement suitable for travel, without any danger of losing your place in the calendar.

Here’s the result. You can see at a glance that it’s showing the 5th month, and the 11th day:

When one considers the dodecahedron, with its 12 faces, and 30 edges, it seems perfectly natural to associate that with calendars & timekeeping. Even without any mystical element, it’s a good fit, and I’d be happy to use it as a desk ornament.

But then if you throw in a bit of a religious or mystical connection, (e.g. a strong cultural attachment to the 12 signs of the zodiac, the apparent shape of the celestial sphere, a somewhat overenthusiastic devotion to the platonic solids, etc.), then I can easily imagine the dodecahedron being not only a functional village calendar, but also an object of reverence, both functional and cultural. Would the winding of the new day have a ritualistic aspect to it? I can imagine that getting to day 15 (and celebration entering the new fortnight) being accompanied with some joy. I can certainly imagine that recording getting to the end of winter being a time of celebration.

This might easily have been carried as a sceptre or staff by someone of importance (chief, village elder, druid?). Imagine a farmer scurrying up and asking them how long till they can plant their crops. They inspect this mysterious and arcane object at the end of their staff carefully, before nodding sagely and answering “Exactly 17 days”. Wow, clearly they’re a wise and erudite chap!

There’s one more bonus to this hypothesis. Although Roman Dodecahedrons are by far the most common shape, there’s at least one example of a Roman Icosahedron which has been found as well:

https://en.wikipedia.org/wiki/Roman_dodecahedron#/media/File:2018_Rheinisches_Landesmuseum_Bonn,_Dodekaeder_&_Ikosaeder.jpg

An icosahedron also has 30 edges. Is there also a similar natural way to wind it? Yes, here’s one approach:

Note that this is essentially the same as the dodecahedron winding (with layers of 5,5,10,5,5). This is not surprising, as the icosahedron and dodecahedrons are ‘duals’ of each other, and from memory the midpoint of each edge is the same between both?

But here’s the nifty thing. Since the 20 faces of the icosahedron are not a useful number for counting months, we can predict that any Roman Icosahedrons found will not have the same series of varying diameter holes on their faces. (Instead, to count months, the obvious way would be to use the 12 nodules instead, e.g. by tying a ring, ribbon or other marker to the nodule, or just to start the string winding there).

This is indeed what we see on the example of the Roman Icosahedron above. The 20 faces do not have varying diameter holes, but instead have what appears to be the same pattern repeated on faces (presumably just decorative).

In the next blog post I’ll examine the strengths and weaknesses of this hypothesis, as well as proposing some ways of falsifying it.

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## 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).

## 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.

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

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.

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.

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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:

Then the last 5 minutes are displayed as yellow:

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

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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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.

We used the pattern and instructions from here:

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:

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:

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.
• 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:

• 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]

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…

## Waves at Bessel-on-Sea

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

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:

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: :

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

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].

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:

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:

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 :

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:

Files here for anyone that wants to make their own:

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

## The Crystallographer’s Watch

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.

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

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.

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.

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.

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)