Roadside Picnic – A Nitpick on How to Shoot the Earth

This blog post is all about a line from the Russian scifi novel “Roadside Picnic“, written in 1971 by the Strugatsky brothers. The book itself went on to influence huge amounts of science fiction and cinema, including the 1979 film “Stalker” by Tarkovsky. (And later a number of video games along the same lines).

The rough premise of the novel is that Aliens have briefly visited earth, before departing again. We get this beautiful framing of the Aliens from the learned professor character:

A picnic. Picture a forest, a country road, a meadow. Cars drive off the country road into the meadow, a group of young people get out carrying bottles, baskets of food, transistor radios, and cameras. They light fires, pitch tents, turn on the music. In the morning they leave. The animals, birds, and insects that watched in horror through the long night creep out from their hiding places. And what do they see? Old spark plugs and old filters strewn around… Rags, burnt-out bulbs, and a monkey wrench left behind… And of course, the usual mess—apple cores, candy wrappers, charred remains of the campfire, cans, bottles, somebody’s handkerchief, somebody’s penknife, torn newspapers, coins, faded flowers picked in another meadow

And the novel, set 30 years after the visitation, covers the lives of the various people who now live in a world with dangerous and incomprehensible artifacts, originating in these bizarrely contaminated “zones”.

The quote in particular that I was struck by is this (emphasis mine):

“The Pilman Radiant is simplicity itself. Imagine that you spin a huge globe
and you start firing bullets into it. The bullet holes would lie on the surface in a smooth curve. The whole point of what you call my first serious discovery lies in the simple fact that all six Visitation Zones are situated on the surface of our planet as though someone had taken six shots at Earth from a pistol located somewhere along the Earth-Deneb line. Deneb is the alpha star in Cygnus. The Point in the heavens from which, so to speak, the shots came is the Pilman Radiant.”

and later on (emphasis mine):

“It suddenly occurred to me that Harmont and the other five Visitation Zones — sorry, my mistake, there were only four other sites known at the time — that all of them fit on a very smooth curve. I calculated the coordinates and sent them to Nature.”

at first glance that seemed to make perfect sense to me. I could imagine in my head a diagram much like this one, which shows the ecliptic and the tilt of the earth offset from that. What could be more natural than for an external celestial event than to lay upon a Great Circle path across the Earth?

587px-Earths_orbit_and_ecliptic

But the more I thought about it, the more I realised that it doesn’t make sense. If you fired shots from Deneb at Earth, then there’s no reason why they actually need to lie on any single curve.

There’s at least 3 ways I can think of to interpret the professor’s discovery.

The first way is that all three shots were fired at the same time, and they happened to intercept Earth. (i.e. they were not aiming at Earth’s centre of mass or anything). This makes most sense if the aliens didn’t even stop, and the unearthly condition of the “zones” is just what happens when ships powered by some reality-alterning warp drive technology passing through ordinary matter.

Roadside picnic interpretation 1 - v01

This would result in 6 sites, all along parallel lines. However there is no curve which explains or allows you to predict the others. (Hypothetically, if you knew about site E, F & D, this would allow you to calculate C. However A & B would not be determinable without knowing at least one of the pair. )

So this interpretation is not a good fit for the story.

The second way to interpret the scientists words are that all the “shots” were aimed at the centre of the earth, but fired at different times. This is consistent with the mentions of spinning in the description. And perhaps the alien technology relies on using the earth somehow. (e.g. they need to aim the teleporter at our gravity well, or something) The results would look like this;

Roadside picnic interpretation 2 - v01

This would result in all the sites having the same latitude North or South. All the “entry wounds” would be in one hemisphere, and all the “exit wounds” would be in the opposite hemisphere.

However this doesn’t fit the story at all, since all the sites having a common latitude is not something you need to do any indepth calculations on, or fit a curve to. And there’s no real need to write to Nature to inform them of your amazing discovery, since everyone with a table of the coordinates would immeidately see the pattern themselves.

The third way to interpret the scientist is this. The shots are fired at different times, and intersect the earth without being aimed at the centre. This results in all the lines having the same angle to the axis of Earth’s rotation, but that’s about it:

Roadside picnic interpretation 3 - v01

There’s no curves, or great circle routes, or really anything you can do with this model. True, if you were informed of the location of a new site [and the time of the hit], it allows you to infer that there is a sister site on the other side of the planet to investigate. But that’s not the sort of calculation the professor described.

My conclusion is either;

  1. This was something quickly thought up for a scifi-story, and 50 years later I’m over-analyzing it.

2. The Dr Pilman character made a mistake. And that his greatest achievement “The Pilman Radiant” is actually a coincidence or misinterpretation, and humanity knows less than we think about the aliens visitation. In fact, given how apathetic and nihilistic his character is about the whole situation — He says at one point that most important discovery that Humanity has made is just that the visitation has occurred. And that it’s not important who they were, or where they come from, etc. — I think he would be rather delighted to know that his main idea is wrong.


Edit — 1 hour after posting the original. Jess and Scott helped me realise that if there was a laser beam stretching across the cosmos, and Earth “wandered into” that beam as part of its orbit around the sun, then it would indeed make a smooth curve. 

So if we have the point of view of the “giant matter transmitter canon” on Deneb, and Earth moves into that beam from left to right: 

Roadside picnic interpretation 4 - v01

Then subsequent shots will hit the earth at the same angle, and different latitudes, and the longitude will depend on how far the Earth has rotated in the day/night cycle compared to how fast it moved in the annual orbit around the sun. 

Roadside picnic interpretation 4b - v01

If we look up the speed of the Earth at the equator, it’s about 0.47km/sec. And the speed that the Earth orbits the sun is around 30km/sec. So clearly the picture on the right is much closer to the truth. 

Darn. So the events described in the novel actually make sense. I’m kind of disappointed. 

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Backrooms Async Lab Diorama – One Day Build

Here’s a quick one day build project, inspired by Kane Pixel’s awesome youtube channel, and this video in particular. 

I lasercut the enclosure out of 7mm plywood. And used cheap adhesive acrylic mirrors on all four sides of the inner room. Then I lasercut some textured wallpaper pieces, and spent a bit of time arranging them carefully so that no matter what angle you view it from, you can’t see the entry threshold inside. 

The lighting inside is neopixel (WS2812) controlled by an arduino nano. I used blue lighting for the “lab” section and yellow lighting for the backrooms, but it’s all controllable. I was going to add fluorescent flicker to the yellow portion, but wasn’t sure if that was cannon or not…

Shoot me an email if you want the files to make your own. 

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Do flat tyres make your speedo lie? (Part 2)

OK, I decided to do a bit more testing, using the CANBUS data of the car, given that a few people have pointed out that “Indirect TPMS” (Tyre Pressure Monitoring Systems) exist, and use the ABS wheel encoders to infer if a tyre is flat. I was really puzzled by these systems, since it looked like my first round of tests proved that (on my tyres at least) it shouldn’t work.

(In theory Indirect TPMS also uses Fourier analysis of the encoder data to look for resonance. Which I’m sure exists. But I’d be fairly skeptical that normal tyres show enough of a radius change for flatness to be reliably detecting without looking at Fourier stuff)

I did two runs. One with all tyres full, and the other with the rear-left tyre deflated to 150kPa (65% of normal pressure). That level is low enough that I can feel a fairly strong pull as the car “wants” to go to one side, but it’s high enough that I can be fairly sure my experiment won’t break my tyres and cost me money…

I made sure to take the same route, starting and ending point, and used a stretch of road that was quite straight for most of the way. CANBUS frames were logged to my laptop, then decoded in Python.

Here’s the results. As you can see there’s only fairly subtle differences in the speed, but it’s very hard to tell when compared with the speed differences that occur during turns:

Tyre speeds - full v01

Tyre speeds - flat v01

A better way to analyze it is to compare the ratio of the wheels (Front Left / Front Right), and see how that varies when compared to the steering wheel angle. Thanks to the CANBUS data I can easily plot that and see where the centre of mass of the data points is:

Ratio of wheel speeds - full v01

Ratio of wheel speeds - flat v01

You can actually see that there probably is a difference, but it’s a damn subtle one. Deflating one tyre almost enough to be a safety issue caused maybe a 1% difference.

Realistically I should do this several times to average out and see if the effect persists. This test involved me driving down the street and having to wait for traffic and lights, with lots of interruptions and turning around. So maybe the results would be different if performed again. But I’m happy to have narrowed down the size of the effect (if any) to be very small.

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Do flat tyres make your speedo lie?

This is a question that I’ve seen debated a few times, both in person and on the internet. It’s not quite in the same league as the immortal argument starter “airplane-on-a-treadmill”, but it has many of the same features.

To be clear about what I’m talking about, the question is whether having either flat or overinflated tyres would cause your car’s speedometer to display a significantly incorrect value (let’s say at least 5%). I’ve seen people advocating that people need to check their tyres, otherwise they could get false speeding tickets. For the sake of this discussion we’ll consider the simple case where the vehicle is traveling in a straight line, and the tyres operating with constant contact with the road, with no slip. Cornering and such will be left out.

At the heart is a couple of conflicting intuitions as to what happens.

The first intuition is that the height of the tyre, and hence the radius, will change as air pressure changes.
Intuition 1 change in radius v01

The second intuition is that regardless of the radius of the tyre, the circumference stays the same. And since each part of the tyre rolls against the ground, it doesn’t matter if it’s smushed down into a tank-track, the length covered by one rotation stays the same:
Intuition 2 constant circumference v01

The third intuition of interest, is that the tyre will physically stretch as more air is inserted, thus changing both the radius and circumference:

Intuition 3 Balloon effect v01

These intutions pull in different ways, and focusing on only one can lead to the conclusion that the answer is easy and obvious. I was kinda fed up with seeing people on the internet arguing about it, and basically no-one doing the actual test. So I decided to do my own experiments and see what the results were.

I started by making a simple model to test the assumptions. I used a hot wire cutter to make two foam discs, and used a lasercut circle of wood in the middle as a hub. I then marked out a line on the edge, and measured how far it takes the tyre to roll one revolution along the table. As I thought, the results were significantly different if the wheel was “smushed” down by pressure, than if the wheel was rotating with no load:
Foam tyre compressed v01

But then I realised that modern tyres have steel bands “radials” in them, which act like the hoops on a barrel, and prevent expansion. So to replicate this, I wrapped one of my fake tyres in packing tape (which has amazing tensile strength):
Prototype tyres v01

When I did the tests again, I could clearly see that the difference between a loaded and unloaded tyre almost disappears:
Rolling fake tyres v01

So, does the real car behave in the same way as the fake tyres? In order to test this I needed to measure the car’s reported speed (from the speedometer), as well as a source of ground truth. I spent a while trying to get OBD-II data decoding on my car working well enough to log directly, but found it very frustrating dealing with the various devices.

I then realised that I was over-complicating this, and with a bit of searching I found an app for my phone that put a GPS overlay onto the camera feed. Then all I needed to do was tape my phone pointing at the speedo, and drive around. Then I could later look at any frame of the video and have a synchronized readout of both gauges.

(I’ve since figured out what I was doing wrong, and realised that rather than dealing with OBD protocols, I can sniff data directly from my car’s CANBUS at much higher rate. But the rest of this analysis is done with the video data).

I decided to try several “runs” in the car, with the tyres at various pressures far below and above normal ratings. My car is 230kPa, so I tried every pressure from 150-275kPa in 25kPa increments. This represents 65% to 120% of normal. (I didn’t really want to go much underneath 150kPa, as the tyres already looked fairly saggy to my eyes, and I didn’t want to risk damaging them and needing to buy a full replacement set).

I decided on a nice, fairly straight stretch of road with a high-ish speed limit ( 70km/h), with a service station at each end. That way if I had a problem with leaks or flats something I’d never t be far from help. Before each run I raised or lowered all four tyres to the same pressure, confirmed it with a handheld gauge, and drove, making sure to consistently record from the same starting point. Speed during the run will vary because there were a couple of traffic lights on the route, but I tried to stay just below the limit at a nice constant 60km/h to get as close as possible to measuring steady state.

After recording all the videos, I went back to the computer and started looking at the footage. Straight away I realised that it’s quite annoying to constantly pause and unpause video to make notes. Also it’s very difficult to pause at a regular gap (e.g. every 2 seconds), and by doing so I’d be introducing a lot of uncertainty into the measurement. This was a problem because there’s obviously a lag between the car’s speedo and the GPS, that I need to correct for.

To get good raw data out of the video, I used the free software “ffmpeg” in order to automatically “slice” the video at regular intervals into a folder full of still frames. According to my notes the magic incantation I used was:
“ffmpeg -i video_150kpa.MOV -filter:v fps=fps=1 ffmpeg%03d.png”

I then sat down in excel and transcribed the images to a spreadsheet. Once I had that I could analyze it in Jupyter notebooks with Numpy, Pandas, and Matplotlib.

The first problem was that there is clearly a scaling issue, as welll as a delay between the speedo velocity and the GPS. (Not unexpected, GPS tends to lag since it relies on calculation of recieved signals).
Scaling error v01

I calculated the mean squared error of the signal after applying various amounts of delay. You can see it’s quite consistent at 1sec for all the runs:
GPS Time lag v02

The second bit surprised me, the scaling always seems to be off by around 11%.
Speedo scaling v02(Later, when I analyzed raw CANBUS frames from the vehicle, I could see the speedo CANBUS values are actually different from the indiviudal tyre speed CANBUS values by a nice round 10%. I strongly suspect Toyota makes the speedo intentionally read exactly 10% fast to encourage more economical driving).

At any rate, if we now replot the data using the values of 1sec delay and 110% scaling, we can see the following:
Delayed and scaled data v01

I can’t see a difference in the above plot. But just to be sure, here’s a heatmap of all the data points, with GPS speed vs speedo speed. The diagonal line is the expected correlation, and the outer two lines are 2% higher and lower. You can see that vast majority of the data’s mass is within +-2% of the expected value.
Heatmap plot v02

So it looks like there’s basically no difference between ridiculously flat and ridiculously inflated tyres as far as your speedo readout goes. Maybe other tyres would behave differently, but I doubt it.

This was a fun experiment to do, and it didn’t take more than a couple of evenings.

[Edit; I now have done more tests in Part 2!]

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Display box for a KP1-4 High Voltage Variable Cap

I was recently at the Central Coast Amateur Radio Club’s annual get together. And I saw this beautiful glass vacuum tube there.

It’s a vintage Russian KP1-4 high voltage capacitor. Basically two metal plates in a vacuum separated from each other by a small distance. What makes this unit cool is that it has a dial on the top to allow you to vary the spacing, and hence the capacitance. There’s some flexible vacuum bellows, and a leadscrew to allow you to change what’s inside the high vacuum glasswork without breaking the seal.

I figured the unit was going to be a bit fragile to have on the shelf by itself, and also tricky to connect to if I wanted to experiment. So I made this lasercut plywood box to hold it, and have acrylic windows to stop prying fingers getting near the HV. There’s a pair of connectors on the side so it is somewhat functional. (Although I’m sure the bandwidth of my banana plugs would be terrible if I actually tried to use this for radio stuff).

Interestingly I can’t see any “getters” inside the vacuum enclosure. Perhaps there is a hidden one, or maybe it’s just that as the whole assembly stays cold (no thermionic emission), it is less sensitive to bad vacuum than other tubes are.

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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

Hardware of the spiral visualizer – Uses Teensy 3.6 for everything and acts as both MIDI input & output devices.
(Ignore the letters on my keyboard. It was a joke from a previous project)

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:

https://github.com/mechatronicsguy/SpiralMusic_python/blob/main/README.m

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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:

Face graph of a hypercube, done in origami and string style

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

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Roman Dodecahedrons – Part IV

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:

C:\Users\gsmith\AppData\Local\Microsoft\Windows\INetCache\Content.Word\Month_length_variation_horiz_Egyptian_v01.png

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):

C:\Users\gsmith\AppData\Local\Microsoft\Windows\INetCache\Content.Word\Month_length_variation_horiz_RomanRep_v01.png

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:

C:\Users\gsmith\AppData\Local\Microsoft\Windows\INetCache\Content.Word\Month_length_variation_horiz_Coligny_v01.png

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:

C:\Users\gsmith\AppData\Local\Microsoft\Windows\INetCache\Content.Word\Roman dodecahedron calenda v01.jpg

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).

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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