Antikythera Mechanism

[David TocherParticipant @davidtocher94033]

Some preliminary results from a least squares fit to the largest sector gives an interesting answer. John Kinsella, a whizz with MATLAB, did a least squares fit to find the centre (xc, yc) and radius (R) of the 37 holes (xi,yi) in sector S3. I downloaded the measurements used in the BHI paper from

https://dataverse.harvard.edu/file.xhtml?persistentId=doi:10.7910/DVN/VJGLVS/WIQJHP&version=3.0

John used a gradient method (Gauss-Newton) to minimise the sum of the squares of (xi -xc)^2+(yi-yc)^2-R^2 wrt xc, yc & R.

Results using optimal xc, yc, R:

mean of angles in degrees: 1.00577.
St dev of angles in degrees: 0.0929292.

R: 77.3073

To me that look like 360 holes round the circumference. The SD (variance) is quite large which reflects the imperfect marking out and drilling the holes.

We need to do a bit more works testing these results against the other sectors.

The inter-hole angles based on the calculated centre are;

Angles in degrees:
0.97
1.08
1.02
0.90
1.01
0.94
0.98
1.04
1.22
0.91
1.04
0.88
1.07
1.07
1.07
0.95
0.98
1.07
1.02
0.95
1.14
0.91
1.09
0.90
1.07
0.96
1.07
0.96
1.07
0.89
1.19
0.83
1.06
0.97
1.10
0.84

Edited By David Tocher on 18/10/2021 19:44:59

[SillyOldDufferModerator @sillyoldduffer]

Posted by David Tocher on 18/10/2021 19:30:59:.

…

To me that look like 360 holes round the circumference. …

360 is possible: it's the number of days in the ancient Egyptian calendar year.

Dave

[Michael GilliganParticipant @michaelgilligan61133]

Posted by SillyOldDuffer on 18/10/2021 20:28:56:

Posted by David Tocher on 18/10/2021 19:30:59:.

…

To me that look like 360 holes round the circumference. …

360 is possible: it's the number of days in the ancient Egyptian calendar year.

Dave

.

Particularly interesting, given that the team on the BHI paper settled for 354

MichaelG.

[David TocherParticipant @davidtocher94033]

I think that a Bayesian approach would be better – look at the possible number of holes which could be one of the set of likely candidates such as (but not restricted to); 354 Athenian#, 360 Babylonian&Egyptian, 365 solar, 366 sidereal (assuming it's related to an annual cycle or some sort), assign some prior probability to each and see how the evidence changes these probabilities.

# BHI value

Edited By David Tocher on 18/10/2021 22:52:00

[David TocherParticipant @davidtocher94033]

A correction to the S3 results; number of holes 357.8

The next largest sector S1; number of holes 360.7

Sector S1
xc: 80.1557 yc: 136.902 R: 78.5604.
mean of angles in degrees using least sqs est of xc, yc, & R: 0.998133.
St dev of angles in degrees: 0.0810382

I think it's clear the results from this approach show the BHI estimate is too low.

[Michael GilliganParticipant @michaelgilligan61133]

For those whose Statistical expertise is as rusty as mine; this is a convenient aide memoire:

**LINK**

https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_OpenIntro_Statistics_(Diez_et_al)./03%3A_Distributions_of_Random_Variables/3.01%3A_Normal_Distribution

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Plug-in David’s values for Mean[s] and Standard Deviation[s] and the difficulty becomes apparent.

I maintain that the data-set is not sufficiently robust for a confident estimate of the number of holes in the full circle.

MichaelG.

[FarmboyParticipant @farmboy]

I still agree with MichaelG that it is almost impossible to acurately reverse engineer the full disc from this small segment but I had to try, within my limits, to explore things further, purely for my own satisfaction.

Since I have very little understanding of statistical analysis or trigonomotry I resorted to basic geometry as originally suggested by Neil. I loaded the photo into TurboCad and drew a cross-hair over the apparent centre of every tenth hole from 2 to 72. I then drew a series of overlapping chords spanning sets of 20 holes (2-22, 12-32, etc.) and drew radius lines. Next I drew arcs centred on the intersections of various pairs of radii. The arc which apeared, to my eye, to most closely follow the holes has its centre a little below the one originally marked on the photo. Using this new centre I then measured the angular displacement between 2 and 72, which was 71.36 degrees. I believe this gives an angle per step of a little over 1.0194, suggesting a full circle of just over 353 holes.

I'm certainly not qualified to challenge the experts but I would suggest that the marked centre on the photo is at least open to question. Given the range of possible results found by different experts perhaps this is not surprising.

Mike.

[FarmboyParticipant @farmboy]

Having now downloaded and started to read the BHI paper it seems my trial-and-error findings are worth even less than I thought The apparently intact section between breaks is only around half of what I assumed so distortion probably accounts for most of the difference. There are in effect several centres, each relating to only a part of the segment.

But I enjoyed the exercise

[David TocherParticipant @davidtocher94033]

The standard statistical analysis for estimating the population mean from a sample is;

The mean error can be assumed to be zero. The sample SD can be used as to estimate population SD. The standard error of the sample mean with a sample size n is SE=SD/SQRT(n-1)

Plug in the numbers for S3;

sample stats; mean=1.00577, sample SD=0.0929292, n=37

SE=0.0929292/SQR(36)=0.015.

With this sample the confidence limits for the true hole spacing is; sample mean+/-1.96SE=1.00577+/- 1.96*0.015= 1.00577+/-0.03 degrees

We are confident, at the 95% level, that the true inter-hole angle lies between 0.97577 (N=348) to 1.03577(N=369) which as Michael wrote doesn't tell us much!

This ignores the fact that the true mean is not a random continuous variable but is discrete and restricted to plausible values.

[Michael GilliganParticipant @michaelgilligan61133]

This is so far beyond my comprehension of statistics that I hesitate to even mention it … But:

I have just been ‘attending’ a [Royal Microscopical Society] Zoom presentation about ‘Pattern Matching’ at the molecular level in biological samples.

The young genius who is doing this work mentioned his use of the Akaite Information Criterion

If David, or any other statistical expert, has the necessary skills … this may, I suspect, help with the guessing-game that this incomplete ring of holes presents.

Alternatively, I am quite happy to be told that it is irrelevant

MichaelG.

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Ref: __ **LINK**

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

[David TocherParticipant @davidtocher94033]

I hate to be thought of as an expert especially after M Gove's comments! My background is chemical engineering and Operational Research not statistics.

There is only one model to explain the observed data. The holes are on the perimeter of a circle. The two parameters are the number of holes (N) in the complete wheel and the radius.

A quick read of the article quoted suggests is best suited to comparing difference models rather than selecting the best parameters of a model.

I have tried a least squares fit but the data is very noisy. A Bayesian approach looks at the possible alternatives (i.e. restricting the possible value of N to likely candidates) and see which the data best supports. The quoted article in wiki outlines the Bayesian method.

If the weather remains awful I'll have another look at the problem in the next few days.

[Michael GilliganParticipant @michaelgilligan61133]

Thanks, David

In my innocence, I was wondering if someone cleverer than I could interpret different numbers of holes as being different ‘models’

It would appear not … so I think the case rests for the moment.

MichaelG.

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P.S. __ for a crumb of comfort, read this:

https://www.spectator.co.uk/article/fact-check-what-did-michael-gove-actually-say-about-experts-

Edited By Michael Gilligan on 01/11/2021 19:38:29

[CalumParticipant @calumgalleitch87969]

Akaike, not Akaite. AIC is for selecting between different statistical models where there are many different possible factors that a particular model may use. For example, a model that assesses your risk of crashing a car (for insurance purposes, say…) could use your age, gender, hair colour, location, car type, engine size, and so on and so on and so on. However, the more factors you use, the more your model is just a complicated representation of your original data. AIC is used when you have a selection of similar models and want to select between them.

[Michael GilliganParticipant @michaelgilligan61133]

Posted by Calum Galleitch on 01/11/2021 22:40:18:

Akaike, not Akaite.

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My apologies … I stand duly chastised

MichaelG.

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As for ‘usage’ … The reason I thought AIC might be helpful is that its use was featured here in relation to Pattern Extraction from ‘sparse data-sets’ :

https://www.rms.org.uk/rms-event-calendar/2021-events/imaging-oneworld-modelling-and-determining-3d.html

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Edit: __ This will explain better than I could ever hope to do:

https://www.researchgate.net/publication/346647969_Nanoscale_Pattern_Extraction_from_Relative_Positions_of_Sparse_3D_Localizations

Edited By Michael Gilligan on 02/11/2021 07:56:05

[CalumParticipant @calumgalleitch87969]

That's a rather different meaning of sparse – from the statistician's point of view, our Antikythera problem here has an abundance of data, we just don't like the answer. Sparse is more like (to continue the car insurance analogy) knowing age for some, gender for others, make of models for some, and not having a full set of factors for most of your datapoint.

One of the problems with statistics in general is that it's a very new subject (Akaike himself died only in 2009) and beyond the basics, it gets fearsomely mathematical very quickly. Moreover, in many domains it's a skill that is called on infrequently. I have a fairly solid grounding in conventional and Bayesian statistics, but I'm still hesitant to weigh in on this problem.

[Michael GilliganParticipant @michaelgilligan61133]

If you say so, Callum

I freely admit that I am totally out of my depth … but when he was ‘deducing’ how many axes of symmetry there ‘must’ be : I thought that perhaps the technique was adaptable to the number of holes in the Antikythera circle.

AIC was just a final straw at which I thought we might possibly clutch.

My own efforts have demonstrated [to my own satisfaction] that, using the available data, we cannot reliably deduce the number of holes in that circle … I’m content to leave it at that unless/until further details emerge.

MichaelG.

[SillyOldDufferModerator @sillyoldduffer]

Posted by Michael Gilligan on 02/11/2021 17:05:38:

… using the available data, we cannot reliably deduce the number of holes in that circle … I’m content to leave it at that unless/until further details emerge.

David Tocher said: We are confident, at the 95% level, that the true inter-hole angle lies between 0.97577 (N=348) to 1.03577(N=369) which as Michael wrote doesn't tell us much!

And earlier, Neil pointed out there are four likely candidates:

365 days = 1 year

354 days = 12 lunar months

355 days = 13 sidereal months (the time for the moon to return to the same place in the sky)

360 degrees

As all the candidates are all in David's target area, it seems that the actual number can't be deduced because more information is needed. Perhaps the archaeologists could have another go at measuring the mechanism. Otherwise, I think we're stuck too.

Dave

[David TocherParticipant @davidtocher94033]

Mods – can you delete the post 02/11/2021 23:24:07? I've noticed a mistake that requires a rethink.

[not done it yetParticipant @notdoneityet]

We-e-e-ll,

I’ve just copied that posting so that I could compare with later thoughts to know/understand where the mistake was made….

[Neil WyattModerator @neilwyatt]

The plot continues to thicken!

[David TocherParticipant @davidtocher94033]

I used the least squares best fit of the hole position data, available from the Harvard website quoted in the BHI paper, for S1, S2 and S3 to fit the points on a circle and I assigned a probability of the true value of N for N=354, 355, 360, 365 & 366 at 0.13 and 0.01 for all other N (for N from 331 to 370). I ignored the other sectors as the number of point was quite small

The analysis modified these prior probabilities by the information obtained from the three sectors using Bayes formula.

N posterior probability

354 0.257

355 0.304

360 0.223

365 0.028

366 0.015

The probabilities for other values of N are less than their initial values.

The obvious conclusion I'd draw from these calculations is that a solar period of 365/366 is highly unlikely but the other three are more or less equal contenders for the true value for N.

I need to modify the least squares fit to force the radii for each sector to be the same. The initial analysis gave different radii which may or may not influence the results I obtained.

[david bennett 8Participant @davidbennett8]

Yes, an interesting puzzle. But I am miising the point of why the number of holes matter. Surely they are there for minor adjustments of the calendar dial? There would hardly be the same number

of holes as divisions on the dial plate if ony fractions of a day were needed to be adjusted e.g 1/4 of a day. Perhaps they were a kind of vernier adjustment .I repeat, I must be missing something obvious.

dave8

Edited By david bennett 8 on 06/11/2021 01:46:01

[david bennett 8Participant @davidbennett8]

P.S — see also post by pgk pgk 07/10/21 12:10:45  He beat me to it.

dave8

Edited By david bennett 8 on 06/11/2021 02:37:34

[Michael GilliganParticipant @michaelgilligan61133]

Posted by david bennett 8 on 06/11/2021 01:44:25:

Yes, an interesting puzzle. But I am miising the point of why the number of holes matter. […]

I repeat, I must be missing something obvious.

dave8

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Have you read the BHI papers, David ?

MichaelG.

[Michael GilliganParticipant @michaelgilligan61133]

Posted by John Haine on 05/10/2021 12:29:16:

Not quite to Neil's question, but there is an open-access copy of two papers on the mechanism on the BHI website, where they present the evidence and arguments for 354 holes. Clickspring, a/k/a C. Budiselic, is one of the authors.

**LINK**

https://bhi.co.uk/wp-content/uploads/2020/12/BHI-Antikythera-Mechanism-Evidence-of-a-Lunar-Calendar.pdf

Edited By John Haine on 05/10/2021 12:29:31

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As our discussion is revived, I’m re-posting John’s link for convenient reference ^^^

MichaelG.

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Edit: __ also featured here :

https://www.model-engineer.co.uk/forums/postings.asp?th=171503

Edited By Michael Gilligan on 06/11/2021 08:30:08

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