Antikythera Mechanism

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Just 351 that are all [ very probably ] in the wrong places

First hole = Last hole … So that one should be right.

MichaelG.

I was going for 352 as in your recent post. 351 is 9*39,which gets us nowhere, except you could do 39 by trial and error, followed by 11 ditto

Edited By duncan webster on 09/10/2021 19:39:17

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So was, I Duncan !

… All I was saying was that 351 of them will [very probably] be in the wrong place.

Although Dave has moved this to a problem of placing 352 holes … The real [Antikythera] issue is estimating the number of holes in a circle, based on a relatively small sample thereof.

The level of accuracy required to do that prediction successfully, to the nearest integer value, is astonishing.

From my table … if 352 is the right answer: we can only get it if we know the angle per step is greater than 1.020° and less than 1.026°

As I am sure you will realise … this is a VERY small tolerance.

MichaelG.

Added to the problem the original might have been distorted in the sinking of the ship, whatever point it got broken, the stresses of corroding etc.
However I would also expect to go to more depth than just a single x-ray of the object as nowadays there is more sophisticated 3D x-ray imaging, multiple wavelength, and possibly going into neutron scans.
That data would need to be examined for each hole effectively treating the light to dark transitions at the edge as if it were a sloping crater, then decide on possible values for how far down the crater you take as the edge. Process that round the hole and decide where the centre is depending on what you have taken as an edge value.
Once you have the multiple possible centres for each hole you can then process the array of holes.

Having recently had a presentation on machine learning (aka neural network) I feel this is just the sort of problem it can work on, almost instantly testing hundreds of variables from the data I outlined above and giving you a probability of the fit to each of your hole counts. Surely someone has done this as a 3rd year university project.

Even back when Michael Wright was working on it he was using X-ray tomography for imaging the object in 3D, and I'm sure they are using more sophisticated studd now.

I haven't really been following this, but are people saying that the project written up in HJ which I linked to, one of the authors being Clickspring, which concluded that the device incorporated a lunar calendar, is wrong?

Edited By John Haine on 09/10/2021 20:46:19

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Have you looked at the paper that Neil linked ?

The image is a composite from multiple ‘Computed Tomography’ scans

My worry is that the authors seem to have built the composite in Photoshop, and then picked-off values from the screen. As I have said repeatedly … I am not convinced that the accuracy of their data points is sufficient to make the ‘count’ estimate that they have provided.

Look at the “point” that they are using to define location #1 … That’s why I chose to start at #2

MichaelG.

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Edited By Michael Gilligan on 09/10/2021 21:31:58

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Not in so many words, John

What I am saying, however, is that [based on the evidence I have seen] … I do not find the estimate of the number holes on one particular ring to be convincing.

… It could conceivably still be the right answer … but , I would suggest ; only if the holes in the missing portion are positioned closer together than those in the image.

… it’s a matter of acknowledging uncertainty !!

MichaelG.

I think that determining the number of holes in this mechanism is unlikely ever to be solved from the currently available (physical) fragment. If in the future more of the mechanism was ever retrieved then it might be possible to count more accurately.

It seems to me than no amount of measuring, statistics, probability or other methods will give us the exact answer because we know there is a variation in hole spacing in the ones we can see and there may be similar variations in the missing ones.

Ian P

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Thank You, Ian

That’s what I have been trying to demonstrate !

MichaelG.

John,

I have just had another look at the BHI articles and, although the statistical analysis seems exemplary, I remain concerned about the quality of the input data.

[ remember the aphorism Garbage IN — Garbage OUT ]

On p3 they state:

“ Our procedure first set the scale in Fiji to match the provided reference of 50 μm per pixel (20 pixels per mm). ”

and then, on p4

“ Note that while the image's resolution limits direct measurement to only two decimals, we present all data to three decimals to reduce information loss when rounding mean measures of hole locations. ”

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I took a different [and perhaps better?] approach : I first enlarged the published image by 500% and saved it as .png to avoid additional .jpeg artefacts … and then measured angles, instead of XY coordinates.

Honestly : I am not convinced that measuring inter-hole distances to only two decimal places of mm is good enough for this exercise.

MichaelG.

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Edit: __ just re-posting this image, for ease of reference: 

Edited By Michael Gilligan on 09/10/2021 23:30:09

A quick calculation :

Radius 77.493
resolution of inter-hole measurement 0.01mm

0.01 ÷ 77.493 = 0.00012904391364
atan (0.00012904391364) = 0.0073936716

So each 0.01mm of their inter-hole measurement represents more than 0.007° of angle.

Now look back at my tabulation of hole counts.

MichaelG.

True.

One of the things it does is derive the sidereal movement from the solar day, so it might be redundant.

Neil

It contains a lunar calendar, but if these holes are for the master motion stepping the device around, it makes logical sense for it to fit the calendar year, with each step a solar day, rather than one of the derived motions.

I think that's what Mike Edmunds feels, more or less, he's writing a paper suggesting that errors in the manufacture mean we can't be certain of the actual hole count.

That's why he was interested in the possible accuracy with which the holes could have been drilled.

Neil

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

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

Dave

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

MichaelG.

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

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.

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.