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- 1 Am I missing something?
- 2 Mathematical Problem (for reference in discussion)
- 3 Original research tag
- 4 From User talk:220.127.116.11: Archimedes Codex: Suter board vs. Codex board for Stomachion
- 5 From Usertalk: IanMacM: Stomachion: Netz-Acerbi-Wilson article in SCIAMVS 5
- 6 Exchange with Slashme and David Eppstein
Am I missing something?
In trying to see how many ways the 14 irregular pieces could be put together to make a square, it now appears that Archimedes anticipated aspects of combinatorics.
- I am sorry, but I fail to see how this follows from the translated text provided in the article. Where does it say Archimedes was specifically trying to see how many ways the square can be constructed? And did he get a result, or even a constraint on the result? -- Cimon Avaro; on a pogostick. (talk) 11:39, 13 December 2007 (UTC)
I seem to recall seeing something in print saying he actually got a particular very large number. Maybe you're missing something simply because no one's inserted what you're missing into the article, and should have. I'll see if I can find anything. Michael Hardy (talk) 20:18, 13 December 2007 (UTC)
- One of the links provided at the bottom of the article makes oblique reference to some palimpsest or another having words that are suggestive of the combinatorics angle to the original text, but the whole thing seems very conjectural indeed. Very little actual text from archimedes himself is spelled out in that. Mostly it appears that the guy "recreated" archimedes thought processes based on vague clues in the palimpsest, without providing clear connection to the text. It may be that that abstract merely fails to mention the specific textual bits, or that there are none. I don't know. But the linked to abstract is far from conclusive in my view. -- Cimon Avaro; on a pogostick. (talk) 07:19, 14 December 2007 (UTC)
Mathematical Problem (for reference in discussion)
In the fragment in Arabic translated by Heinrich Suter it is shown that each of the pieces has an area that is a rational fraction of the total area of the dissected parallelogram. In the Greek version of the treatise, some investigation is made as to the sizes of the angles of the pieces to see which could go together to make a straight line. This might have been preparatory to consideration of the number of ways the pieces might reform some prescribed shape, for example the box in which the pieces were contained, although there is not enough of the Greek text remaining to be sure. If this is the case, then Archimedes anticipated aspects of combinatorics. Combinatorics often involves finding the number of ways a given problem can be solved, subject to well-defined constraints. For example, the number of ways of reforming a square using the pieces as proposed by Suter is 536 without distinguishing the result up to rotations and reflections of the square, but allowing the pieces to be turned over. One such solution is shown in colour to the right. However, if pieces are not allowed to be turned over, for example, if obverse and reverse can be distinguished or, in the case of Suter's pieces, on account of the sharpness of some angles, the corresponding number is 4. If rotations and reflections are treated as distinct, these numbers rise to 17,152 and 64, respectively. The counts of 4 and 64 may be verified easily as lower bounds by elementary group theory, but were confirmed as exact by Bill Cutler shortly after his determination of the count 536 and by the same computer program.
The Greek text of the fragment can also be found in the Bibliotheca Augustana website.
This is an English translation, with added disambiguation in brackets, of the text of the fragmentary Arabic manuscript (translated from Heinrich Suter's German translation in: Archimedis opera omnia, vol. 2, p. 420 sqq., ed. J. L. Heiberg, Leipzig 1881, as published in the Bibliotheca Augustana website):
" We draw a [rectangular] parallelogram ABGD, we bisect BG in E and draw EZ perpendicular to BG [to intersect AD], we draw the diagonals AG, BZ [intersecting AG at L], and ZG, we also bisect BE in H, and draw HT perpendicular to BE [to intersect BZ], then we put the ruler at point H and - looking to point A - we draw HK [to intersect BZ], then bisect AL in M, and draw BM. So the A-E rectangle is divided into seven parts.
Now we bisect DG in N, ZG in C, we draw EC and attaching the ruler to the points B and C we draw CO [to intersect DG], furthermore CN. Thus the rectangle ZG is also divided in seven parts, but in another way than the first one. Therefore, the whole square has fourteen parts.
We now demonstrate that each of the fourteen parts is in rational relationship to the whole square.
Because ZG is the diagonal of the rectangle Z-G, the triangle DZG is half of this rectangle, that means 1/4 of the square. But the triangle GNC is 1/4 of triangle DZG, because, if we extend the line EC, it comes to point D, and that means triangle GDC has half area of the triangle DZG and is equal to the two triangles GNC and DNC taken together; that means triangle GNC is 1/16 of the square. If we presume that line OC is orientated to point B, as we have drawn it before, so the line NC is parallel to BG, which is the side of the square and of the triangle OBG, so we get the proportion
BG : NC = GO : NO.
But BG is four times NC, and in the same way GO four times NO; therefore is GN three times NO, and triangle GNC = 3 ONC. However, as we have shown, triangle GNC is 1/16 of the square, that means triangle ONC = 1/48 of the square. Furthermore, as triangle GDZ = 1/4 of the square, and therefore GNC = 1/16 of that triangle and NCO = 1/48 of that, it remains for the quadrilateral DOCZ = 1/6 of the square’s area. According to the proposition that line NC [extended] intersects [ZE at] point F, and GE is parallel to CF, [and labelling the intersection of AG and CE as Q,] we get the proportion
EC : CF = EQ : CQ = GQ : FQ.
Because EQ = 2 CQ and GQ = 2 FQ, triangle EQG is double to the two triangles GCQ and EFQ. It is clear, that triangle EGZ = 2 times triangle EFG, because ZE = 2 FE. As the triangle EGZ = 1/4 of the square, that means triangle EFG = 1/8 of the square. This triangle is three times as big as each of the two triangles EFQ and GCQ, so each of these two triangles = 1/24 of the square A-G. And the triangle EGQ is double to each of the two triangles EFQ and GCQ, so it is = 1/12 of the square. Furthermore because ZF = EF, triangle ZFG = triangle EFG. If we now take away triangle GCQ (= triangle EFQ), it leaves quadrilateral FQCZ (= triangle EGQ), therefore quadrilateral FQCZ = 1/12 of the square A-G.
We have now divided the rectangle Z-G in 7 parts, and go on to divide the other rectangle.
Because BZ and EC are two parallel diagonals, and ZF = EF, therefore triangle ZLF = EFQ, and also triangle ZLF = 1/24 of the square A-G. Because BH = HE, triangle BEZ is four times the triangle BHT, because each of them is rectangular. As triangle BEZ = 1/4 of the square ABGD, triangle BHT = 1/16 of that. According to our proposition the line HK [extended[ intersects point A, so we get the proportion
AB : HT = BK : KT.
Because AB = 2 HT, and BK = 2 KT and BT = 3 KT, triangle BHT is three times the triangle KHT. However, because triangle BHT = 1/16 of the whole square, triangle KHT = 1/48 of that. Triangle BKH is double the triangle KHT, so = 1/24 of the square. Further, as BL = 2 ZL, and AL = 2 LF, triangle ABL is twice the triangle ALZ, and ALZ double the triangle ZLF. However, because triangle ZLF = 1/24 of the whole square, triangle ALZ = 1/12 of that, so triangle ABL = 1/6. But triangle ABM = triangle BML, so each of these two triangles = 1/12 of the square. It leaves the pentagon LFEHT = 7/48 of the entire square.
We have now also divided the square AE into 7 sections, therefore, the whole figure ABGD in 14 parts. Each of these fourteen parts is in rational relationship to the whole, and that is what we wanted."
However, Suter was translating unpointed Arabic (that is, the older style of Arabic without consonant pointing) in which, as he concedes (at fn 30), equals and twice are easily confused. At a crucial juncture (fn 6) in his translation he ignores this in making his figure a square; instead he makes a typographical error, equating, not the sides, but a side and a diagonal, in which case the figure cannot be a rectangle. Suter does know that, for the areal results discussed in the text, it suffices that the figure be a parallelogram. Suter may have been under the impression that the figure had to be a square, so made it one.
The dissection lines given by Archimedes' Stomachion from the Archimedes Palimpsest, as agreed by Heiberg and Dijksterhuis, are a subset of those of Suter's square board when the latter is subjected to a lateral stretch by a factor of two. This was recognised by Richard Dixon Oldham, FRS, in a letter to Nature in March, 1926. The letter triggered a wave of interest in the dissection puzzle, with kits for sale and a feature article in The New York Times that August; Popular Science Monthly had related items in its issues for November, 1926 and February, March, May and June, 1927, including Stomachion competitions with cash prizes. As each of the two unit squares is cut diagonally, the pieces can be arranged in a single square of side the square root of two, after the manner deployed by Socrates in Plato's Meno. Thus, the 14 pieces still form a square, even when the dissected board is not a square. Naturally, the count of solutions also changes, but here we now have three possible boxes: the two unit squares side by side; the two unit squares one on top of the other; and this single square of side the square root of two. The single square can be formed from four congruent right isosceles triangles, but this can also be accomplished with three right isosceles triangles, two congruent and the third double their area, as in Tangram, as well as in a third way where only one right isosceles triangle is formed. (Bill Cutler has made a comprehensive study of the associated counts comparable to those mentioned for the Suter board.) With two exceptions, the pieces can be formed from right angled triangles with legs in the ratios 1:1, 1:2 and 1:3, with some pieces similar to others, possibly at different scales; this observation provides a comparatively easy means to determine the proportional areas of the pieces.
Suter's translation can now be checked against the fragment of Archimedes' Stomachion in the Archimedes Palimpsest, which was presumably not open to Suter himself when he made the translation. Heiberg, followed by Dijksterhius, seem to have thought that the two texts, the Greek discussing angles, the Arabic areas, were so different there was no relation between the two, so may have felt there was nothing to check. But, in Suter's square, the diagonals cross at right angles, whence the nature the angles examined in the first proposition of Archimedes' Stomachion, whether acute or obtuse, is immediate, making it puzzling why Archimedes would prove this as a proposition. Instead, the first proposition, if it is seen to relate to the underlying figure of the dissected Stomachion board, sets up two squares side by side, in which setting the nature of the angles is more sensitive, but still tractable. The two squares separately suggest two frames of an iterative process. If this process is continued, rational approximations of the square root of two are obtained; now known, in the modern context of continued fractions, as the Pell numbers, they were familiar to the Greeks in the terminology of Theon of Smyrna as the side-diameter numbers. At each step of the process, the nature of the angles switches back and forth, highlighting the significance of the first proposition. Whether or not this has any bearing on Archimedes' Stomachion is another matter, but it does provide a geometrical means to handle recurrence relations if you do not have algebraic notation, such as subscripts, and the geometry is fully within Archimedes' capabilities. The same technique readily produces the bounds for the square root of three that Archimedes assumes without comment in Measurement of a Circle; if it were applied to the dissection of a square proposed by Suter, analogous bounds for the square root of five are obtained. The geometry can be worked independently of knowledge of Pell's equation or of the properties of convergents of a continued fraction, but to similar effect.
The dissection of the two squares side by side can also be seen as a layered composite or collage of instances of the diagrams for Elements II.9, 10. Viewed numerically, these propositions provide the link between the legs of right angled triangles in the unit square grid and those of right angled triangles in the overlaid diagonal square grid where the right angled triangles share their hypotenuse, although Euclid proves them in terms of geometrical lines. Thus, pairs of successive side numbers in the overlaid grid are associated with pairs of successive diameter numbers in the unit square grid, as remarked by Proclus, rather than side and diameter numbers being paired. The ratios within these pairs are then rational approximations for the tangent of π/8, in agreement with the iterative process of the previous paragraph, which is, in effect, an angle bisection algorithm. This reminds us that a natural setting for the side-diameter numbers is a nest of regular octagons. The analogue for the square root of three is not an angle bisection algorithm, but it does produce rational approximations to the tangent of π/12, with the regular dodecagon as a setting. Instead of simply approximating the square roots in each case, we can approximate the corresponding regular polygons. But, again, whether this was actually Archimedes' agenda is another matter.
Irrespective of the parallelogram adopted for the Stomachion board, the presence of several centroids of triangles on the board as further points of it might seem another Archimedean theme, not least as they facilitate the computation of the relative areas of the pieces. Most obviously, Q is the centroid of the right triangle GZE, so the pieces in that triangle are either one sixth or one third of it. But similarly K is the centroid of right triangle EAB, so triangle HKB has area one sixth of it. In On the Equilibrium of Planes, Archimedes makes use of the concurrence of the three medians of a triangle in the centroid, but does not prove it. As it happens, for both K and Q, we see two medians intersecting in them, but the pairings are not the same, although the triangles GZE and EAB are translations of one another.
Original research tag
Several unstated presumptions to obtain the number 17,152, the most troubling being that we reach for a Stomachion board proposed in a questionable translation of another text which does not match the Archimedes Codex. It is unclear where Dr. Netz ever comes to grips with this problem.
Unfortunately the source you are using is not reliable in this case precisely because of tacit assumptions. To be reliable in combinatorial mathematics you have to state the assumptions. The point about turning over pieces simply was not mentioned. The numbers 64 and 4 which you have now edited out were obtained by Bill Cutler who did the original research back in 2003. Somehow, they have just never been mentioned. But why not check with Bill Cutler at <firstname.lastname@example.org>. If you really want to stick with Dr. Netz, you need to edit out the phrase about turning over pieces, because he never mentioned it explicitly. That is why the source is tendentious and so less than reliable.
I have taken the liberty of undoing your edit in good faith while accepting your edit itself was in good faith. The problem for you is that what got formally published in 2003 was selective: it left out qualifying phrases that make the published counts precise; and it left out the alternative counts, when the qualifying phrases do not apply. You will notice that later on it is mentioned that there are three other boxes for the Codex board, so there are at least six other counts. Bill Cutler has those worked out, too, but they are not yet in general circulation. Naturally, I am encouraging him to publish his entire suite of results. Then they can be inserted, too. But the 64 and 4 have been known for six or seven years.
- As the saying goes, I don't make the rules around here. Wikipedia articles have to stick to what has been published and peer reviewed. The material added to Ostomachion is unsourced commentary, which is against guidelines. Other editors are justified in removing material of this kind. See also Wikipedia's guidelines on original research.--♦IanMacM♦ (talk to me) 20:15, 17 June 2010 (UTC)
That is your problem or rather the problem for Wikipedia, which is why I flagged the matter for you in the first place. I am grateful to you for your speedy response, as that will surely encourage others to pay closer, more critical attention. As you will see, attribition of combinatorial counts to Bill Cutler has now been included in the body of the text, while external links that allow readers to inspect the works of Suter, Gow and Ball have been added. Notice that the published materials are largely silent on these sources, while mention of Oldham has only recently been included in the website of Chung (and Graham); Gina Kolata's article in The New York Times in December, 2003 did not pick up on the backstory of Oldham's article there in August, 1926. But what is commentary? Is it commentary to observe that Sutter has a problem with his fn6: presumably he is thinking to tell us that the sides of the board are equal; instead he equates side and diagonal? Is it commentary to observe that Suter is making the false presumption that, if the pieces form a square, the board has to be a square, when perhaps the most celebrated and memorable scene from Greek mathematics is Socrates instruction of the slave boy in Plato's Meno? The entire published and peer-reviewed literature can fall into a trap like that. Fortunately in this case, Oldham rescued us from that trap in 1926. But you would hardly know that from published and peer-reviewed material this last decade or so, just as you would not know about Gow or Ball.
Might it not be useful to flag specific passages that are felt to need improvement or verfication, for example, by showing them in italics or red, so as to pinpoint more exactly where further work might usefully be done? What more do you want to see done on this entry?
- The article needs more sourcing, as the "Mathematical problem" section still reads as a commentary. Any criticism of the work on the problem by Reviel Netz would need reliable sourcing. Could you look out some more sourcing in this area?--♦IanMacM♦ (talk to me) 08:10, 20 June 2010 (UTC)
From User talk:18.104.22.168: Archimedes Codex: Suter board vs. Codex board for Stomachion
The Archimedes Palimpsest is known to scholars as the Archimedes Codex. If you read the entry Stomachion, you will see that comparison is already made between the Arabic and the Greek texts. It is the Arabic text that is the more complete, so complete that Suter produced a board for the Stomachion, which, in turn, was taken over by Netz, seemingly without comment or critical reflection on Suter's translation, even although Netz was handling the Greek text, not the Arabic text. However, the Greek text, as established before Netz, and seemingly agreed by him, suggests that Archimedes was studying a double square, not a single square. That is why the Suter board has to be stretched by a factor of two to bring it into line with the figures implicit in the Codex; the unpointed Arabic text supports this reading, in that, as Suter allows elsewhere in his translation, equals and twice are easily confused. It should be understood that Suter gives no reason for making his board a square. Equally, Netz gives no reason for seemingly ignoring the diagrams implicit in the Greek text for Stomachion, although elsewhere in his study of the Codex he places great enphasis on the diagrams as especially faithful to Archimedes. To sum up, while the Arabic text is the more complete, the Greek text provides a check - and, indeed, a corrective. But Netz goes the other way.
From Usertalk: IanMacM: Stomachion: Netz-Acerbi-Wilson article in SCIAMVS 5
In this scholarly article Netz and his co-authors repeat what are essentially the figures for Stomachion that have been agreed since Heiberg first published on the Archimedes Codex. It is clear that they set up a double square, not a single square. Netz, in his popular account of the Codex with Noel, berates Heiberg for his neglect of figures. Netz also argues that we come as close as is likely to Archimedes' original figures in the Codex. It is therefore a complete mystery why Netz opts in that book for the Suter Board, rather than the Codex Board. The Suter Board is associated with an unpointed Arabic text and is most obviously inconsistent with the Greek text because the diagonals of a square cross at right angles, rendering the surviving propositions in the Greek text trivial if they referred to the Arabic text. Suter conceded elsewhere that twice and equals are easily confused and that seems to be the explanation of why the Suter Board is mistaken. Netz does not argue for the Suter Board, he just adopts it; this is also what he did in working with Diaconis, Graham and others. It was Oldham, in a letter to Nature in March, 1926, who pointed out the reconciliation of the two texts. But he did not go into the geometrical or linguistic objections to the Suter Board.
I do not know whether you have a serious interest in the matter. I also do not know what you can do about the Wikipedia articles touching on Archimedes and Stomachion under Wikipedia policies, since it is certainly true that Netz has published his ideas in a popular book. It is just that his own scholarly article undermines the book, which, being a popular work, was probably not, strictly speaking, peer reviewed.
However, if you are interested, and would like to be sent the relevant material for your own inspection, do please indicate how that might be arranged.
Exchange with Slashme and David Eppstein
Slashme, in August, 2012, deleted a passage from Ostomachion on the grounds that it missed the point, but did not indicate what the point missed might be. It is not clear, from his qualifications and expertise, whether Slashme has any special expertise in enumerative combinatorics. On the other hand, David Eppsten, who has also engaged recently in editing this entry, certainly does. So, I should like to remind both Users of the passage in question in seeking their consent to restoring the passage, perhaps with some further comment:
- So, there are at least four different answers that we might give just considering Suter's proposal. Clearly, to count, you have to know what counts. When, as here, the number of outcomes is so sensitive to the assumptions made, it helps to state them explicitly. Put another way, combinatorics can help sharpen our awareness of tacit assumptions. If, say, answers like 4 or 64 are unacceptable for some reason, we have to re-examine our presumptions, possibly questioning whether Suter's pieces can be turned over in reforming their square. As emerges below, there is also some objection to Suter's proposal which would render this combinatorial discussion of the Suter board academic.
As it seems to me, the point the writer intends is that a problem in enumerative combinatorics has to be well-posed in order for it to be possible to answer it and that wide variation in potential answers is an indication that the problem has not been well posed. What is missing here is the further observation that Netz, in the referenced book with Noel, just jumps into his conjecture that Archimedes was doing high-level enumerative combinatorics leading to a large number, which is then confirmed by producing a suitable large number, but without any discussion of the underlying assumptions need to produce that large number. Two possible reasons for this reticence occur to me: (i) the writer wished to avoid being unduly adversarial; and (ii) the writer considered Netz' adoption of the (flawed) Suter Board, again without discussion why this Board is privileged over that emerging from the Archimedes Codex, a more serious obstacle.
So, I propose for your consideration and, as I hope, approval restoration of the deleted passage, with the further observation of how it relates to Netz' presentation in his book with Noel.
Now, also missing in the entry is discussion of Netz more scholarly discussion, jointly with Fabio Acerbi and N. G. Wilson, that came out in SCIAMVS in 2004:
- Towards a reconstruction of Archimedes' Stomachion, SCIAMVS, 5 (2004), 67-97.
The account here is decidely lower key than the earlier book and, if anything, supports the thrust of the Wiki entry in faulting the account in the book. In the first place, we see, not the Suter Board, but the outlines of a board that is two squares side by side, just as Heiberg had suggested (although in the book, Netz takes Heiberg to task for neglecting figures). Secondly, following the discussion in footnotes referring to Suter, the article recognises Suter's own admission of the provisional nature of his reconstruction (but not the typo in fn 6, where Suter has AD = DB, where presumably AD = AB is intended, not Suter's later conncession that, in the unpointed Arabic of his text, twice and equals are easily confused, not that Suter recognizes that this opens the possibility that the sides might be related as AD = 2AB, as seen in the Archimedes Codex). Thirdly, the authors have studied Heiberg and Suter sufficiently thoroughly to be able to say where Heiberg diverges from Suter. Fourthly, the authors even have reference to the note on Lucretius in 1956 by H. J. Rose from which they could have been led, by equally close reading, to Oldham's letter to Nature in 1926, although, particularly for a senior classicist, such as N. G. Wilson, Rose's standard Handbook of 1934, would be the more obvious source of acquaintance with Oldham's letter. For further reference here, we can consult Suter's article of 1899:
Comparison of book and paper does invite question about Netz' approach to scholarship? As it happens, an extended answer has been given by Netz' co-author, Fabio Acerbi, whose own work delving into Ancient Greek enumerative combinatorics seems, by Netz' own account, to have been an inspiration for Netz to emulate and equal.
- Archimedes and the Angel: Phantom Paths from Problems to Equations, Aestimatio, 2 (2005), 169--226; see esp. 178--179.
- The pointis not even whether Netz' approach should be labeled as history of mathematics, or whether, more likely, he is inventing a new genre ... Netz' book simply raises problems of methods: ...
Netz' earlier showmanship in publicizing his conjecture on Archimedes' Stomachion, namely that it was an exercise in high-level enumerative combinatorics, was sprung on a less suspecting audience.
- Please feel free to edit the article in any way that you feel improves it, but the section that I removed seemed to me to add hardly anything to the discussion. I see you tried to paraphrase it above, so maybe you can rewrite it in a way that is more direct and less subject to interpretation. You might also like to ask for a review of your text at Wikipedia:WikiProject_Mathematics. As you say, I'm not an expert. --Slashme (talk) 12:22, 27 February 2013 (UTC)
It is good of Slashme to be so accommodating. Possibly the whole text of this entry might usefully be reviewed at Wikipedia:WikiProject_Mathematics, as there is no point in tinkering only to be challenged on commentary or original research, as I see has happened before.
- I was not the person who pointed out the original research issues, and I've just reread the section that I deleted within context, and it is essentially a very wordy repetition of what had been covered already. Although I am not a professional mathematician, I can quite easily follow the arguments here, and I pruned that section because it added nothing to the article. The article is still very verbose, and could be improved by critically reviewing the text and pruning it down. As Pascal said: "I would have written a shorter letter, but I did not have the time."
- I won't be seriously editing this article, so don't worry about my opinion, and certainly don't wait for my input before editing it. Be bold and keep cool. If you really feel that the exact text that I cut is essential to the readers' understanding, put it back. Read WP:Edit warring to make sure that you know what to avoid, but certainly replacing text that another editor has removed once, while adding an explanation on the talk page of the article, is 100% OK. --Slashme (talk) 06:10, 8 March 2013 (UTC)
The edit I have made just now turns blue a swathe of red links left by EdwardLane simply by adjusting the text to pick up content already in Wikipedia (all these adjustments are fairly slight and rather obvious). I shall wait a month or so before attempting any more substantive revision.
- Thanks for fixing the redlinks - there are still several that link to disambiguation pages that I ought to fix, the redlinks were not that obvious to me (I failed a degree in maths and came to the page as a result of the did you know months ago)- or at least not where to find them quickly in wikipedia (I only had time for a quick run through trying to spot things that were perhaps not obvious to a 'normal reader') EdwardLane (talk) 15:05, 5 March 2013 (UTC)
- Cite error: The named reference
Archwas invoked but never defined (see the help page).