4umi.com/play/stomachion/

Stomachion

4umi play

The stomachion is perhaps the world's oldest puzzle. It was described by the mathematician Archimedes of Syracuse (Αρχιμήδης ο Συρακούσιος) in the 3rd century BC, but his words have survived only fragmentarily. It consists of 14 pieces drawn from a square. Throughout the ages, people have amused themselves with the design of new shapes with these pieces. Animals, buildings, stars and other interesting shapes can be made with them, not unlike the Chinese tangram. Although it was alternatively known as Loculus Archimedius or ‘Archimedes' box’ in Roman times, he was probably not the inventor of the puzzle, but his examinations of its geometrical properties made it a popular toy. Try it yourself, play the game.

Several authors of antiquity have left traces of the puzzle's existence in their work. In the fourth century, the Roman poet Ausonius wrote in his Cento Nuptialis that the Greek called it Ostomachion, a word meaning a fight (μάχη, mákhion) with bones (ὀστέον, ostéon) in reference to the pieces which were often made out of ivory. The Romans themselves knew it as Stomachion, an apparent corruption after a misleading translation into Arabic (where vowels are not written) and confusion with the Greek word for abdominal pain (or stomach ache), which was also used figuratively as meaning “that which drives me mad” (ὀστομάχιον, stomachêin). Whatever the origin of the game and its name, here is a stomach turner.

2222 years

The first printed reference to the game of tangrams is a book published in China in 1813, although its roots may lie in the furniture used during the Song dynasty in the 12th century. The history of the stomachion puzzle is not only more than twice as long, it is surrounded by mystery every step of the way. When Archimedes died in 212 BC at the hands of a rookie Roman soldier, his writings were considered timeless already, but the Stomachion treatise has been the least understood and was almost lost. For ages it was known only from indirect references and incomplete translations into Arabic and Latin. In the 20th century new texts in the original Greek language were discovered in the Archimedes Palimpsest, and they contain words that lead to the idea that his main concern was not the creation of new forms as previously believed, but the question in how many ways these pieces can be combined back into a square. There is not enough remaining text to be entirely sure, but if this is the case, Archimedes anticipated the field of combinatorics long before that science came into its own during the age of the computer.

Many good websites are devoted to the palimpsest, the puzzle and the person behind them. This site aims to deepen and further broaden the understanding of the Stomachion puzzle and facilitate a way to see all solutions with and without context in a format that is completely customizable to suit many different environments. The Stomachion generator is a PHP-script that accepts multiple parameters for sizes, colours, text overlays and an autorefresh feature. Now anyone can have a daily stomachion on their blog!

1 square

Stomachion
A stomachion square.

The square can be divided in twelve equal parts horizontally and vertically to find the positions of the intersecting lattice points of the dividing lines. Strangely, the corners of the puzzle pieces fall on these points in every configuration. This is certainly not the case for every dissection of a square, but it is unclear whether Archimedes was attracted by this interesting property. In the traditionally known text, he calculates how the sizes of the pieces relate to the whole, and which angles fit together to make a straight line.

There are three sets of pieces which can only go together in the creation of new squares, because they have unique angles which cannot be compensated by other pieces, so they are usually replaced straightaway with single pieces when determining the combinatorical properties of the puzzle. The pair in the top right corner becomes the combined yellow piece in the coloured stomachion on this page, the two very small triangles bottom left become one of the lightblue pieces, and the two longer triangles on the left form the orange triangle. This way the problem is reduced to only eleven pieces.

The remaining text fragments do not reveal whether Archimedes' calculations arrived at the correct answer, or, for that matter, any at all. Only in October 2003, after the palimpsest research had stirred a buzz in the mathematics community, the issue was finally settled by mathematician, puzzle maker and computer programmer Bill Cutler, who wrote a piece of software that automatically scanned all promising constellations for squares. A few weeks later, a team of Californian university professors, including Fan Chung and Ron Graham, provided a theoretical framework to get a grip on the plethora of solutions, supporting his conclusion. It turns out there are 536 unique stomachion squares. If derivations such as rotated and flipped versions are counted separately, the number adds up 32-fold to 17152. If the lightblue and the combined lightblue triangles are considered exchangeable, the number is halved to 268.

14 pieces

The pieces have been ordered according to their relative sizes, an arbitrary choice. The area is given as a fraction of the whole square. The angles in degrees are rounded to full numbers, because some contain an irrational fraction even when expressed in radians proportional to π (pi). In the following table γ (gamma) represents a 45° angle, or ¼π radians. The angle δ (delta) of 26.565051...° is the one whose arctangent is ½.

Properties of the 11 pieces
 ID PieceShapeAreaAnglesBuildNotes
A  tetragon9/48=3/16 63+90+90+117°2γ - δ, 2γ, 2γ, 2γ + δunion of two pieces
B  triangle8/48=1/6 27+45+108°δ, γ, 3γ - δunion of two pieces, larger version of pieces I and J
C  pentagon7/48 72+135+90+90+153°γ + δ, 3γ, 2γ, 2γ, 4γ - δ 
D  tetragon4/48=1/12 27+126+72+135°δ, 4γ - 2δ, γ + δ, 3γ 
E  triangle"45+63+72°γ, 2γ - δ, γ + δcopy of F
F  """"copy of E
G  "3/48=1/16 27+63+90°δ, 2γ - δ, 2γunion of two pieces, copy of H
H  """"copy of G after the merger
I  "2/48=1/24 27+45+108°δ, γ, 3γ - δcopy of J, smaller version of triangle B
J  """"copy of I, smaller version of triangle B
K" "" 18+54+108°γ - δ, 2δ, 3γ - δthe only piece whose sides all have fractional lengths

268 arrangements

By reversing or rotating a symmetrical subset of pieces, one square turns into another. Repeating with another segment then gives a third square. Continuing this way, it is possible to travel through 266 distinct squares. The remaining 2 form a miniloop on their own. This amazing property of the stomachion is only vaguely understood, but the study of graphs and Hamiltonian cycles, as they are called in mathematical jargon, allows some logic to be applied. There is a collection of 24 arrangements, the core squares forming a Hamiltonian path on their own, that are each made up of four triangular subsets of pieces and from which most others can be deducted. Possible shapes for the flipped region are isosceles triangles, trapeziums, parallelograms and rectangles.

<< Square 1 >>

See the combined 11 or all 14 pieces in arrangement number or play/stop a slow/fast looping animation.

Many different paths could have been chosen, this line up has all the logic and surprises one might have come to expect from the stomachion. The ‘default’ arrangement of the pieces, described in the classical texts and commonly used in representations of the puzzle, appears as number 205, on its side and in reverse.

64 symmetries

Every unique configuration can be rotated in four directions, and each of those configurations can be mirrored. Thus the number of arrangements is increased by a factor of 4 * 2 = 8.

Four rotations.
Four mirrored rotations.

Exchanging two identical triangles doubles the result, and as there are three such pairs of triangular twins (seen here in lightgreen, lightblue and darkblue colours, labeled large, medium and small below), the number is doubled twice more. And 2 * 2 * 2 = 8.

Six exchanged twins.

This makes a grand total of 8 * 8 * 268 = 17152 solutions for the stomachion puzzle.

17152 solutions

A system has been devised to extend the numbering to include all 17152 solutions. With it, the generator can be instructed to generate any square. In this scheme, number 269 is the same as number 1 in the Hamiltonian path above, but rotated 90° to the right (clockwise), number 270 is square 2 rotated right 90°, number 3 rotates into 271, number 4 into 272 and so on. The next series, which starts at 268 × 2 + 1 = 537, is the same again, but this time rotated 180°, the one after that is the fourth rotation at 90° to the left (counterclockwise). The next four series are the mirrored versions of the first four, so the flipped (vertically mirrored) version appears in the seventh series as rotated 180° mirrored (orientation 6 if the first is labeled 0). The next eight series are the same, except for the switching of the large twinned triangles. The next sixteen series are the same again, but with exchanged medium triangles. And so on. In the established order of combinations, the large pair is switched every other series (variations 1, 3, 5 and 7 if the first is labeled 0), the medium pair in the second half of every half (2, 3, 6 and 7) and the small pair in the complete second half (4, 5, 6 and 7).

All 64 combinations are listed below. The series number can be used to find the ID of any square. It is multiplied by 268 and added to the ID of the unrotated, unvaried counterpart square in the basic series of the first 268 constellations. The series number is made up of the orientation number and eight times the variation number as shown.

The 17152 Stomachion squares
IDSeriesOrientationVariation
RotationMirrorLargeMediumSmall
8×0+0=001 - 268-----
8×0+1=01269 - 536right----
8×0+2=02537 - 804round----
8×0+3=03805 - 1072left----
8×0+4=041073 - 1340-yes---
8×0+5=051341 - 1608rightyes---
8×0+6=061609 - 1876roundyes---
8×0+7=071877 - 2144leftyes---
8×1+0=082145 - 2412--yes--
8×1+1=092413 - 2680right-yes--
8×1+2=102681 - 2948round-yes--
8×1+3=112949 - 3216left-yes--
8×1+4=123217 - 3484-yesyes--
8×1+5=133485 - 3752rightyesyes--
8×1+6=143753 - 4020roundyesyes--
8×1+7=154021 - 4288leftyesyes--
8×2+0=164289 - 4556---yes-
8×2+1=174557 - 4824right--yes-
8×2+2=184825 - 5092round--yes-
8×2+3=195093 - 5360left--yes-
8×2+4=205361 - 5628-yes-yes-
8×2+5=215629 - 5896rightyes-yes-
8×2+6=225897 - 6164roundyes-yes-
8×2+7=236165 - 6432leftyes-yes-
8×3+0=246433 - 6700--yesyes-
8×3+1=256701 - 6968right-yesyes-
8×3+2=266969 - 7236round-yesyes-
8×3+3=277237 - 7504left-yesyes-
8×3+4=287505 - 7772-yesyesyes-
8×3+5=297773 - 8040rightyesyesyes-
8×3+6=308041 - 8308roundyesyesyes-
8×3+7=318309 - 8576leftyesyesyes-
8×4+0=328577 - 8844----yes
8×4+1=338845 - 9112right---yes
8×4+2=349113 - 9380round---yes
8×4+3=359381 - 9648left---yes
8×4+4=369649 - 9916-yes--yes
8×4+5=379917 - 10184rightyes--yes
8×4+6=3810185 - 10452roundyes--yes
8×4+7=3910453 - 10720leftyes--yes
8×5+0=4010721 - 10988--yes-yes
8×5+1=4110989 - 11256right-yes-yes
8×5+2=4211257 - 11524round-yes-yes
8×5+3=4311525 - 11792left-yes-yes
8×5+4=4411793 - 12060-yesyes-yes
8×5+5=4512061 - 12328rightyesyes-yes
8×5+6=4612329 - 12596roundyesyes-yes
8×5+7=4712597 - 12864leftyesyes-yes
8×6+0=4812865 - 13132---yesyes
8×6+1=4913133 - 13400right--yesyes
8×6+2=4013401 - 13668round--yesyes
8×6+3=5113669 - 13936left--yesyes
8×6+4=5213937 - 14204-yes-yesyes
8×6+5=5314205 - 14472rightyes-yesyes
8×6+6=5414473 - 14740roundyes-yesyes
8×6+7=5514741 - 15008leftyes-yesyes
8×7+0=5615009 - 15276--yesyesyes
8×7+1=5715277 - 15544right-yesyesyes
8×7+2=5815545 - 15812round-yesyesyes
8×7+3=5915813 - 16080left-yesyesyes
8×7+4=6016081 - 16348-yesyesyesyes
8×7+5=6116349 - 16616rightyesyesyesyes
8×7+6=6216617 - 16884roundyesyesyesyes
8×7+7=6316885 - 17152leftyesyesyesyes
Orientation
 ID Description
0No rotation, no reflection
1Rotated 90° right, no reflection
2Rotated 180°, no reflection
3Rotated 90° left, no reflection
4No rotation, reversed
5Rotated 90° right, reversed
6Rotated 180°, reversed
7Rotated 90° left, reversed
Variation
 ID Switched triangles
0None
1Large pair
2Medium pair
3Large and medium pairs
4Small pair
5Large and small pairs
6Medium and small pairs
7All three pairs

Appendix: Cento Nuptialis

Roman poet Decimus Magnus Ausonius (ca. 310 – ca. 393) mentions the Stomachion in his description of a type of poetry in which various meters are put together, like the pieces of the puzzle, in the introduction to his book Cento Nuptialis (A Wedding Cento):

Et si pateris, ut doceam docendus ipse, cento quid sit, absolvam, variis de locis sensibusque diversis quaedam carminis structura solidatur, in unum versum ut coeant aut caesi duo aut unus et sequens cum medio, nam duos iunctim locare ineptum est, et tres una serie merae nugae. Diffinduntur autem per caesuras omnes, quas recipit versus heroicus, convenire ut possit aut penthemimeris cum reliquo anapaestico, aut trochaice cum posteriore segmento, aut septem semipedes cum anapaestico chorico aut post dactylum atque semipedem quidquid restat hexametro: simile ut dicas ludicro quod Graeci ostomachion vocavere. Ossicula ea sunt: ad summam quattuordecim figuras geometricas habent. Sunt enim aequaliter triquetra vel extentis lineis vel eiusdem frontis vel rectis angulis vel obliquis: isoscele ipsi vel isopleura vocant, orthogonia quoque et scalena. Harum verticularum variis coagmentis simulantur species mille formarum: helephantus belua aut aper bestia, anser volans et mirmillo in armis, subsidens venator et latrans canis, quin et turris et cantharus et alia huiusmodi innumerabilium figurarum, quae alius alio scientius variegant. Sed peritorum concinnatio miraculum est, imperitorum iunctura ridiculum. Quo praedicto scies, quod ego posteriorem imitatus sum.

Hoc ergo centonis opusculum ut ille ludus tractatur pari modo sensus diversi ut congruant, adoptiva quae sunt ut cognata videantur aliena ne interluceant: arcessita ne vim redarguant, densa ne supra modum protuberent hiulca ne pateant. Quae si omnia ita tibi videbuntur ut praeceptuni est, dices me composuisse centonem.

And if you will suffer me, who need instruction myself, to instruct you, I will expound what a cento is. It is a poem compactly built out of a variety of passages and different meanings, in such a way that either two half-lines are joined together to form one, or one line and the following half with another half. For to place two (whole) lines side by side is weak, and three in succession is mere trifling. But the lines are divided at any of the caesurae which heroic verse admits, so that either a penthemimeris (--) can be linked with an anapaestic continuation (--), or a trochaic fragment (--) with a complementary section (--), or seven half-feet (--) with a choric anapaest (--), or after a dactyl and a half-foot (--) is placed whatever is needed to complete the hexameter: so that you may say it is like the puzzle which the Greeks have called ostomachia. There you have little pieces of bone, fourteen in number and representing geometrical figures. For some are equilateral triangles, some with sides of various lengths, some symmetrical, some with right angles, some with oblique: the same people call them isosceles or equal-sided triangles, and also right-angled and scalene. By fitting these pieces together in various ways, pictures of countless objects are produced: a monstrous elephant, a brutal boar, a goose in flight, and a gladiator in armour, a huntsman crouching down, and a dog barking — even a tower and a tankard and numberless other things of this sort, whose variety depends upon the skill of the player. But while the harmonious arrangement of the skilful is marvellous, the jumble made by the unskilled is grotesque. This prefaced, you will know that I am like the second kind of player.

And so this little work, the Cento, is handled in the same way as the game described, so as to harmonize different meanings, to make pieces arbitrarily connected seem naturally related, to let foreign elements show no chink of light between, to prevent the far-fetched from proclaiming the force which united them, the closely packed from bulging unduly, the loosely knit from gaping. If you find all these conditions duly fulfilled according to rule, you will say that I have compiled a cento.

(Translation based on the one by Hugh G. Evelyn White, published London, 1919. The word ‘cento’ is also Latin for ‘patchwork’.)

Appendix: Timeline

The Stomachion has survived the millenia by little more than chance, it seems. The continuous cultivation of interest in and understanding of its mathematical contents, needed for its sustainment, has been threatened by warfare, decline and barbarism more than once. Archimedes wrote his treatises in the form of letters, mostly to his friends in Alexandria. When the library of Alexandria was burned and the school closed, somehow its surviving inheritance was shipped to Byzantium, but when and by who this was done is open to debate. There it appears that a book containing treatises by Archimedes was copied in the sixth century during the time of Isidore of Miletus, and once again in the ninth century thanks to Leo the Geometer. This copy ended up in Italy where it was translated into Latin in the thirteenth century. Now lost, it is known as Codex A. Late in the tenth century, judging by the script, another copy was made. This is the Archimedes Palimpsest, technically known as Codex C. Another Byzantine book, Codex B, containing treatises about optics and mechanics, was also available to the translator of the thirteenth century, but is now lost as well. We are left with copies of copies, and Codex C. All we know today about Archimedes was known and appreciated in the Byzantine world around the new millenium, and probably more. Two centuries later however, it had all been swept away.

DateEvent
ca. 350 BCXenocrates of Chalcedon (Ξενοκράτης ο Χαλκηδόνιος) considers a combinatorical problem when he calculates that
the letters of the Greek alphabet allow the formation of 1002000000000 syllables.
ca. 287 BCArchimedes born in Syracuse, an independent Silician Greek city-state in the line of fire between Carthage and Rome.
YouthProbably studies at the Mouseion of Alexandria, famous for its great library, learning from pupils of Euclid.
Later lifeInvents defense systems for Syracuse, Archimedean screw, writes letters on papyrus about the displacement of
water, the ratio of volume of a cilinder compared to a sphere, the calculation of π (pi) and and and Stomachion.
Autumn 212 BCArchimedes killed by a Roman soldier, his books and instruments carried to Rome.
48 BCLibrary of Alexandria accidentally burned down by Julius Ceasar, according to Plutarch and others.
1st centuryRoman poet Bassus writes De metris, saying the stomachion game helps children train their memory.
2nd centuryIntroduction of the parchment codex replacing papyrus scrolls.
4th centuryRoman poet Caius Marius writes in Ars Grammatica there are as many meter types in poetry as stomachion figures.
11 May 330Byzantium proclaimed capital of the Eastern Roman Empire by emperor Constantine the Great, whose name stuck.
ca. 370Roman poet Ausonius composes his Cento Nuptialis for emperor Valentinianus, mentioning the stomachion as a game
of creating different shapes.
392Serapeum library in Alexandria, successor of the Mouseion, destroyed by a Christian mob after Emperor Theodosius I
orders the destruction of pagan temples, according to Gibbon.
425Pandidakterion (Πανδιδακτήριον) founded in Constantinople by Eastern Roman emperor Theodosius II as a center of
learning in 15 Latin and 16 Greek faculties.
439Mar Saba monastery founded in Palestine by Saint Sabbas the Sanctified.
ca. 510 onwardsEutocius of Ascalon writes commentaries on Archimedes and Apollonius, which he dedicates to Anthemius of Tralles.
532 - 537Hagia Sophia ('Holy Wisdom', from the Greek Ἁγία Σοφία), built in Constantinople by Anthemius of Tralles and Isidore
of Miletus, who taught Archimedes in Alexandria and Constantinople, as a tribute On the Sphere and the Cylinder,
would remain the largest cathedral in the world until the completion of Seville Cathedral in 1520.
Early 800'sStomachion and other works translated into Arabic by astronomer Thābit ibn Qurra (ثابت بن قرة بن مروان) in Baghdad.
9th centuryIntroduction of the miniscule (lowercase) letter form in manuscript production.
850'sLeo the Geometer appointed head of the University of Constantinople, who is remembered in a complimentary note
added to the end of Quadrature of the Parabola in codex A.
ca. 975Circa fourth or fifth copy made of Archimedes' writings, now on goatskin parchment but maintaining the format of
columns used on papyrus scrolls, most likely in Constantinople, certainly in the Byzantine Empire.
1187Gerardus Cremonensis translates On the Sphere and the Cylinder from Arabic into Latin in Toledo.
12 April 1204Constantinople sacked during the Fourth Crusade, its libraries burned and perhaps all but three books on Archimedes
destroyed, now known as codex A and B, which are taken to Italy as spoils of war, and C, which disappears.
ca. 1229Some scientific books taken apart, centuries old text scraped off, euchologion written on top, with prayers typical of
the Jerusalem region.
18 February 1229Jerusalem obtained by treaty by the Holy Roman Emperor Frederick II of Germany from Muslim Sultan Malik al-Kamil
during the Sixth Crusade.
18 March 1229Frederick II crowned King of Jerusalem in the Church of the Holy Sepulchre.
14 April 1229Euchologion dedicated and signed by ‘presbyter Ioannes Myronas’ on Easter Saturday.
ca. 1230 - 1830Used in services at the Mar Saba monastery near Betlehem, gathering candle wax, losing several pages at the end.
The Stomachion introduction is now the last page.
15 July 1244Jerusalem razed by Karizimian Tartars, who were invited by the Sultans to recapture it.
1269Parts of codex A and B, not including Stomachion, translated into Latin by Dominican monk Willem van Moerbeke at
the pontifical court of Viterbo, 100 km north of Rome.
1311Codex B last seen in the papal library in Viterbo.
ca. 1458Codex A translated into Latin by Jacobus Cremonensis for Pope Nicholas V.
29 May 1453Constantinople, capital of the Byzantine Empire, captured by Sultan Mehmed II of the Ottoman Empire.
ca. 1449 - 1468Codex A copied into codex E in Venice, now in the Biblioteca Marciana.
ca. 1468Cremonensis copied by German mathematician Regiomontanus with help of codex E.
1491Codex A copied into codex D in Venice for the Biblioteca Laurenziana in Florence after a tip from Angelo Poliziano.
16th centuryMar Saba ex libris added to codex C.
1503First publication in print of Archimedes' works, Measurement of a Circle and Quadrature of the Parabola, by
astronomer Luca Gauricio in Venice.
1544First complete edition printed by Thomas Gechauff Venatorius in Basel, with the Greek text based on a Nürnberg
manuscript copied from A earlier in the century but with interpolations from B, and the Latin from Regiomontanus.
1544Codex A copied into codex H by Christopherus Auverus for the library of Francois I in Fontainebleau.
1564Codex A last seen when its owner, Cardinal Rodolfo Pio da Carpi, dies and his collection passes to the Biblioteca
Estense in Modena, 550 km north of Rome.
1625Mar Saba monastery bought by the Patriarch of Jerusalem.
Circa 1830'sMar Saba library collection moved to the Library of the Patriarchate of Jerusalem, and then to the Metochion
("daughter") of the Holy Sepulchre, owned by the Patriarchate, in Constantinople.
Early 1840'sCodex C noted by German biblical scholar Constantin Tischendorf in the Metochion as an "interesting palimpsest
containing some mathematics".
1899Cataloged by Greek paleographer Athanasios Papadopoulos-Kerameus in the Metochion, who copies some of the
mathematical undertext, which German scholar H. Schöne will bring to the attention of Archimedes expert Heiberg.
1899Swiss orientalist Heinrich Suter publishes a German translation of Stomachion from a 17th century Arabic manuscript.
1906Palimpsest studied and photographed by Danish philologist Johan Heiberg, identified as unique source of
On Floating Bodies, The Method and Stomachion in original Greek, making headlines the following year.
1908Palimpsest last seen by Heiberg.
1910 - 1915Transcription by Heiberg published, uncovering circa 80% of the text.
Early 1920'sMetochii Constantinopolitani manuscripts moved to National Library in Athens, although several never arrive.
ca. 1923Palimpsest "bought from a monk" by French businessman Marie Louis Sirieix on a trip to the Orient.
1932Parisian art dealer Salomon Guerson, born 1872 in Turkey, has a manuscript identified by the curator of the
Huntington Library in Los Angeles as the "manuscript of Archimedes described by J.L. Heiberg".
1934Guerson offers his manuscript for sale for $6,000.
After 1937Forgeries painted on top of four leaves, covering part of The Method but not Stomachion, containing a green
pigment which first became available in Germany in 1938.
1942Guerson escapes from Paris and relocates back to Istanbul.
16 July 1942First roundup of Jews, especially foreign-born, by the occupying Nazi regime in Paris.
Around 1940'sPalimpsest stored in unhealthy damp conditions, gathering mold.
1956Palimpsest inherited by Anne Guersan after her father Sirieix dies.
1960'sInspected by philologer Jean Bollack of Lille University after Guersan grows concerned about the book's state.
1970'sInspected by classicist Abraham Wasserstein in Leicester and Joseph Paramelle of the Parisian Institut de Recherche
et d'Histoire des Textes, after which it is restored by Etablissement Mallet in Paris with PVA glue and Blu-Tack.
1971A manuscript owned by Cambridge University is identified by Oxfordian classicist Nigel Wilson as a leaf missing from the
palimpsest transcribed by Heiberg, and is subsequently traced back to the estate of Tischendorf.
1980'sPalimpsest offered for sale to numerous libraries and museums around the world.
1991Offered for auction to Christie's, who appraises it at around 1 million dollars.
1998Examined by Nigel Wilson and photographed under UV-light by Roger Easton of RIT for Christie's sale catalogue.
October 1998Greek patriarch and government try to prevent sale, claiming it was stolen, but lose in court.
29 October 1998Palimpsest sold by Sirieix's grandson Robert Guersan at auction by Christie's in New York to American philantropist
"Mr. B." for $2,202,500, bidding against the Greek general consul for his ministry of Culture.
15 January 1999Lent to Walters Art Gallery in Baltimore for conservation and research, where three pages seen by Heiberg are found
to be missing.
1999 - 2003Disassembled at an average rate of one leaf per fifteen days.
2000Heiberg's photographs, separated from his other papers and believed lost, found in the Royal Library in Copenhagen.
2000 - 2006Imaging experiments at the Rochester Institute of Technology and John Hopkins University.
31 October 2003Stomachion puzzle solved by Bill Cutler and his computer.
November 2003Stomachion graph theory published by Fan Chung and Ron Graham.
2005 - 2006Imaging experiments at Stanford Synchroton Radition Laboratory.
29 October 2008Tenth anniversary of the research, dataset released, uncovering circa 95% of the text.
A page from the palimpsest today, with Archimedes' writing showing in vertical lines behind the horizontal Christian prayer text. The word comes from from the Greek "again" (παλιν, palin) "I scrape" (ψαω, psao).

Appendix: Archimedes' works

The titles for the different writings of Archimedes have been established by the early scribes, allowing for variations among copies and translations. The parabola was thus named by Apollonius in the 1st century AD. Archimedes used the old term 'section of a right-angled cone'.

Extant works by Archimedes

  1. On the Sphere and the Cylinder (De Sphaera et Cylindro), in two books,
      shows the surface area of a sphere is 4 pi r2, and the volume of a sphere is two-thirds that of the cylinder in which it is inscribed
  2. Spiral Lines (De Lineis Spiralibus),
      develops many properties of tangents to the spiral of Archimedes
  3. Conoids and Spheroids (De Conoidibus et Sphaeroidibus),
      finds the volumes of solids formed by the revolution of a conic section (circle, ellipse, parabola, or hyperbola) about its axis
  4. Measurement of the Circle (Dimensio Circuli),
      shows pi to be between 3 10/70 and 3 10/71
  5. The Sand Reckoner (Arenarius),
      shows how to deal with large numbers in a calulation of the number of grains of sand needed to fill the universe, and contains the most detailed surviving description of the heliocentric system of Aristarchus of Samos
  6. Planes in Equilibrium (De Planorum Aequilibriis, or, De Planis Aeque Ponderantibus), in two books,
      finds the centers of gravity of various plane figures and conics, and establishes the "law of the lever"
  7. Quadrature of the Parabola (Quadratura Parabolae),
      finds the area of any segment of a parabola, first by "mechanical" means (Archimedes' "Method") and then by rigorous geometry
  8. The Method,
      describes the process of mathematical discovery
  9. On Floating Bodies (De Corporibus Fluitantibus), in two books,
      finds the positions of various solids floating in a fluid, and establishes Archimedes' principle that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object
  10. The Cattle Problem (Problema Bovinum)
  11. Stomachion
  12. Eutocius' three commentaries, reviewing On the Sphere and the Cylinder, Measurement of the Circle and Planes in Equilibrium, are usually included with the complete works of.

Contents of codex A, in likely original order

  1. On the Sphere and the Cylinder
  2. Measurement of the Circle
  3. Conoids and Spheroids
  4. Spiral Lines
  5. On the Equilibrium of Planes I
  6. On the Equilibrium of Planes II
  7. The Sand Reckoner
  8. Quadrature of the Parabola
  9. Eutocius' three commentaries

Contents of codex B, in unknown order

  1. On the Equilibrium of Planes I
  2. On the Equilibrium of Planes II
  3. On Floating Bodies I
  4. On Floating Bodies II
  5. Quadrature of the Parabola

Contents of codex C, the palimpsest, in likely original order

  1. On the Equilibrium of Planes I (?)
  2. On the Equilibrium of Planes II
  3. On Floating Bodies I
  4. On Floating Bodies II
  5. The Method
  6. Spiral Lines
  7. On the Sphere and the Cylinder
  8. Measurement of the Circle
  9. Stomachion
  10. Commentary (Alexander of Aphrodisias)
  11. Diondas (Hyperides)
  12. Timandros (Hyperides)
  13. Menaion (unknown)
  14. Pantoleon (unknown)

Contents of codex F and siblings, in order

  1. de sphaera et cylindro I
  2. de sphaera et cylindro II
  3. de dimensione circuli
  4. de conoidibus
  5. de lineis spiralibus
  6. de planis aeque ponderantibus
  7. arenarius
  8. quadratura parabolae
  9. Eutocius' three commentaries

Lost works by Archimedes

  1. On Polyhedra, mentioned by Pappus
  2. On the Measure of a Circle
  3. On Plynths and Cylinders
  4. On Surfaces and Irregular Bodies
  5. Mechanics
  6. Catoptrica (κατοπτρικά), on properties of mirrors, mentioned by Theon of Alexandria
  7. On Sphere-Making, mentioned by Pappus
  8. On the Length of the Year

Appendix: Reference

Stomachion

Archimedes

Palimpsest

Greek-Orthodox Patriarchate of Jerusalem v. Christie's, Inc.

More