Stomachion

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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

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.

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.

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):

(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.

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

• Joe Marasco, Stomachion, inspired Kate Jones to make a puzzle
• Kadon Enterprises, Archimedes' Square, puzzle makers, asked Ed Pegg for a solution
• Ed Pegg's Math Games: The Loculus of Archimedes, Solved, thought of Bill Cutler
• Bill Cutler Puzzles, Inc., had a computer program
• A tour of Archimedes’ Stomachion by Fan Chung and Ron Graham
• In Archimedes’ Puzzle, a New Eureka Moment, Gina Kolata, New York Times, 2003
• Chris Rorres: Archimedes' Stomachion, concise overview including Pick’s theorem
• Logelium, surveying 637 convex forms

Archimedes

• The Works of Archimedes. Edited in Modern Notation with Introductory Chapters, T. L. Heath, 1897
• The method of Archimedes, recently discovered by Heiberg; a supplement to the Works of Archimedes, 1897, T. L. Heath, 1912
• A history of Greek mathematics, Vol. II, T. L. Heath, 1921 (also here and here)
• The Impact of Archimedes on Medieval Science, Marshall Clagett, The History of Science Society, 1959
• The Origins of Mathematical Physics, Reviel Netz, 2000
• The Works of Archimedes: On the Sphere and the Cylinder, by Reviel Netz, 2004
• What's Happening in the Mathematical Sciences, by Dana Mackenzie, 2009

Palimpsest

• archimedespalimpsest.org, the project website
• The Walters Art Museum, Baltimore, US
• Archimedes Palimpsest in Google Books
• archimedes-lab.org, original texts
• Bibliotheca Augustana, original texts
• Folio 177 verso and 172 recto in Google Maps, with a Stomachion diagram in the upper half on the right
• Archimedes: The palimpsest and the tradition, Nigel Wilson, in: Byzantinische Zeitschrift, 1999
• Exploratorium: Ancient Writings Revealed, webcast, April 2006
• Modern X-ray technology reveals Archimedes' math theory under forged painting, Stanford Report, 9 August 2006
• Imaging of Historical Manuscripts, Roger L. Easton, Rochester Institute of Technology, 2006
• Revealing the Secrets of the World's Greatest palimpset, book review, Current World Archaeology, 2007
• Revolutionary? Authentic? Stolen? The Story of the Archimedes Manuscript, book review, Spiegel Online, 2007
• Een perkamentje van twee miljoen, book review (Dutch), H. Brandt Corstius, NRC, 2007
• Archimedes in Bits: The Digital Presentation of a Write-Off, William Noel, The Walters Art Museum, 2009

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

• Conflict Of Laws In Litigation Between Original Owners And Good-Faith Purchasers Of Stolen Art, Patricia Youngblood Reyhan, 2001
• The Legal Implications and Mysteries Surrounding the Archimedes Palimpsest, Katherine Carver, American Journal of Legal History no. 47, April 2005

More

• Istanbul under allied occupation, 1918-1923, Bilge Criss, 1999
• Archimedes Death Ray Test, product engineering class 2.009, MIT, October 2005
• Wolfram MathWorld: Combinatorics, an introduction