STUDI • MARIANO TOMATIS ANTONIONO

TORNA ALL'INDICE

Euler's Knight's Tour on the Grand Parchemin

THE GRAND PARCHEMIN WAS CREATED AFTER 1759 • JUNE 27, 2007

Euler is the source for the Grand Parchemin

The Knight's Tour is a mathematical problem involving a knight on a chessboard. The knight is placed on the empty board and, moving according to the rules of chess, must visit each square exactly once. Many variations on this topic have been studied by mathematicians, including Euler in 1759.

He published many solutions to the problem in "Solution d'une question curieuse que ne paroit soumise a aucune analyse" ("Solution of a curious question which does not seem to have been subjected to any analysis"), one of his more famous papers. It was the first mathematical paper on knight's tours. This paper, although presented in 1759, did not appear in print until 1766 in the Memoires de l'Academie Royale des Sciences et Belles Lettres (1759) 15, pages 310-337, Berlin: 1766.

In page 312 Euler published the first "cyclic" tour: the final cell (64) can be linked with the first one (1) so virtually "closing" the path:

A modified version of this tour was used to encrypt the Grand Parchemin by his author.

Here is the path in a more "visual" form; follow the lines on the right chessboard:

The author of the Grand Parchemin "closed" the path in a cyclic way (see the red line):

The chessboard has been signed "Euler" in red in order to make the reflexions simpler.

The chessboard was then rotated of 180 degrees:

And the path was re-opened in a random place:

This was used as a first chessboard. The second path was "calculated" by just "mirroring" the first one. The two chessboards were put one next to the other:

The epitaph of De Nègre tombstone...

CT GIT NOBLE MARIE DE NEGRE D ARLES DAME D HAUPOUL DE BLANCHEFORT AGEE DE SOIXANTESEPT
ANS DECEDEE LE XVII JANVIER MDCOLXXXI REQUIESCAT IN PACE PS PRAECUM

...was anagrammed in order to get:

BERGEREPASDETENTATIONQUEPOUSSINTENIERSGARDENTLACLEFPAXDCLXXXIPAR
LACROIXETCECHEVALDEDIEUJACHEVECEDAEMONDEGARDIENAMIDIPOMMESBLEUES

And this text was written following the two paths:

If you read it on the two chessboards, you get:

XNLSPANNASITTIATEXRRPBTEUCAEENIRXTGEENDELORSIAAOELEFSDQRPEDCUPGX
AIEMUIDOCEJDNMEGMCOCEEPDSHRXAIADHATMOAESEBICELERNEEAIEEDLVEVULDC

The author used a Vigenère table like this and the reversed epitaph of De Nègre tombstone as key:

MUCEARPSPECAPNITACSEIUQERIXXXLOCDMREIVNAJIIVXELEEDECEDSNATPESETN
AXIOSEDEEGATROFEHCNALBEDLUOPUAHDEMADSELRADERGENEDEIRAMELBONTIGTC

Consider the first letters (X and M): find the column M, look for the X on that column and write down the letter at the beginning of that row (J).

Repeat if for all the 128 letters and you get

JRINOHXTJNFSDTQZDTYMGFCZCSCGGBSOSGNZUQODBFIVKUNJZHZCNZXDOJMXBKLI
ZKUXBDZJXXIIUXYBEZABRCKZGLCGEHRZCMSIUURADXDJXGPMJZUHHQZQJGPBLEIZ

Using on this string the same Vigenère table and the key:

MORTEPEE (repeated 16 times)

the author got:

VCPSJQROVYMYYDLTPEFRBOXTODJLBKNJFQUEPAJYNPPBFEIELRGHIIRYBTTCVTGD
LUCCVMTEJHPNPGSVQJHGMLFTSVJLZQMTOXANPEMUPHKORPKHVJCMCATLVQXGGNDT

After having added the two words AD GENESARETH in the center of the string:

VCPSJQROVYMYYDLTPEFRBOXTODJLBKNJFQUEPAJYNPPBFEIELRGHIIRYBTTCVTGD
AD GENESARETH
LUCCVMTEJHPNPGSVQJHGMLFTSVJLZQMTOXANPEMUPHKORPKHVJCMCATLVQXGGNDT

he got the 140 letters to be put inside the original latin text:

JESUSERGOANTESEXDIESPASCHAEVENITBETHANIAMUTFUERATLAZARUSMORTUUSQU
EMSUSCITAVITJESUSFECERUNTAUTEMEICAENAMIBIETMARTHAMINISTRABATLAZAR
USVEROUNUSERATEXDISCUMLENTILUSCUMMARTAERGOACCEPITLIBRAMUNGENTINAR
DIPISTICIPRETIOSIETUNXITPEDESJERUETEXTERSITCAPIIRISSUISPEDESERTIE
TDOMESIMPLITAESTEXUNGENTIODAREDIXATERGOUNUMEXDISCIPULISEIUXIUDAXS
CARJOTISQUIERATCUMTRADITURUSQUAREHOCUNBENTUMNONVENITTRECENPISDENA
RISETDATUMESTEGENISEDIXIUTEMHOCNONQUIADEEGENISPERTINEBATADCUMSEDQ
UIAFURERTETLOCULOSHABENSEAQUAEMITEBANTURPOXRRABETDIXITERGOJESUSSI
NEILLAMUTIXDIEMSEPULTURAEMEAESERNETILLUDPAUPERESENIMSEMPERHABETIS
NOBTISCUMMEAUTEMNONSEMPERHABETISCOGNOVITERGOTURBAMULTAEXIUDAEISQU
IAILLICESTETVENERUNTNONPROTERIESUMTANTUMSEDUTLAZARUMVIDERENTQUEMS
USCIAVITAMORTUISCOGITAVERUNTAUTEMPRINCIPESSACERDOTUMUTETLAZARUMIN
TERFICERENTQUIAMULTIPROPTERILHUMABIBANTEXUTAEISXETCREDEBANTINIESUM

You can create your own parchment with this tool.

Many thanks to Marco Cipriani who published the reference to Euler's work here.

Oulipian influence?

Many clues indicates that Philippe de Cherisey was the author of this Parchment, not only his papers Pierre et Papier and CIRCUIT.

The use of anagrams and of the Knight's tour is a typical "constrained writing technique", used in France in Sixties (the Great Parchment appeared in print in 1967) by Oulipo members. Oulipo stands for "Ouvroir de littérature potentielle", which translates roughly as "workshop of potential literature". It is a loose gathering of (mainly) French-speaking writers and mathematicians, and seeks to create works using constrained writing techniques. It was founded in 1960 by Raymond Queneau and François Le Lionnais. Other notable members include novelists like Georges Perec and Italo Calvino, poets like Oskar Pastior or Jacques Roubaud, also known as a mathematician.

The Vigenère table used is a sort of lipogram (writing that excludes one or more letters: in this case, the letter W), like George Perec's La disparition. It is curious the fact Perec dedicated to letter W another book, W ou le souvenir d'enfance ("W or the memory of childhood"). One can suppose that removing the letter W has some freudian meaning, linked with removal of childhood...

The Knight's Tour was used by other members of Oulipo: the most notable example is the 10×10 Knight's Tour which sets the order of the chapters in Georges Perec's novel La Vie mode d'emploi ("Life: A User's Manual").

Notes on Georges Perec's novel

In CIRCUIT, each chapter corresponds to a Tarot card. It is the same costrained technique used by Oulipian member Italo Calvino in The Castle of Crossed Destinies, a 1973 novel that details a meeting among travelers who are inexplicably unable to speak after traveling through a forest. The characters in the novel recount their tales via Tarot cards, which are reconstructed by the narrator.

All these clues should be considered with care, because we have no traces of the parchment older than 1967, and Philippe de Cherisey hypotesis is so far the most convincing. (1)

1.Translation by Mariano Tomatis with the precious help of Marcus Williamson.

© 2017 Mariano Tomatis Antoniono