An illustrated philatelic introduction to 4 × 4 Latin squares in Europe: 1283–1788 (with some comments about le fauteuil Dagobert 1 and about philatelic Latin squares in Europe: 1984–2007) George P. H. Styan June 19, 2009 1 For presentation at the 18th International Workshop on Matrices and Statistics: IWMS’09, Smolenice Castle, Smolenice, Slovakia, 23–27 June 2009, and based on Report 2009-02 from the Department of Mathematics and Statistics, McGill University, Montréal. Special thanks to Peter D. Loly for introducing me to philatelic Latin squares and to Oskar Maria Baksalary for his careful reading of these overheads. Many thanks go also to BiStamp.com, Anthony Bonner, Christian Boyer, Ka Lok Chu, eBay.com, Jose Garrido, Google search, Groth AG: WWF Conservation Stamp Collection, Ian Kimmerly Stamps, Alexander Kovačec, Michelle Kuan, Marlen Stamp & Coin Ltd., Phil-Ouest : Les timbres de France website, Friedrich Pukelsheim, Simo Puntanen, Klaus Schmidt, Scott Standard Postage Stamp Catalogue, Marc Steurs WWF stamp catalogue online, Evelyn Matheson Styan, Götz Trenkler & Wikipedia. This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada. George P. H. Styan Latin squares in Europe 1283–1788 Introduction: Latin squares and magic squares on amulets page SW2 In “The history of latin squares”2, Lars Døvling Andersen points out that: The first known occurrence of Latin squares seems to be their use on amulets3 in certain Arab and Indian communities from about the year 1000. The nature of the sources makes the dating difficult. Many amulets contain not Latin, but magic squares, and were worn to fight evil spirits, show reverence for gods, celebrate the sun and the planets. 2 Chapter on “The history of latin squares” by Lars Døvling Andersen, intended for publication in The History of Combinatorics, edited by Robin J. Wilson. Preliminary version, [Report] R-2007-32, Department of Mathematical Sciences, Aalborg University (Aalborg Øst, Denmark), 32 pp., 2008. 3 An amulet (from Latin amuletum, meaning “an object that protects a person from trouble”), consists of any object intended to bring good luck and/or protection to its owner. The stamps are from Thailand: 2005, Scott 2178a–2178e (from left to right) Buddha warrior amulet votives: Phra Ruang Lang Rang Puen, Phra Hu Yan, Phra Chinnarat Bai Sema, Phra Mahesuan, Phra Tha Kradan. (A votive is an object given to a god at a temple or shrine as a token of devotion.) Scott catalog numbers are as given in the Scott Standard Postage Stamp Catalogue (James E. Kloetzel, ed.), pub. annually by Scott Publishing, Sidney, Ohio. George P. H. Styan Latin squares in Europe 1283–1788 MAGIC MATRICES: April 12, 2007 page 15 MAGIC MATRICES: April 12, 2007 page 15 Introduction: Magic square(s) on Dürer’s Melencolia I page SW3 A (classic) magic square4 is a k × k array filled with the symbols 1; 2;:::; k2 such that the sum of the numbers in every row, column and in each of the two main diagonals is the same. A Latin square is a k × k array filled with the symbols 1; 2;:::; k such that each of the numbers occurs precisely once in every row and in every column. A completed 9 × 9 Sudoku grid is a special kind of 9 × 9 Latin square. 4 The Otto Schäfer Museum in Schweinfurt (Germany) has prints of two versions of Dürer’s copper-plate engraving Melencolia I with a magic square in the upper right corner. The stamp is from Aitutaki (Cook Islands) 1986, Scott 391. We try to illustrate this talk, as much as possible, with postage stamps. George P. H. Styan Latin squares in Europe 1283–1788 Philatelic Latin squares page SW4a Postage stamps are occasionally issued in sets of k different stamps and printed in a k × k array of k2 stamps containing k of each of the k stamps. Sometimes the array forms a philatelic Latin square: each of the k stamps = appears exactly once in each# 165 in Abkhazia. row Beetles and in each column as with30/01/09 k5:20 PM 4 and Albania 1999:5 Mickey mouse Albania 2000: Donald duck these four 4 × 4 arrays of 16 stamps fromAbkhazia # 165 Europe: 1989–2000. Beetles 5 The stamps are (left to right) from Isle of Man 1989, Scott 399–402 (Manx seabirds); Portugal 1997, Scott 2153–2156 (Pyrenean desman); Abkhazia 2000, BiStamp 165 (beetles); Albania 2000, Scott 2615 (Donald Duck). The Manx seabirds depicted are the puffin, black guillemot, cormorant and kittiwake; the Pyrenean desman (Galemys pyrenaicus) is a small semi-aquatic mammal that lives in the the wetlands and temperate areas of the Pyrenees. George P. H. Styan Latin squares in Europe 1283–1788 http://www.bistamp.com/inter/BiStamp.nsf/2483bb849b4b993a42256a28005508b0/a1d4aaa1e28e54e2c2256ad100355f9e!OpenDocument Page 1 of 2 Philatelic Latin squares (PLS) page SW4b # 165 in Abkhazia. Beetles 30/01/09 5:20 PM Albania 1999: Mickey mouse Albania 2000: Donald duck Abkhazia # 165 Beetles The first and second sets E1; E2 are both in Sudoku form, the third set (beetles) E3 is a 1-step forwards circulant Latin square, and the fourth set of (Donald Duck) E4 is a 1-step backwards circulant. To-date we have identified 160 4 × 4 PLS from 1970–2009 (with 24 from Europe: 1984–2007) including 14 embedded in a sheetlet of size p × q; with p + q ≥ 9: There are 24 distinct Latin squares of size 4 × 4 in standard-form and we have found PLS for 15 of these 24. The 4 most popular of these 15 PLS are currently those illustrated here: E1 (Manx seabirds) with 47, E2 (desman) 8, E3 (beetles) 33, and E4 (Donald Duck) 53, which total 141 (out of 160). George P. H. Styan Latin squares in Europe 1283–1788 http://www.bistamp.com/inter/BiStamp.nsf/2483bb849b4b993a42256a28005508b0/a1d4aaa1e28e54e2c2256ad100355f9e!OpenDocument Page 1 of 2 4 × 4 Latin squares in Europe: 1283–1788 page SW5 WeLlull’s will Latin discuss square theseL1 in 4an× epitaph4 Latin found squares in Cornwall in Europe: 1283–1788:page 10 1 2 3 4 1283: Ramon Llull’s Prima Figura Elementalis,2 1 4 3 Llull’s criss-cross Latin square of type L1 = is (also) to be 1704: Poignard’s Jupiter Latin square,3 4 1 2 4 3 2 1 found1708: in anEpitaph epitaph to to Hannibal Hannibal Bassett Bassett, (1687–1708), in a churchyard in Mawgan-in-Meneage,1723: Ozanam–Grandin Lizard Peninsula, solution Cornwall, to the Magic England. Card Puzzle, 1788: CrettéHanniball de Palluel’s [sic] Ba experimentalet [sic] here Inter’d design doth with lye 16 sheep, Who dying lives to all Eternitye 1791: Crettéhee de Palluel departed and thisLe life fauteuil the 17th Dagobert. of Ian 1709/8 in the 22th year of his age ∼ A lover of learning image in (row 1, columnimage in 3) (row should 1, column be the 3) should be the spades ,king of spades The 1725 statement of and solution (with typo) to the Magic Card Puzzle. The images are apparently of Shall wee all dyeplaying cards in the French Portrait de Paris style. Wee shall dye all all dye shall wee dye all wee shall. FIGURE 2: A solution to the magic card puzzle from Récréations mathématiques et physiques, Tome quatrième, George P. H. Styan Latin squares in Europe 1283–1788 George P. H. Styan Latin squares in Europe 1283–1788 Ramon Llull’s Prima Figura Elementalis: 1283 page SW6 In Europe the first appearance of a Latin square might be in the set of four 4 × 4 arrays for fire, air, water, and land which comprise the “First Elemental Figure” diagram in the Ars Demonstrativa [v. 1, pp. 318–320] by Ramon Llull6 (1232–1316), apparently first published in 1283. 6 Ramon Llull (in Catalan) is Ramón Llull or Raimundo Lulio (in Spanish), Raymond Lully or Raymond Lull (in English), or Raimundus Lullus or Raymundus Lullus (in Latin). We will use the spelling: Ramon Llull. George P. H. Styan Latin squares in Europe 1283–1788 Ramon Llull (1232–1316) page SW7 Ramon Llull was a Majorca writer and philosopher born in Palma, Majorca, now part of Spain. He wrote the first major literary work in the Catalan language. Recently surfaced manuscripts show him to have anticipated by several centuries prominent work on electoral systems, see, e.g., Hägele & Pukelsheim7. http://www.asmalldoseof.org/historyoftox/MiddleAges/Ramon_Llull.jpg 7 Llull’s writings on electoral systems, by Günter Hägele & Friedrich Pukelsheim, Studia Lulliana, 41, 3–38, 2001. The stamp, which depicts a detail from Llull’s De Nativitat, is from Andorra (Spanish Admin.) 1987, Scott 182. George P. H. Styan Latin squares in Europe 1283–1788 Llull’s Figura Ignis is a criss-cross Latin square page SW8 In all 4 “elemental figures”, Ramon Llull used the criss-cross Latin square pattern as exhibited by the matrix 0 1 1 2 3 4 B2 1 4 3C L = B C ; 1 @3 4 1 2A 4 3 2 1 which we call “criss-cross” since all the forwards-diagonal elements and all the backwards diagonal elements are equal. Llull’s Latin square matrix L1 is formed by coding 1 = Ignis (fire), 2 = Aër (air), 3 = Aqua (water), 4 = Terra (land). George P. H. Styan Latin squares in Europe 1283–1788 Properties of Llull’s criss-cross Latin square matrix L1 page SW9 01 2 3 41 B2 1 4 3C We have coded the criss-cross Latin square matrix L = B C 1 @3 4 1 2A 4 3 2 1 to be in standard-form—the first row is 1, 2, 3, 4.