An Illustrated Introduction to Some Old Magic Squares from India
In this article we consider old magic squares from India associated with 1. Daivajna Varāhamihira (505–587 AD) and his Bṛhat Saṁhitā [39]: magic perfume; 2. Khajuraho 945 AD: Sir Alexander Cunningham (1814–1893) [14]; 3. Dudhai (Jhansi district) early 1
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Abstract In this article we consider old magic squares from India associated with 1. Daivajna Var¯ahamihira (505–587 AD) and his Br.hat Samhit¯ ˙ a [39]: magic perfume; 2. Khajuraho 945 AD: Sir Alexander Cunningham (1814–1893) [14]; 3. Dudhai (Jhansi district) early 11th century: Harold Hargreaves (b. 1876) [27]; 4. T.hakkura Pher¯u (fl. 1291–1323): Gan.itas¯arakaumud¯ı: The Moonlight of the Essence of Mathematics [1]; 5. Simon de la Loubère (1642–1729): Monsieur Vincent, Surat [3,15]; 6. Major-General Robert Shortrede (1800–1868) [16], Gwalior 1483 [11, (1842)]; Andrew Hollingworth Frost (1819–1907) [23], Nasik [17, (1877)]; 7. N¯ar¯ayan.a Pan.d.ita (fl. 1340–1400): Gan.ita Kaumud¯ı [2, (1356)]; 8. Srinivasa Aiyangar Ramanujan (1887–1920) [34,35,40,43]; Prasantha Chandra Mahalanobis (1893–1972). Magic squares were once part of occult philosophy, but more recently, however, they form part of recreational mathematics. For the past 50 years or so, they have been studied in a matrix-theoretic setting. Our main interest is in the history and philosophy of magic squares and the related magic matrices and in the related bibliography and biographies. We try to illustrate our findings as much as possible and, whenever feasible, with images of postage stamps and other philatelic items.
G.P.H. Styan (B) Department of Mathematics and Statistics, McGill University, 805 ouest rue Sherbrooke Street West, Montreal, Quebec, Canada H3A 0B9 e-mail: [email protected] K.L. Chu Dawson College, 3040 Sherbrooke Street West, Westmount, Montreal, Quebec, Canada H3Z 1A4 e-mail: [email protected] R.B. Bapat et al. (eds.), Combinatorial Matrix Theory and Generalized Inverses of Matrices, DOI 10.1007/978-81-322-1053-5_18, © Springer India 2013
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Keywords Sir Alexander Cunningham (1814–1893) · Andrew Hollingworth Frost (1819–1907) · Harold Hargreaves (b. 1876) · Simon de la Loubère (1642–1729) · Prasantha Chandra Mahalanobis (1893–1972) · N¯ar¯ayan.a Pan.d.ita (fl. 1340–1400) · T.hakkura Pher¯u (fl. 1291–1323) · Srinivasa Aiyangar Ramanujan (1887–1920) · Major-General Robert Shortrede (1800–1868) · Daivajna Var¯ahamihira (505–587 AD) · Bibliography · Biography · Chota Surang shrine · Dudhai (Jhansi district) · Franklin’s most magically magical of magic squares · Gwalior magic squares · Khajuraho magic square · Magic matrices · Magic perfume · Nasik (pandiagonal) magic squares · Recreational mathematics · Rhomboidal magic squares Mathematics Subject Classification (2010) 00A08 · 01A32 · 15A24
1 Introduction I want to tell you about some old magic squares from India and to show you images of some related postage stamps. But first a little history. Legend has it that the very first magic square, the Luoshu, was discovered in China near the Luoshu River about 4000 years ago on the back of a turtle or tortoise! The magic square known as Luoshu = Luo River Writing is based on the classic 3 × 3 fully magic matrix ⎛ ⎞ 4 9 2 L = ⎝3 5 7⎠ . 8 1 6
In a classic fully magic matrix, the numbers in all
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