Science in the Universe of the Matrix Elements 1..n 2 Windsor 2007 June 1-3

1. Science in the Universe of theMatrix Elements 1..n2Windsor 2007 June 1-3 Peter Loly & Ian Cameron With Walter Trump, Adam Rogers & Daniel Schindel Critical funding from the Winnipeg Foundation in 2003.

3. Overview • Focus on sequential integer square matrices with matrix elements 1..n2 (but make use of general properties of real square matrices). • Factor out the magic eigenvalue for semi-magic squares from characteristic polynomial, and thus for magic squares. • Singular value decomposition (SVD) analysis. • Compound squares – Kronecker product. • Maple, Mathematica, Scientific WorkPlace and MATLAB used where appropriate.

4. 12,544x12,544 compound magic square

5. Pathway patterns

6. The Manitoba Magicians[with apologies to the International Brotherhood of Magicians (IBM), founded in Winnipeg in 1922] • John Hendricks, meteorologist (retired to Victoria, British Columbia) • Frank Hruska, Chemistry, University of Manitoba • Vaclav Linek and John Cormie (antimagic squares) University of Winnipeg • Peter Loly, retired December 2006, Senior Scholar, U Manitoba • Ian Cameron, Planetarium and Observatory supervisor, U Manitoba • Marcus Steeds, Frantic Films • Wayne Chan, Centre for Earth Observation Science (CEOS) • Adam Rogers, graduate student, genetic algorithms for astrophysics • Daniel Schindel, Michigan State, East Lansing, nuclear theory • Matthew Rempel, second degree in engineering • Russell Holmes, PhD Princeton • Gideon Garland, mass spectroscopy, Israel.

7. Modern Combinatorics • Persis Diaconis (and company) • “Following MacMahon [45] and Stanley [52], what is referred to as magic squares in modern combinatorics are square matrices of order k, whose entries are nonnegative integers and whose rows and columns sum up to the same number j.” • www.emis.ams.org/journals/EJC/Volume 11/PDF/v11i2r2.pdf • Counting integer points in polyhedral cones (de Loera, Beck, Ahmed, ...). Difficult to handle matrix elements 1..n2.

8. Doubly stochastic matrices • Shin, Guibas and Zhao, CS dept., Stanford: footnote 5: • “The doubly-stochastic matrix is a N x N non-negative matrix, whose rows and columns sum to one.” http://graphics.stanford.edu/projects/lgl/papers/sgz-ipsn-03/sgz-ipsn-03.pdf

9. Normal (Classic) Magical Squares“magic” n-sum =n(n2 +1)/2 • Semi-magic (SM) • All rows and columns • Magic squares (MS) include both diagonals • Antipodal constraint (local) associative or regular MS • Global constraints: pandiagonal MS, bent diagonal (Franklin), complete (McClintock), complement and pandiagonal. • “not even semimagic” • Non-magic pandiagonal • No rows or columns • Example: serial squares of any order • Example: logic squares of orders n=2p • i.e., n=2,4,8,16,32

10. Bageltorus topologyrubber sheet geometry • Take a square sheet • Join a pair of opposite edges to form a cylinder • Bend the cylinder until its ends join • Also known as periodic boundary conditions. • Useful for thinking about pandiagonals.

11. How Many Normal Squares? • 1/8(n2)! distinct 1.. n2 squares • for n = 2 => 4.3.2/8 = 3 distinct squares • 3 x 3: now 9.7.6.5.4.3.2 = 45,360 • 4 x 4: 57,657,600 times 3-by-3 count = 2,615,348,736,000 (2.615..* 1012) • After listing the n =3 squares, we add constraints to reduce these numbers!

12. Three 2-by-2’s S2 “Serial”, upper left, is pandiagonal and regular. “Cyclic”, upper right, is affine. “Scissors” lower left also affine.

13. 4-by-4 serial S4 and logic L4 squares • Both are pandiagonal non-magic • Serial squares exist for any order • Logic squares or order 2p derive from Karnaugh maps and Gray code, e.g., edges: {0,1}; {00, 01,11,10} [Loly and Steeds, 2005] (incremented to 1.. n2) • Complementing alternate cells with 17, i.e., 17-x, yields a pandiagonal magic square (Meine and Schütt, Siemens)

14. Serial Squares of order ncharacteristic polynomial xn-2(x2+ αnx+ βn ) • Loly and Steeds, “A New Class of Pandiagonal Non-Magic Squares, Int. J. Math. Ed. Sci. Tech. 36 (2005) 375-388. [IJMEST] • Sloane Integer Sequence A006414 (unique from first 4 terms, checked 11) [http://www.research.att.com/~njas/sequences/] • Walsh and Lehmann, “Counting rooted maps by genus. III: Non-separable maps, J. Comb. Th. Ser. B 18 (1975) 222-259

15. Serial and Logic squares: rank 2

16. Magic Square Counts: Trump (c. 2002)

17. Backtracking • Schroeppel 1972 - See Gardner, M., Mathematical Games, Scientific American (1976) 118-122. • Pinn, K. and Wieczerkowski, C., 1998, Number of Magic Squares from Parallel Tempering Monte Carlo, International Journal of Modern Physics C, 9(4), 541-546. • Trump, W. Notes on Magic Squares and Cubes, www.trump.de/magic-squares • Schindel, D. G., Rempel, M. and Loly, P., 2006, Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS, Proceedings of the Royal Society A: Physical, Mathematical and Engineering, 462, 2271-2279. • (Screen savers)

18. Vector Spaces of Magical Squares • General 3x3 Lucas 1891 • General 4x4 Bergholt 1910 • John Tromp & Peter Loly - Haskell/Maple

19. Bergholt 19104-sum: A+B+C+D

20. Rotation • A square has 8 phases obtained from any one by rotations and reflections. • It is convention to select one, but it turns out to be illuminating to study a second phase, either a 90° rotation or a flip. • This is illustrated next for the archetypal LoShun = 3 magic square from China – probably 2 millennia old.

21. Lo-shu (A,B: Frank Hruska 1991)[SVD: 15, 4√3, 2√3 – Loly 2007]

22. MATLAB’s magic(n) • Separate algorithms for producing one square in each case [Cleve Moler, MATLAB’s Magical Mystery Tour, Winter 1993, MathWorks Newsletter 7(1) 8-9] • Odd n – regular (associative) NONSINGULAR • Singly even – NOT regular, SINGULAR • Doubly even – regular, SINGULAR • 1995 Kirkland and Neumann give EV’s and SVD formulae for n = 4k [Lin. Alg. and its Appls. 220: 181-213].

23. Mattingly singular even order regular magic squares • 1999 (preprint) Mattingly proof of singularity for even order [Am. Math. Monthly 107 (2000) 777-782] • 1999 Loly already had studied all singular squares in the 4th order set of 880 by Dudeney group, finding examples with just one non-zero eigenvalue.

24. 5th order regular magic squares • Mattingly had conjectured odd orders were non-singular. • 2003 Schindel and Loly, using programming ideas from an n=6 hybrid backtracking code of Walter Trump, regenerated the 5th order set of some 275 million magic squares and found that of the 48,544 regular magic squares, 652 were singular with 2 zero EVs, AND four had 4 zero EVs. • The 16 ultramagic squares [Suzuki] are non-singular.

25. 7th order ultramagic squaresWalter Trump • Trump found 20,190,684 of these squares, of which Schindel found 20,604 to be singular, with a pair of zero eigenvalues. • Trump also found that less than 0.06% of 7th order regular squares are singular. • Kerry Brock, “How many singular squares are there?” Math. Gaz. 89 (Nov. 2005) 378-384.

26. W-42 Götz Trenkler • “On the Moore-Penrose inverse of magic squares” • We leave this aspect of singular magic squares to GT.

27. Dudeney Groups Henry Ernest Dudeney (1857-1930)

28. 8804th order magic squares

29. Talks W-44 and W-08 • Kimmo Vehkalahti will talk about the 640 singular magic squares in more detail than I can include in my talk. • La Lok Chu, in joint work with George Styan, will talk about various issues for the 880, including their work on odd powers of certain remarkable cases. • PDL will focus on a few examples and SVD.

30. SVD • Since we have found many magic squares with only a single non-zero eigenvalue, but rank 3 (or more), SVD values give more information (akin to an X-ray). • Further, motivated by analytical results of Kirkland and Neumann, we turned to examining the eigenvalues of ATA, the (observability) Gramian matrix, a symmetric matrix, where the square root of its (positive) eigenvalues gives the SVD values, with the largest being the linesum eigenvalue.

31. all regular (group 3)ATA’s are bisymmetric (M=JMTJ) Example: F790 regular • EV: 34,0,0,0 • NO CHANGE ON ROTATION • SVD: 34, 8√5, 2√5, 0 • EV(ATA ): 1156, 320, 20, 0 • F803 has same EV’s N.B. F803 has a different ATA matrix, but same EVs, SVD

32. Rotation of F109group 1, pandiagonal Charpoly (Maple): x(x-34)(x2-64)=0 eigenvalues: 34, ±8, 0 rank 3 SVD: 34, 16.4924, 8.2462, 0 34, 4√17, 2√ 17, 0 Rotated F109: Charpoly: x3(x-34)=0 eigenvalues : 34, 0, 0, 0

33. F175 & F790group 3, regular • F175 • EV’s : 34, ±8, 0 • RF175: 34, ±8i, 0 • SVD 34, 8√5, 2√5, 0 • F790 • EV: 34, 0, 0, 0 • No change on rotation • SVD 34, 8√5, 2√5, 0

34. F181 & F268 – nonsingularFULL RANK Group 11 F181: 34, -8,4±2i√2 RF181: 34, -5.30783, 2.65391±5.3972i SVD 34, 17.442, 5.6569, 1.9460 Group 7 F268: 34, -11.3873, 9.26392, 2.1234 RF268: 34, -12.6382, 11.0315, 1.60667 SVD 34, 15.646, 9.6428, 1.4847

35. Parameterization – Vector Spaces • Bergholt had 8 variables which reduce with further constraints. • For groups 1,2,3 we have found 4 dimensional spaces, and have used Maple to factorize their characteristic polynomials. • We have also found algebraic formulae for the eigenvalues of the Gramian matrix, ATA, which gives the squares of the SVD values, for groups 1 and 3.

36. Parameterization of Regular 4’s (48)

37. Woodruff 1916 (n=8)x5(x-260)(x2-8736)=0 SVD260, 129. 06, 72.0

38. Regular, n=5 (Schindel, Trump) x2 (x-65)(x2-340)=0 SVD 65, 32.948, 14.142, 3.8006 [squared: 4225, 1085.57, 200, 14.444] 2007 Trump has studied all singular 5th order squares x4(x-65)=0 SVD 65, 26.458, 20.837, 12.878 Squared: 4225, 700, 434?, 165?

39. Ultramagic, n=5 (Trump) • Pandiagonal & regular • After factor (x-65): (x4-250x2+12245) • EV’s: 65, ±a, ±b a =√(125-26√5) b =√(125+26√5) • SVD 65, 25.348, 24.450, 7.2249, 2.7347

40. Ultramagic, n=7(Trump) • EV’s: • 0,0,175, ±3, ±i√231 • SVD 175, 74.369, 53.970, 28.031, 20.796, 11.759, 0

41. Compound Squares • Wayne Chan & Peter Loly, Mathematics Today 2002 • Harm Derksen, Christian Eggermont, Arno van den Essen, Am. Math. Monthly (in press) • Matt Rempel, Wayne Chan, and Peter Loly • Adam Rogers’ Kronecker product

42. Compounded Lo-shu(1275 Yang Hui; Cammann)

43. Second Compound Method(1275 Yang Hui; Cammann)

44. Kronecker Product • For 2nd order A, any B

45. 2004 Adam Rogers(4th year Quantum Mechanics) • EN is Nth order square of 1’s • AM and BN are Mth and Nth order squares • Associative Compounding: • RA = EM BN + Nk (AM  EN) • Distributive Compounding: • RD = BN EM + Nk (EN  AM) • Given the EVs and SVDs of A and B, Rogers can find those for both compound methods • (k=2 for squares, 3 for cubes, etc.,)

46. “Franklin” binary • All 2x2’s sum to 2, as do all bent diagonals. • Half rows have sum 1, rows and columns sum 2.

47. Franklin Squares At right – pandiagonal Franklin square PRSA Arno van den Essen’s book The 12th order question http://www.geocities.com/~harveyh/M-p_B-d.htm [6May 2007] http://www.geocities.com/~harveyh/franklin.htm#Comparison Donald Morris n=12 “Franklin” 2007

48. Franklin, McClintock,Ollerenshaw & Brée • Bent diagonal squares, half row/column squares 17.. n = 8, 16 • Complete squares 18.. • Most-Perfect Pandiagonal squares • 2006 exact count of Franklin squares for n = 8: 3*368,640 = 1,105,920 • The problem of n = 12 • 22,295,347,200 complete squares (O&B 1998) • Eggermont – no pandiagonal F’s at n = 12 • Donald Morris – 1/3 rows/cols for n = 12, 1/5 for n = 20, etc.

49. Inertia Tensor • Moment of Inertia of magic squares – Loly Math. Gaz. 2004 • I = ∑mi (ri)2 => In = (1/12)n2(n4 -1) • Only need the semimagic constraints! • Inertia Tensor of Magic Cubes (Rogers and Loly, Am. J. Phys. 2004) • Folding magical squares to create magical cubes: 8*8 square => 4*4*4 cube • Kronecker products • Multiway arrays

50. Issues • Constraint satisfaction problems (CSP’s) • Constraint Logic Programming (CLP), e.g., FormulaOne Compiler • Counting integer points in polyhedral cones (Maya Ahmed, Jesus de Loera, Matthias Beck, …) • Cryptography (O&B, Meine & Schuett) • Dither matrices (patents)