20 Magic Mathematics Based on New Matrix Transformations (2D and 3D) for Interdisciplinary Physics, Mathematics, Engineering and Energy Management Prof. Dr.-Ing. Wolfram Stanek1,2 and Dipl. Ing. Maralo Sinaga3 1University of Applied Sciences Koblenz, 2Guest Lecturer at Swiss German University, BSD City-Jakarta 3Head of Mechatronics Department, Swiss German University, BSD City-Jakarta 1Germany 2,3Indonesia 1. Introduction Mathematics is magic. If we can either use one formula for a wide range of applications or the formula itself will produce magic properties. As one of several introductory examples the generally not well known Leibniz formula for calculating determinants in matrix theory will show that both the well known Laplace laws and Sarrus rules for evaluating matrices are only graphically visualised subsets of this ingenious Leibniz formula. Visualising complex formulas and matrix transformations in 2D and 3D as equivalent graphs is a basic method of the main author in this publication. The huge range of fascinating technical applications based on 2D magic matrices will be sketched: Constant distribution in all directions of numbers, power, energies, element properties, transport, automation, information flows etc or compensation of punctual disturbances without variation of sum of energy or automatic minimization of energy loss remaining constant distribution or both concentration of energies in near field and hiding of energies in far field or solving magic equation systems in mathematics without using back tracking methods etc. 2. Background The extremely complex problem in mathematics of finding a perfect solution of a 4x4x4 - 3D - magic cube (64 unknowns, but 76 equations/conditions) with constant sum in all directions and continuous numbers from 1 to 64 was solved first by the German mathematician W. Trump in the year 2004 (Spectrum of Science, 2008-2): But this world wide first solution of a 4x4x4 magic cube was only based on parallel computations with several computers and extremely time-intensive back-tracking methods with time consuming solution. In contrast to this computer-based solution of 4x4x4 magic cubes in 2004, the main author Prof. Dr. W. Stanek has shown a new analytical method manually solving this problem during a presentation on German MemoMasters 2008 and 2009: Using this analytical method for 4x4x4 www.intechopen.com 378 Products and Services; from R&D to Final Solutions magic cubes, the manual 3D solution lasts a few minutes - applying this algorithm the solution time with MATLAB® needs only fractions of seconds (ca. 0.01 s). The results of these matrix transformations for magic 64-cells-cubes show two main aspects: a. Extremely fast solution of such matrix problems in 3D by immediate transformation from magic 2D matrices to magic 3D cubes with remaining central magic properties. b. New idea solving large sets of linear equations (with also determinant-zero-matrix- property) NOT using conventional equation solvers (Gauss-Seidel, Newton-Raphson etc) and backtracking methods but only simplified geometrical 3D transformations and logic. This magic math algorithm is shown by visualised graph transformations and underlying equivalent structures. 3. Magic square and magic cube (2800 B.C – 2008) Magic squares were known to Chinese mathematicians, and Arab mathematicians, possibly as early as the 7th century, when the Arabs conquered northwestern parts of the Indian subcontinent and to learned Indian mathematicians and astronomers, including other aspects of combinatorial mathematics. The most famous 2D magic squares are the Lo Shu square and the Duerer square. A normal magic square contains the integers from 1 to n2. The constant sum in every row, column and diagonal is called the magic constant or magic sum, S. The magic constant of a normal magic square with continuous numbers depends only on n and has the value: n S= (n2+1) (1) 2 3.1 Lo Shu Square Lo Shu square or the Nine Halls Diagram is the unique normal magic square of order 3 x 3. Lo Shu is part of the legacy of the most ancient Chinese mathematical and divinator traditions, and is an important emblem in “Feng Shui” the art of geomancy concerned with the placement of objects in relation to the flow of 'natural energy'. The Lho Shu square was introduced in 2800 BC. Fig. 1.(a) shows the Loh Shu square used symbolism instead of numbers, and Fig. 1.(b) representing continuous number 1 to 9 this square. The Loh Shu square dimension is n=3, then the magic sum S is 15. 8 1 6 3 5 7 4 9 2 (a) Loh Shu square with symbols (b) Loh Shu square with numbers Fig. 1. Loh Shu square (Wikipedia, 2010) www.intechopen.com Magic Mathematics Based on New Matrix Transformations (2D and 3D) for Interdisciplinary Physics, Mathematics, Engineering and Energy Management 379 3.2 Duerer magic matrix The Renaissance engraving “Melancholia I” was developed by the German artist, painter, and mathematician Albrecht Duerer (in the year 1514). This image is filled with mathematical symbolism and in the upper right corner of the first picture a square can be seen. The Fig. 2(a) shows an enlarged view of the Duerer‘s square cropped from the image. This square is known as a magic square and was believed by many in Duerer's time to have genuinely magical properties. It does turn out to have some fascinating characteristics worth exploring. 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 (a) (b) Fig. 2. Duerer square, Melancholia I, 1514, (Wikipedia, 2010) The Duerer's square in Fig. 2(b) is filled up with continuous numbers 1 to 16. The square dimension is n=4, then the magic sum S of Duerer's square is 34. 3.3 Sudoku Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contain all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which typically has a unique solution. Completed puzzles are always a type of Latin square with an additional constraint on the contents of individual regions. For example, the same single integer may not appear twice • in the same 9x9 playing board row • in the same 9x9 playing board column or • in any of the nine 3x3 subregions of the 9x9 playing board. The puzzle was popularized in 1986 by a Japanese puzzle company and became an international hit. An extended example of the Sudoku game is shown in the Fig. 3, where a 9x9 square A is developed from a 3x3 square. Then the other cells are filled by using a regular shifting Fig. 3. Sudoku 9x9 from 3x3 square and bi-magic square (www.multimagie.com, 2009) www.intechopen.com 380 Products and Services; from R&D to Final Solutions method. From the square A, then square B is constructed by using rotating technique, and finally the bi-magic square C (with continuous number 1 to 81) is developed through addition of A and B, example applying the formula C=9· (A–1)+B. This magic square is bi- magic (or multi-magic) if it remains magic after each of its numbers have been squared. This was introduced by Tarry and Cazalas. All cells content are squared, resulting in magic sums of C=369 and D=20049. 4. Components of creative intelligence, Leibniz matrix and new solution technique. The following MATLAB® program cutoff to calculate the determinant of n x n matrix should be a central background of this publication. According to the famous Leibniz formula for determinant calculation of any n x n matrix, the fact shows that the Method of Sarrus and Method of Laplace to solve the determinant of the matrix use similar concept of the Leibniz formula with different visualisations. Both Sarrus and Laplace Methods could be structurised and visualised in mnemotechnique method by the main author. Finding the determinant of nxn matrix using the Leibniz formula is shown in the equation (2): ⎛ n ⎞ det(A)=∑⎜⎜sgn(σ)∏ai,σ(i)⎟⎟ (2) ⎝ i=1 ⎠ Based on this point, W.Stanek provides an algorithm to obtain any 3D magic structure, especially 64 cells cube, primarily the visualised structure of the algorithm. At the same time it will be shown that in a linear equation with a lot of unknowns (64 unknowns and 76 equations) is first solved by using the logical method and visual solution. This problem was first solved in 2004 by using computers, working in parallel and based on the back tracking algorithm method. The equation is represented in MATLAB® function and algorithm shown as: % MATLAB and Leibniz formula provide the same result % det(M1) = -360 for 3x3 Loh Shu Matrix % For magic 4x4 Duerer Matrix both methods yield det(M2) = 0, too detA_Matlab = det(M1) % Leibniz-Formula for all nxn-Matrices, here only shown for 4x4-Matrix : % With n=4 follow 4! = 1x2x3x4 = 24 Terms for solutions of det(A) % With a14=0; a24=0; a34=0; a44=1; a41=0; a42=0; a43=0; % Leibniz formula also for 3x3-Matrices like i.e. magic 3x3 Loh Shu Matrix. detA_Leibniz = (a11*a22*a33*a44 + a11*a23*a34*a42 + a11*a24*a32*a43... - a11*a22*a34*a43 - a11*a23*a32*a44 - a11*a24*a33*a42... + a12*a21*a34*a43 + a12*a23*a31*a44 + a12*a24*a33*a41... - a12*a21*a33*a44 - a12*a23*a34*a41 - a12*a24*a31*a43... + a13*a21*a32*a44 + a13*a22*a34*a41 + a13*a24*a31*a42... - a13*a21*a34*a42 - a13*a22*a31*a44 - a13*a24*a32*a41... + a14*a21*a33*a42 + a14*a22*a31*a43 + a14*a23*a32*a41... - a14*a21*a32*a43 - a14*a22*a33*a41 - a14*a23*a31*a42) % NOTE: Leibniz is central for all det(A)-calculations by Sarrus and by Laplace www.intechopen.com Magic Mathematics Based on New Matrix Transformations (2D and 3D) for Interdisciplinary Physics, Mathematics, Engineering and Energy Management 381 % 3x3 matrices calculated by Sarrus Rule directly from Leibniz Formula % nxn matrices (n=3, 4, ...) by Laplace Rule directly from Leibniz, too % Both rules of Sarrus and Laplace are visualised structures of the % Leibniz Formula detA_Leibniz (above shown for 4x4-Matrices) % This Leibniz Formula is ingenious as basis for Sarrus, Laplace etc From equation (2) following equation (3) can be derived. It is possible to expand a determinant along a row or column using this formula, which is efficient for relatively small matrices. To do this along row i, say, we write: n n det(A)=∑Ai,j⋅Ci,j =∑Ai,j⋅(−1)i+j⋅Mi,j (3) j=1 j=1 where the Ci,j represents the i,j element of the matrix cofactors, i.e. Ci,j is ( − 1)i + j times the minor M , which is the determinant of the matrix that results from A by removing the i-th i,j row and the j-th column, and n is the length of the matrix. The determinant of a 2x2 matrix A is calculated by: M=⎡⎢a11 a12⎤⎥ det(A)= a11 a12 a a a a ⎣ 21 22⎦ 21 22 det(A)=(a11⋅a22−a21⋅a12) For a of 3x3 matrix, the determinant is calculated by using the Sarrus method, derivated from Leibniz’s formula: ⎡a11 a12 a13⎤ a11 a12 a13 ⎢ ⎥ A=⎢a21 a22 a23⎥ det(A)= a21 a22 a23 ⎢a a a ⎥ a a a ⎣ 31 32 33⎦ 31 32 33 - - - a a a a a 11 12 13 11 12 det(A)= a21 a22 a23 a21 a22 a a a a a 31 32 33 31 32 + + + det(A)= (a11⋅a22⋅a33+a12⋅a23⋅a31+a13⋅a21⋅a32−a31⋅a22⋅a13−a32⋅a23⋅a11−a33⋅a21⋅a12) To find the determinant of a 3x3 matrix according the Laplace formula shown in the equation (3): a a a 11 12 13 a a a a a a det(A)= a21 a22 a23 =a11 a22 a23 −a12 a21 a23 +a13 a21 a22 a a a 32 33 31 33 31 32 31 32 33 www.intechopen.com 382 Products and Services; from R&D to Final Solutions det(A)={a11(a22⋅a33−a32⋅a23)−a12(a21⋅a33−a31⋅a33)+a13(a21⋅a32−a31⋅a22)} (4) Because it is dealing with a 3× 3 matrix, it sets up the 3× 3 sign matrix. This is always a “checkerboard” matrix that begins with a “+” sign in the upper left corner and then alternates signs along rows and columns. The Leibniz formula is the root of the Sarrus formula and the Laplace formula. The regularity of the Leibniz-, Laplace-, and Sarrus-determinant calculation was the basis for the main author developing magic matrices and cubes through visualised transformation of the cell contents (shifting, rotating and reflecting or mirroring). 5. Computer solution It was only possible to solve a 4x4x4 cube (76 equations with 64 unknown) by using a several computers, working in parallel, and based on the backtracking algorithm method. This was shown by the German mathematician Walter Trump in year 2004. New Idea The 4x4x4 cube (76 equations with 64 unknown) will be solved now by using only logical thinking and geometrical methods (bending surfaces). Assume a box, with 4x4 cells on each side is opened into a 2 dimensional plane as shown in the Figure 4; the number must be match each other to the side plane. The following sequence is used to solve the magic-matrix and cube respectively. Fig. 4. Solution idea based on “box-exploding” (Stanek, 2009) www.intechopen.com Magic Mathematics Based on New Matrix Transformations (2D and 3D) for Interdisciplinary Physics, Mathematics, Engineering and Energy Management 383 Step 1. Start with any magic 4x4 matrix M1, Step 2. M1 is reflected in all sides of the box. Step 3. Use the logic method, match all the edge cells Step 4. From M1 until M4, magic cube 2 is constructed using surface transformation, bending, or reflecting (mirroring). The computer based magic cube solution, which is discovered in 2004 with highest degree of perfection is first in 2008 analytically solved by the main author. The solution ideas and the important steps using the Stanek Method to solve the magic cube will be shown in the following pages and a pattern solution is attached. 6. Short information: magic cube with Stanek-Method analysis In the above shown graphical method the solution of magic cube is explained. All related data of „Sudoku to the power of 3“application will be simple and always enough to solve or to construct in the 4 main layers. 6.1 Example: magic + ultra-magic square and cube A magic square is bi-magic (or multi-magic) if it remains magic after each of its number has been squared and an ultra magic square has more extended properties. The following 4x4 square shows an example of ultra magic square (Fig. 5.(a)). (a) (b) (c) Fig. 5. (a) Perfect square or ultra-magic square with continuous number 1 to 16 all rows, columns, diagonals, and four neighbor cells always resulting magic sums = 34 (b) Magic Cube developed from magic square, with continuous number 1 to 16 in 3D, resulting a perfect magic sum=34 in rows, columns, diagonals (c) Magic Cube developed from magic square, with number 1 to 64 in 3D, resulting a perfect magic sum=130 in rows, columns, diagonals in x-,y-, and z-planes (Stanek, 2009) www.intechopen.com 384 Products and Services; from R&D to Final Solutions The sum of all numbers in the horizontal, the vertical as well as in the diagonals are equal and also the sum of all four neighbor cells, which form a square are always constant to 34. The square pattern in Duerer’s square and Loh Shu’s square are easy to memorised. Starting with predetermined Loh Shu’s pattern, a 9x9 squares can be constructed easily by applying a shifting, reflecting method. A magic cube with 4x4x4 dimension can be developed for instance from the known Duerer’s square pattern, or extended from ultra magic square, converted by reflecting, shifting and bending. 6.2 An ultramagic square as the base of Stanek Cube Developments Each of 48 given ultra-magic matrices or squares results in a new magic cube with a maximum possible degree of perfection. Through simple transformation, (shifting, rotating and reflecting) it is easy to construct other ultra magic matrices. Fig. 5. 48 possible magic squares, constructed through shifting, rotating and reflecting. 7. Solution by using mnemonic scheme for Stanek Cube No.1 + No.2 On the Fig. 4 it is shown how to generate and to develop a magic square and magic cubes from a given ultra-magic square. An example is the square number 22 shown in the Fig. 5. www.intechopen.com Magic Mathematics Based on New Matrix Transformations (2D and 3D) for Interdisciplinary Physics, Mathematics, Engineering and Energy Management 385 (given by the audience at MemoMasters – MindFestival 2009 to Prof. W. Stanek for manual solution). Starting by choosing this ultra magic square number 22 as the start matrix M1, the next ultra magic square M2 is created by using the reflecting (mirroring) method. Then from matrix M2 we can construct the next matrix M3 by transposing and at the same time reflecting the content of cells and finally the matrix M4 is generated from M3 using reflecting in each four neighbor cells (number 1 to 16). The resulting ultra-magic matrices M1 until M4 are used to develop the next layers; creating the magic cube layers with a sketching pattern (+0, +16, +32, +48), as shown in the Fig.6. Fig. 6. Pattern Solution to solve a magic cube (Stanek, 2009) www.intechopen.com 386 Products and Services; from R&D to Final Solutions 7.1 First comparison: computer and analytical solution with highest degree of perfection A comparison between a solution which was reached by using computer backtracking method (W. Trump, 2004) and an analytical solution using logic and brain memory shows that the Stanek analytical solution delivers the highest perfect precision of magic cube (MemoMasters 2008). Fig. 7. (a) Magic cube, computer solution by W.Trump (Spektrum Wissenschaft, 2008) (b) Analytical solution by Prof. Wolfram Stanek. This result for the specific analytical cube was predictable, since a start ultra-magic matrix was chosen. The result in Fig. 7.(a) was shown in “Spektrum der Wissenschaft”, 2008 and the Fig. 7.(b) was represented on MindFestival in August 2009. In all the transformations of planes in the space partially some magical sum properties are lost. If many magical properties in the initial matrix are available (at ultra magic matrix of the case), the result of the magic cube (1-64) has a relatively highest degree of perfection. 7.2 Second comparison: minimising the deviation of space diagonals with computer and analytical methods By choosing the Duerer’s matrix, which is not perfect, a magic matrix is created as the start matrix in developing other magic cubes as shown in the Fig. 8. The Fig.9 shows the difference between W.Trump’s magic cube which is solved by computer and the main author’s analytical solution by starting with the magic matrix in the www.intechopen.com