Showing papers on "Magic square published in 1999"

Islamic patterns : an analytical and cosmological approach

Abstract: The point of departure the manifestation of shape magic squares pattern and cosmology the pentagon the tetractys the mathematics of two-dimensional space-filling the circle and cosmic rhythms specimen Islamic patterns.

The projective geometry of Freudenthal's magic square

Abstract: We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial.

On the volume of the polytope of doubly stochastic matrices

Abstract: We study the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices (real nonnegative matrices with row and column sums equal to one). We describe two methods. The first involves a decompos ition of the polytope into simplices. The second involves the enumeration of “magic squares”, that is, n × n nonnegative integer matrices whose rows and columns all sum to the same integer. We have used the first method to confirm the previously known values through n = 7. This method can also be used to compute the volumes of faces of Bn For example, we have observed that the volume of a particular face of Bn appears to be a product of Catalan numbers. We have used the second method to find the volume for n = 8, which we believe was not previously known.

On squares of squares

Abstract: 0. There is a long and intriguing history of the subject of magic squares, squares whose row, column, and diagonal sums are all equal. There has recently been some interest in whether there can exist a three-by-three magic square whose nine elements are all perfect squares; the problem seems first to have been raised by LaBar [5]. The answer is of course yes, for example   52 12 72 72 52 12 12 72 52  

Magic “Squares” Indeed!

Abstract: 6392 + 1742 + 8522 = 9362 + 4712 + 2582 (counter-diagonals) 6542 + 7982 + 2132 = 4562 + 8972 + 3122 (diagonals) 6932 + 7142 + 2582 = 3962 + 4172 + 8522 (counter-diagonals). This property was discovered by Dr. Irving Joshua Matrix [3], first published in [5] and more recently in [1]. We prove that this property holds for every 3-by-3 magic square, where the rows, columns, diagonals, and counter-diagonals can be read as 3-digit numbers in any base. We also describe n-by-n matrices that satisfy this condition, among them all circulant matrices and all symmetrical magic squares. For example, the 5-by-5 magic square in (1) also satisfies the squarepalindromic property for every base.

Power of a determinant with two physical applications

Abstract: An expression for the kth power of an n×n determinant in n2 indeterminates (zij) is given as a sum of monomials. Two applications of this expression are given: the first is the Regge generating function for the Clebsch-Gordan coefficients of the unitary group SU(2), noting also the relation to the 3 F2 hypergeometric series; the second is to the even powers of the Vandermonde determinant, or, equivalently, all powers of the discriminant. The second result leads to an interesting map between magic square arrays and partitions and has applications to the wave functions describing the quantum Hall effect. The generalization of this map to arbitrary square arrays of nonnegative integers, having given row and column sums, is also given.

Method and device for transforming code

Abstract: PROBLEM TO BE SOLVED: To solve the problem that a conventional method of selecting values of a transformation table to be used for code transformation has not specifically been prescribed, and that a biased transformation result possibly influence a transformation characteristic depending on set positions of the figures (or signs or the like) of the transformation table. SOLUTION: Input information P is transformed into line position designating information RP by a line position designating part RCr of a line-column designating part RP, and is also transformed into columnar position designating information CP by a columnar position designating part RCc. An encoding part obtains transformed information C from a magic square M by designating a line position and a columnar position of the magic square M. Here, when a value of an nth order magic square is assumed to be from 0 to n2-1, the input information P takes a value from 0 to n2-1; a value of the line position designating information RP takes a quotient of P/n; and a value of the columnar position designating information CP takes P mod n (a remainder of P/n). The magic square has an even distribution of elements, and this secures inverse transformation.

On properties of 3 X 3 semi-magic squares

Abstract: In this paper we consider 3 X 3 semi-magic squares understood as 3 X 3 matrices whose rows and columns add up tothe same constant. We identify the most general form of such matrices, calculate their Moore-Penrose inverse, their rank and their eigenvalues.