Magic hypercube

Abstract: This paper systematically surveys the use of different approaches for the magic square (MS) as an indicator of welfare, a formal system of necessary relations to deal with conflicts of socioeconomic objectives. The starting point is the article of Kaldor (1971) followed by contributions by the OECD from the 1970’s resulting in a diagram which allowed a visual diagnosis of macroeconomic performance. Such representations were re-examined by Medrano-B and Teixeira (2013), who introduced a required normalization of the variables. Here, we show that this approach was marred by an oversight, namely the issue of the ordering of variables along the axes. In order to avoid this problem, we propose the use of a new mathematical approach involving a Hypercube Graph, which we call magic hypercube, which produces the same index, regardless the ordering of the variables. An application of the new concept is offered using economic data from Brazil and Chile.

Abstract: Magic squares, a form of recreational mathematics, are a possible source of geometric design. The magic squares are a different kind of number array from those presented by Richard Kostelanetz, recently in Leonardo [1] and from other numerical matrices. A basic magic square (Fig. la) is a series of consecutive numbers from 1 to n2 so arranged that at least the horizontal rows, vertical columns and the two principle diagonals always add up to a constant sum [2-4]. The example shown is a pan diagonal square of sum 65. This constant is five times the center number of the series, 13. The square is named 'pan diagonal' because all broken diagonals also total 65, as 10-3-21-19-12, etc. Magic squares may be of interest to artists for three reasons. First, many squares contain other number groupings within the square besides the required row, column and diagonal sets that add up to a constant sumthese beautiful harmonies of numbers may have a kind of aesthetic appeal. Next, it is possible to place numbers at the intersections of lines or curves of a great variety of geometric plane figures that exhibit a constancy in their summation. Likewise, numbers may be situated on the surfaces of cubes, spheres or other 3-dimensional constructions, or in spatial lattices to total constant sums. Lastly, the natural magic squares of numbers 1 to n2 may yield basic design elements from magic line tracings. These are derived from following the path of the consecutive numbers within the square or the paths of other submultiples of the series. In the 5 x 5 square (shown in Fig. l a) there are other magic groupings or constellations that total 65. Every plus (+ ) or times ( x ) cross set of five contiguous numbers, as 11-18-24-5-7 or 10-24-13-17-1, exhibits this property. Also, any group of four numbers in a square, plus another number separated by one cell diagonally, total 65, as 10-18-11-24-2. In some types of higher order squares, many kinds of such constellations occur at all places within such magic squares. An example of an exotic geometric arrangement with magic number properties is shown in Fig. 2. This is a representation of a 2-dimensional projection of a theoretical 4-dimensional figure, a magic hypercube or tessaract. It totals 34 for various constant groups of four numbers, as the corners of all squares or diamonds which may be regarded as the faces of projections of cubes. Magic lines are exemplified by the tracing shown in Fig. 1(b). This is formed by connecting points representing in place the numbers in Fig. 1(a), in consecutive order (then back to 25 to close the path). Other magic lines may be traced by following only the even numbers, or only the odd numbers, or the submultiples (as 1-5, then back to 1, 6-9, etc.), or by skipping from the initial number of each

Abstract: T he four dimensional objects have been already introduced in [2],the standard tesseract in Euclidean 4-space is given as the convex hull of the points (±1,±1,±1 ±1).That is it consists of the points { (x 1 , x 2 , x 3 , x 4 ) I R 4 , -1 £ x i £ 1}. The vertex cover polynomial of a graph G of order n was introduced in [3] it is defined as the polynomial....