In lieu of an abstract, here is a brief excerpt of the content:

Leonurdo. Vol. I I. pp. 207-209. (':PergamonPress Ltd. 1978. Printed in Great Britain . 10 18 1 14 22 11 24 7 20 3 17 5 13 21 9 23 6 19 2 15 4 12 25 8 16 0 0 2 ~ 4 x /78/070I4207so2.00/0 MAGIC SQUARES: A DESIGN SOURCE Ben F. LaPosky* Magic squares, a form of recreational mathematics, are a possible source of geometric design. The magic squares area differentkind of number array from those presented by Richard Kostelanetz, recently in Leonard0 [I] and from other numerical matrices. A basic magic square (Fig. 1a) 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 [24]. 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 1&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 setsthat 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 latticesto total constant sums. Lastly, the natural magic squares of numbers 1 to n2may yield basic design elements from niagic fine 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. la) there are other magicgroupings 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 groupof four numbers in a square, plus another number separated by one cell diagonally, total 65, as 10-18-1 1-24-2. 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. Magiclinesareexemplifiedby the tracing shown in Fig. I(b). This is formed by connecting points representing in place the numbers in Fig. I(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 I, 6 9 , etc.), or by skipping from the initial number of each *Artist, 301 South 6th Street, Cherokee, IA 51012, U.S.A. (Received 2 Feb. 1978) submultiple group to the next etc., as l-6-ll-I6--2l. Other seriesgroupings arealso possiblefor magiclineuse. A group of basic design elements derived from 4 x 4 magic squares is shown in Fig. 3. These could be combined in various waysto provide all-over patterns for textiles, or used alone or in border repeats. The patterns shown are from different types of 4 x 4 magic squares, and are derived by the use of various submultiple groups of numbers. Claude Bragdon, an architect and designer, produced patterns from magic line tracings, as shown in his books, The Frozen Fountain and Projective Ornament [5, 61. A variety of methods has been used for theconstruction of magic squares and other magic figures. If one follows the consecutive magic line tracing in Fig. l(a), it is possible to see that there is a strategic distribution of numbers...