<div id="LayerE10" style="visibility: visible; overflow: visible; z-index:0; left:10px; top:5px; position:absolute; "><?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> <head> <!-- ======================================================= --> <!-- Created by AbiWord, a free, Open Source wordprocessor. --> <!-- For more information visit http://www.abisource.com. --> <!-- ======================================================= --> <meta content="text/html;charset=UTF-8" http-equiv="content-type" /> <title>Harmonics & Mathematics - Magic Squares.html</title> <style type="text/css"><!-- #toc, .toc, .mw-warning { border: 1px solid #aaa; background-color: #f9f9f9; padding: 5px; font-size: 95%; } #toc h2, .toc h2 { display: inline; border: none; padding: 0; font-size: 100%; font-weight: bold; } #toc #toctitle, .toc #toctitle, #toc .toctitle, .toc .toctitle { text-align: center; } #toc ul, .toc ul { list-style-type: none; list-style-image: none; margin-left: 0; padding-left: 0; text-align: left; } #toc ul ul, .toc ul ul { margin: 0 0 0 2em; } #toc .toctoggle, .toc .toctoggle { font-size: 94%; }@media print, projection, embossed { body { padding-top:1.0000in; padding-bottom:1.0000in; padding-left:1.2500in; padding-right:1.2500in; } } body { text-indent:0in; text-align:left; font-weight:normal; text-decoration:none; font-variant:normal; color:#000000; font-size:12pt; font-style:normal; widows:2; font-family:'Times New Roman'; } table { } td { border-collapse:collapse; text-align:left; vertical-align:top; } p, h1, h2, h3, li { color:#000000; font-family:'Times New Roman'; font-size:12pt; text-align:left; vertical-align:normal; } --></style> </head> <body> <div> <p style="text-align: left;" dir="ltr"><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US"> <table style="border-collapse: collapse;" width="600" cellspacing="0" cellpadding="3" bordercolor="#000000" border="0"><tbody><tr><td width="100%" valign="top"> <div> <span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US"> §50.7. Magic Squares</span> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Ferdinand Maack, the original and indefatigable campaigner for a “scientific xenology,” published a series of articles, “Magisch-Quadratische Studien,” in the journal he founded, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Wissenschaftliche Xenologie</span><span lang="en-US" xml:lang="en-US"> (No. 2, July 1899). The following quote comes from the introduction to the series: “[The magic square] contains an ancient mathematical problem, which is continually attacked from new angles over the passage of the centuries. Even the greatest brains of all nations have occupied themselves with this problem, as the bibliography shows” (Maack gives an extensive bibliography, ibid. pp. 33-38). “Academic journals opened their columns to the magic square; university professors gave inaugural lectures about it; it represents an established category in mathematical dictionaries and encyclopedias. All this would surely not be the case if the 'tetragram' were only a 'learned trumpery of numbers,' an 'arithmetical diversion and amusement,' only a number game, a graphic puzzle. Most authors do claim this. But one can read between their lines that the 'divine square' was more to them than just a simple arithmetical example, that along the way they have the feeling that there might yet be something peculiar behind it.”</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> What is a magic square?</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> The discovery of a secret number-figure, the “Lo-shu,” containing the numbers 1-9, is attributed to the legendary Fou-hi. Fou-hi supposedly saw this figure in a wondrous vision on the shell of a tortoise (Thimus, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonikale Symbolik</span><span lang="en-US" xml:lang="en-US"> I, 101). The illustration that the Chinese writers drew from this figure, which have been found on a four to five thousand-year-old Chinese tablet (the Lo-shu), and which is probably the oldest example of a magic square, is the number-square represented in knots:</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">8 3 4</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">1 5 9</span></p> <p style="text-align: center;" dir="ltr"><span lang="en-US" xml:lang="en-US">6 7 2</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">If one writes out the progression of numbers 1-9 in the form shown in Fig. 457:<br /> </span></p> <p style="text-align: center;" dir="ltr"><img src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/0.png" alt="" /></p> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 457<br /> <br /> </span></div> <p style="text-align: center;" dir="ltr"><img src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/1.png" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US">Figure 458<br /> <br /> </span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">and adds on the numbers standing outside the square thus, in these empty fields, as is indicated by the numbers in parentheses, then Fig. 458 appears, i.e. precisely this ancient Chinese tetragram. This is one of the simplest methods of constructing the magic square. The arithmetical characteristics of the magic square, of which there are an infinite number with various indexes, is in the following: the sums of all vertical, horizontal, and diagonal rows are the same, likewise the sum of each pair of numbers opposite one another; and the parallel main and secondary diagonals form an equidistant series. Larger squares show further mathematical curiosities. But these are only the arithmetical attributes. Each magic square is also in equilibrium. If, for example, we set up the Natural Square (N.S.) that belongs to every magic square, i.e. Fig. 457, with weights according to the measures of the numbers, for example 1 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US"> on the topmost, 4 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US"> and 2 </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">g</span><span lang="en-US" xml:lang="en-US"> on the next fields, etc., and then hang the N.S. by a thread in the middle, then the system tips over; if we do the same with the magic square (Fig. 458), it stays in balance. Thus, the problem of magic squares is a problem of equilibrium. Furthermore, Maack discovered (ibid.) that all magic squares can be developed from the natural squares through “torsion of the magical system,” i.e. through rotations of certain inner configurations. If one writes the Natural Square and Magic Square of Figures 457 and 458 as follows (Figure 459 and 460):</span></p> <p style="text-align: center;" dir="ltr"><img src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/2.png" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US"> Figure 459<br /> <br /> </span></p> <p style="text-align: center;" dir="ltr"><img src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/3.png" alt="" /> <span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US"><br /> Figure 460</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">then the shaded fields are symmetrical around the middle field. The sum of each pair of opposite shaded fields is twice the value of the middle field in both squares (Figure 459 and 460). F. Maack calls these shaded and non-shaded fields, which thus appear both in the magic square and the natural square, “magic systems.” If one now imagines the middle field as an axis, the magic square (Fig. 460) appears from the natural square (Fig. 459) through torsion of the magic system. Namely, Fig. 460 appears from Fig. 459 in such a way that first the shaded system of Fig. 459 must be rotated 225° to the right, then the non-shaded system of Fig. 459 must be rotated 45° to the right.</span></p> <p style="text-align: center;" dir="ltr"><img src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/4.png" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US"> Figure 461</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> If we now question the “harmonics” of the magic squares, an equal intervallic quality in this simple example (here the octave would be set at 15) catches our attention. As a comparative theorem, the “complete P” would first come to mind. But there is an even closer parallel. If we construct the corresponding magic square, analogous to the above specifications, from the natural seven-square (Fig. 461), in which we inscribe the diatonic seven-step scale from 1-7, 8-14, etc., in seven different octaves (Fig. 462):</span></p> <p style="text-align: center;" dir="ltr"><img src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/5.png" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US"> Figure 462</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US">then we are immediately reminded of our diagram of fifths (Fig. 463), in which we find the tone-values of the magic square (Fig. 462) drawn (in a thick line) starting from </span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">1</span><span lang="en-US" xml:lang="en-US">/</span><span style="vertical-align: sub;" lang="en-US" xml:lang="en-US">3</span><span lang="en-US" xml:lang="en-US"> </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f</span><span style="font-family: 'Symbol';" lang="en-US" xml:lang="en-US"></span><span lang="en-US" xml:lang="en-US">. Admittedly, in this fifth diagram, the ratio progression is limited at </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">h</span><span lang="en-US" xml:lang="en-US">, due to the 6 fifth intervals </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">f c g d a e h</span><span lang="en-US" xml:lang="en-US">.</span></p> <p style="text-align: center;" dir="ltr"><img src="/EZ/kayser2/kayser2/imagelibrary/Mathematics/6.png" alt="" /></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold; font-size: 10pt;" lang="en-US" xml:lang="en-US"> Figure 463</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Furthermore, the lines of the equal octave tone-values are different. But yet the ordering of the magic square appears undoubtedly to agree more closely with that of the fifth diagram, and therefore also with that of the “open P.” Mathematically oriented readers who are interested in this problem will certainly find an exciting field of operation on the basis of the above. In itself, the domain of the square (see Bibliography) is already interesting enough, and although it is still considered by many today to be a useless game, since no machines, bombs, or grenades can be manufactured </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">de facto</span><span lang="en-US" xml:lang="en-US"> with its problems, I still believe that any serious mathematician would dismiss such a superficial judgment. Otherwise he would have to think of all of mathematics as a “useless” or “profitable” game, against which admittedly nothing could be said.</span></p> <p style="text-align: justify;" dir="ltr"><span lang="en-US" xml:lang="en-US"> From the many historical writings about the magic square, one more is introduced here that is important for us. Around the year 1000, in Arabic cultural circles, there existed a strange Neoplatonist sect known as the “Brethren of Purity.” It was a secret society that was founded in Basra around the year 960, and they opposed strict Mohammedan faith with free philosophical thought. Within this order, an encyclopedia of 51 treatises was created, which arranged and summarized the knowledge of the times according to its elements. “The center of their thought is the study of the soul, which for them tends towards a doctrine of its development from the lower forms to human individuality, and its regeneration in the perfection of its origin. The worldview at the core of this speculation is the emanation doctrine of the Plotinians, with their universal Intellect and their all-pervading Universal Soul, whose formative activity is evident in nature and in the creation of humans, to which Pythagorean number-symbolism is added to make the emanation system more understandable through number ratios. Also various symbolisms, such as that of letters, tone-harmonies [!], etc., are used to demonstrate that all in the material and spiritual world is an image of the stepwise emanation from the world of pure spirituality” (J. Goldhizer, </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Allgemeine Geschichte der Philosophie</span><span lang="en-US" xml:lang="en-US">, Kultur der Gegenwart, Vol. I, 5, 1909, p. 53). Unfortunately, I have not yet been able to obtain a copy of the 8-volume </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Philosophie der Araber nach den Schriften der Lauteren Brüder</span><span lang="en-US" xml:lang="en-US"> (1858-1886) by F. Dieterici. I have only been able to look at the </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Naturanschauung und Naturphilosophie der Araber im 10. Jahrhundert nach den Schriften der Lauteren Brüder</span><span lang="en-US" xml:lang="en-US"> (Berlin 1861) by the same author, but it was not very fruitful with regard to our harmonic interests. Moritz Cantor, in his </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Vorlesungen über Geschichte der Mathematik</span><span lang="en-US" xml:lang="en-US"> (4</span><span style="vertical-align: super;" lang="en-US" xml:lang="en-US">th</span><span lang="en-US" xml:lang="en-US"> ed., 1922, p. 516), states that “magic squares play a mysterious role in the Arabic philosophical sect of the so-called Brethren of Purity, squares with 9, 16, 25, 36, 64, and 81 fields were especially familiar to them, therefore there must surely have been a method available with which to construct them.”</span></p></div></td></tr></tbody></table><br /> </span></p> <p style="text-align: left;" dir="ltr"></p> </div> </body> </html></div>
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