By Lee C.F. Sallows

Conventional magic squares are squares of numbers within which the rows, columns, and diagonals all upload as much as a similar overall. With a voluminous literature going again a few 2,500 years, the common assumption has ever been that magic squares are inherently arithmetical gadgets. during this leading edge paintings by way of a British engineer, the writer initiates a Copernican revolution in our knowing through exchanging numbers with two-dimensional varieties. the result's no longer in simple terms a unique form of geometrical magic sq. yet a revelation that conventional magic squares are actually higher obvious because the one-dimensional example of this self-same geometrical activity.

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Fig. eleven. five A geo-Eulerian sq.. Fig. eleven. 6 A Eulerian template. ahead of going extra, there's a sophisticated element concerned right here that merits exam. As we've seen, determine eleven. 6 is work eleven. 1 with d = –c. might we now not pass a step extra within the comparable path by way of making a additional template within which b is decided equivalent to -a, besides? the end result may then appear to supply upward thrust to a unique form of geo-Eulerian sq.. the answer's definite. in truth, we already did so. it's determine nine. 2 on web page 25, which was once the detrivialized Latin sq. forming the template for determine nine. three. This increases a good contrast: even if we want to regard determine nine. three as a detrivialized geo-latin sq. (as I do), or as a real geo-eulerian sq. (which I don’t). in spite of the fact that, looking back we see that the addition of 1 (appropriate) latin sq. to a different is in truth a technique of detrivializing them either instantaneously, in order that either interpretations can be considered as valid. the problem is hence might be educational, yet I judged it worthy bringing to the reader’s realization. as a minimum, there's a extra cause of taking a more in-depth examine determine eleven. five, which might be unremarkable have been in now not for the truth that the keys changing a and b may be of virtually any form. determine eleven. eight, or Indian Reservation, illustrates this estate via combining shapes for a and b that perfectly envelope the rest oblong a part of the objective after which abut one another on the way to whole a wigwam, or equilateral triangle. A is hence the small oblong region instantly less than the vertical fit in the (red) most sensible left hand piece, the rest of the piece (i. e. , the (red) left- and right-hand decrease halves of the objective triangle) being the most important a. within the piece to correct of it is a comparable (green) rectangle of size above the slot, appended to that is the (green) most sensible 1/2 the triangle goal, or key b. Key/keyhole c is a really small triangle (pointed tip) appended to the blue peak – three rectangle within the 3rd most sensible row cellphone, and excised from the yellow height-4 rectangle within the fourth best row mobilephone (incised tail). basically, the triangular form of the objective is a completely arbitrary selection which may be substituted for via any wanted replacement. the internal rectangle might also be tucked away in a miles nook as a small, visually insignificant component to a much more problematic objective. I worry that this tortuous description will do little to inspire enthusiasm for Graeco-Latin established geomagics. Given the prominence of Graeco-Latins within the magic sq. literature despite the fact that, a quick touch upon the issues and possibilities they current appeared a suitable subject to be raised during this account, for all its tedious personality. Fig. eleven. 7 ‘Merlin’s Magic’ The sword at the protect is, after all, ‘Excalibur’. Fig eleven. eight ‘Indian Reservation,’ one other geo-Eulerian sq.. 12 Uniform sq. Substrates we have now checked out a couple of examples of 4×4 geomagics in accordance with Latin squares. the method of building used to be a similar in each one case: from an preliminary Latin sq. we produce a geo-Latin substrate to which keys might be extra and/ or keyholes excised to be able to yield a non-trivial geomagic sq..

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