Message: 1 Date: Mon, 05 Mar 2012 20:08:32 -0500 From: David Wilson <davidwwilson@comcast.net> To: Dan Asimov <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Fwd: Re simple perfect squared tori Message-ID: <4F556390.5060208@comcast.net> Content-Type: text/plain; charset=ISO-8859-1; format=flowed

The 70x70 square cannot be tiled with squares 1x1 through 24x24, but maybe the torus?

On 3/4/2012 9:35 PM, Dan Asimov wrote:

...

You're saying this is the smallest solution you know of?

<< That was 24 squares and a side of 181, that's memory for you

Possibly, thanks to Eric Friedman and Guenter Stertenbrink we have these http://www.squaring.net/140x70-2x24squares.png http://www.squaring.net/140x140-4x24squares.jpg so perhaps a torus with 2,3 or 4 squares of each size is achievable I'm looking into using the electrical network method with Thom Sulankes surftri program http://hep.physics.indiana.edu/~tsulanke/graphs/surftri/ to see what else is possible with other surfaces. Squared cylinders and tori were suggested by Brooks et al. (1940, p.339). Michael Goldberg said "The squaring of a torus, however, is accomplished only by a distortion of the area since a torus is not developable." (M. Goldberg, The Squaring of Developable Surfaces, Scripta Mathematica (1952) vol.18 pp. 17-24). David Gale's book 'Tracking the Automatic Ant' has a chapter Tiling a Torus: Cutting a Cake has an intersting discussion of this topic Tiling of Surfaces by Unequal Squares "The question is, or rather was, which rectangles can be tiled by squares no two of which have the same size .... (discussion of squared rectangles and examples of 33x32 and 69x61) ... Now such a "squaring" of a rectangle can be converted in a trivial way to a squaring of the cylinder, torus, Mobius strip, or Klein bottle by the usual identification of opposite sides. But there are also nontrvial squarings of these other surfaces, even simple squarings, meaning squarings in which there is no subset of tiles whose union is a rectangle. Until recently, however, it was not known whether there might be squarings of these surfaces requiring fewer than 9 squares. Then in 1991, Bracewell found a squaring of the Mobius strip using only 8 squares. Very recently the question has been completely settled by S.J. Chapman. Perhaps the simplest but most surprising result is that a 1 x 5 Mobius strip can be tiled by 2 squares as becomes obvious from Figure 9.3. To accommodate this example, one must extend the notion of tiling to allow the mapping of the squares into the surface to self-intersect on their boundaries. Chapman shows that there are no 3- or 4-squarings of the strip but that there is a unique 5-squaring (Fig 9.4) see http://mathworld.wolfram.com/MoebiusStripDissection.html For cylinders, the situation is interesting. Again it turns out that 9 squares are necessary. There are exactly two nontrivial 9-squarings of the cylinder, and these use exactly the squares of Figures 9.1 and 9.2 but in a different arrangement. The tiling corresponding to Figure 9.1 is shown in Figure 9.5. Note for example that, that the squares of size 10 and 15 are disjoint in the rectangle, but they are contiguous on the cylinder. The cases considered so far involve surfaces with a boundary, which forces one to orient the squares with one side parallel to the boundary. This is no longer the case with the torus and Klein bottle. If one allows arbitrary orientation, then, in fact, any two squares can tile some torus. Namely, let the squares have sides a and b, and consider the torus obtained from identifying opposite sides of a square side c, where c^2 = a^2 + b^2. Figure 9.6 shows how to do it and at the same time provides a new (?) proof of the Pythagorean Theorem. A more symmetrical representation is given in Figure 9.7. or here http://mathworld.wolfram.com/TorusDissection.html If one allows only tiles that are parallel to the sides of the square, then it turns out there are no nontrivial 9-squarings of the the torus. Any squaring of the Mobius strip gives a squaring of the Klein bottle. For 6 or fewer tiles these are the only ones, but in the case of 7 or 8 tiles, this is not known. Also it is not known whether there are tilings of the Klein bottle in which the tiles need not be parallel to the sides of the big square. Chapman's techniques for these problems are quite different from and simpler than those of Tutte et al, and depend on a clever encoding of tilings by matrices of 0s, 1s, and -1s." scans of the figures in the chapter; http://www.squaring.net/IMG_0211.JPG http://www.squaring.net/IMG_0212.JPG Geoffrey Morley wrote to me recently; Join opposite longer sides of rectangle 12x26 (11,1x7)(1x8)(7x8,4)(5)(6x7,1)(6), after half-twist, to create a simple perfect squared Möbius band (in imagination, not 3D space!) of order 7 with boundary square of side 6. As with SPSRs, side 5 boundary square may be possible but example hard to find. In my alternative notation each square appears twice: (9,7)(5,2)(3,5,3)(2,9)(7). It is easy to see that a square of side 8 can be added to give a 6 square solution (from an 8x29 rectangle): (9,7)(5,2)(3,8)(8)(5,3)(2,9)(7). These squares can be rearranged to give another solution: (9,7)(5,2)(8,3)(8)(3,5)(2,9)(7). Both have 7 on the boundary. The 7-square solution with 6 on the boundary is (11,7,6)(1,5)(8)(4,8,4)(5,11)(7,1)(6). A square of side 12 can be added to give an 8-square solution (from a 12x38 rectangle): (11,7,6)(1,5)(8)(4,12)(12)(8,4)(5,11)(7,1)(6), also with 6 on the boundary. Stuart