Preparing some exercises for my High School pupils I came across this question: How can you tile a square into three similar (ie., same shape, different size) rectangles? With a bit of algebra it can be easily shown that there is one non-trivial solution (I mean, apart from three equal stripes) involving the Plastic number (aka Padovan constant). It has to be a very old problem but I hadn't been able to find on the web any references or any real example (eg, in architecture) of this "Plastic proportion"... Any hint?
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$\begingroup$ I think that as far as architecture goes a good starting point would be en.wikipedia.org/wiki/Hans_van_der_Laan (but the most promising link on that site is in Dutch). $\endgroup$– Gerry MyersonNov 9, 2015 at 22:29
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$\begingroup$ Here's a citation for the Padovan essay on these matters: Richard Padovan, "Dom Hans Van Der Laan and the Plastic Number", pp. 181-193 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002. An abstract is at nexusjournal.com/the-nexus-conferences/nexus-2002/… $\endgroup$– Gerry MyersonNov 9, 2015 at 22:41
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3$\begingroup$ Not relevant to the question, but I cannot resist mentioning the following amazing result of Laczkovich and Szekeres, Discrete Comput. Geom. 13 (1995), 569-572. Let $x>0$. Then a square can be tiled with finitely many copies of rectangles similar to a $1\times x$ rectangle if and only if $x$ is an algebraic number all of whose conjugates have positive real part. For instance, $x=\sqrt{2}+\frac{17}{12}$ is o.k. but not $\sqrt{2}+\frac 43$. $\endgroup$– Richard StanleyNov 10, 2015 at 14:03
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$\begingroup$ What about another solution, as in the above shape in figure 7, but with R2 = R3? x = 2/3*AB and y = 1/2 AB. $\endgroup$– SamuelOct 10, 2016 at 21:21
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$\begingroup$ I think it's implicit in the question that no two rectangles are to be congruent. $\endgroup$– Gerry MyersonOct 10, 2016 at 22:07
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2 Answers
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(Above, $\psi$ is the plastic number.)
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Here is one reference:
de Spinadel, Vera W., and Antonia Redondo Buitrago. "Towards van der Laan’s Plastic Number in the Plane." Journal for Geometry and Graphics, 13.2 (2009): 163-175. (PDF download.)
(Above, $\psi$ is the plastic number.)
It may be that the problem derives from Martin Gardner:
M. Gardner, "Six challenging dissection tasks," Quantum, 4 (1994), pp. 26–27.
A Gardner’s Workout, A K Peters Ltd., Natick, MA, 2001, pp. 121-128.
But I cannot access either of these easily...
See Gerry Myerson's comment on the Gardner source.
answered Nov 9, 2015 at 20:48
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2$\begingroup$ The Gardner article is on pages 121-128 of the Gardner's Workout book. He attributes the question to "Karl Scherer, a computer scientist in Auckland, New Zealand." He gives the tiling, calling it his independent rediscovery of a solution Scherer had found years earlier. Gardner also points to Ian Stewart's Scientific American columns for June and November 1996. Also, he writes of the "plastic number", "This was the name given to it recently by Richard Padovan, an Italian architect who credited the number's discovery to a French architect in 1924." No citation is given. $\endgroup$ Nov 9, 2015 at 22:24
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2$\begingroup$ Wikipedia essay on Padovan says, "in Padovan's 1994 essay Dom. Hans van der Laan : Modern Primitive he attributed the sequence to Hans van der Laan." Ian Stewart's contribution can be found in his book, Math Hysteria, pages 85-93. He drops a few artistic names (Alan St. George, and Padovan), but gives no references. $\endgroup$ Nov 9, 2015 at 22:33
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I discovered and obtained the plastic-constant-related solution of the division of a square into three similar, mutually non-congruent rectangles independently, and possibly earlier than anyone else, circa 1987 in connection with my researches into the problem of rectangular tripartite similarity divisions of generalized rectangles. In May of 1996 I encountered the article entitled "Tales of a Neglected Number" on pages 102 and 103 of Ian Stewart's Mathematical Recreations column in the June 1996 issue of Scientific American Magazine. My reading of that article disabused me of the mistaken notion that the number (1.324717957) was something that I alone was familiar with and served to spur me to share my discovery and solution of the division of a square into three similar, mutually non-congruent rectangles as well as a number of other plastic-constant-related discoveries with Ian Stewart in a letter dated May 22, 1996 and sent to him in care of Scientific American Magazine. In the November 1996 issue of this same Scientific American Magazine, in the feedback section of Ian' Stewart's Mathematical Recreations column entitled a Guide to Computer Dating, on page 118, a published diagram of my result with the following acknowledgement can be found: John H. Bonnett, Jr. of Livingston, N.J., sent me a wealth of information, and I offer one example. If a square is divided into three similar (same shape, different size) rectangles, as in the figure, then the ratio of the two pieces along the vertical edge is the plastic number.
Unbeknownst to me at the time, the question of this tripartite division evidently had at least two, earlier 'in the print record' occurrences. It appeared in the Canadian Journal of Mathematics: Crux Mathematicorum, Volume 15, #7, September 1989, Problem No. 1350, pages 215 thru 218, posed by Peter Watson-Hurthig, Columbia College, Burnaby, British Columbia, in the following form: (a) Dissect an equilateral triangle into three polygons that are similar to each other but all of different sizes. (b) Do the same for a Square. (c) Can you do the same for any other regular polygon? (Allow yourself more than three pieces if necessary.) It was solved for (b) 'a square' in the case of similar rectangles by L. F. Myers, The Ohio State University and by Richard K. Guy, University of Calgary. Also, it evidently was posed (along with five other dissection tasks) by Karl Scherer some time prior to 1994 and disclosed by Martin Gardner (who evidently also discovered this dissection on his own) in his Mathematical Surprises column in the May/June 1994 issue of Quantum magazine in an article entitled "Six challenging dissection tasks". There it was discussed as correlated with the p^2 = 1.754877666 value (which Gardner proposed calling "high-phi") with no apparent appreciation of this number's relation to Dom Hans van der Laan's plastic constant (p = 1.324717957), now commonly referred to as "psi."
As far as I've been able to ascertain, the relationship of this tripartite division (in particular the ratio of the edge division of the square) to psi, the plastic constant (p = 1.324717957), as fundamental and not to high-phi (p^2 = 1.754877666) had not hither been noted, disclosed, or published in any form prior to Stewart's November 1996 Scientific American column publication of my disclosure to him of this fact and of my disclosure to him of the dissection itself and of my understanding of the logic of the associated p edge ratio plastic rectangle and its p^2 edge ratio gnomon (the very ratio of the tripartite dissected square's similar rectangles).
Gardner's July 18th 2001 publication of this dissection on page 124 of his Workout book was perhaps only the fourth time that this dissection had thus far appeared in print. I have only been able to find a handful of other, subsequent early instances of its appearance in print, including: 1) a paper by Paul Yiu, Department of Mathematics, Florida Atlantic University, Summer 2003, Chapters 1–44, Version 031209 entitled Recreational Mathematics 2003, Project: Cutting a Square into three similar parts, p. 317. 2) a paper by Federico Ardila and Richard P. Stanley (circa 2004) entitled Tilings, page 10. 3) The paper by de Spinadel, Vera W., and Antonia Redondo Buitrago. "Towards van der Laan’s Plastic Number in the Plane." Journal for Geometry and Graphics, 13.2 (2009).
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$\begingroup$ It's a shame Stewart didn't include this in the Tales of a Neglected Number chapter of Math Hysteria, published in 2004. $\endgroup$ Nov 20, 2017 at 5:22
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$\begingroup$ Gerry, for obvious reasons I share your lament that the tripartite, similar, non-congruent rectangle division of a square was not included in the Tales of a Neglected Number Chapter of Stewart's Math Hysteria book. This particular tripartite rectangle dissection of a square is a rather unique little gem that deserves greater visibility and exposure than it has thus far gotten. $\endgroup$ Nov 21, 2017 at 2:08
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2$\begingroup$ The Crux Mathematicorum issue is freely available at cms.math.ca/crux/backfile/Crux_v15n07_Sep.pdf – I note that in addition to the three-rectangle solution, another way of dissecting a square into three similar, noncongruent quadrilaterals is given there. The quadrilaterals are trapezoids, and the side ratios are 1 to 2 to 3, so no irrationals are needed. $\endgroup$ Nov 23, 2017 at 11:32
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$\begingroup$ Curiously, but perhaps not too surprisingly, this same, non-unique result (It can be accomplished in an infinite number of different ways.) with these same 1 to 2 to 3 trapezoid edge ratios and with the figure in exactly the same orientation (eight different orientations were possible) was employed by Gardner (attributing the solution to Scherer) in his Mathematical Surprises column in the May/June 1994 issue of Quantum magazine. $\endgroup$ Nov 26, 2017 at 2:34