in Dissections: Plane & Fancy, by Greg N. Frederickson

- More on William Wallace
- Short Biography for Mr. Lowry
- Michael J. Cohn
- Tarski's work geometric dissections
- General bounds on numbers of pieces
- Hinging one of Hart's dissections
- Hinging two similar circumscribed polygons to two
- More on Harry Hart

William Wallace was an early champion of differential calculus in Great Britain and a translator of French mathematical works. See "Calculus and Analysis in Early 19th-Century Britain: The Work of William Wallace," by Alex D. D. Craik, inHistoria Mathematica 26(1999) pp. 239-267.

Our Mr. Lowry was John Lowry, a mathematician who was born at Cumberland in 1769. For some time he was an excise officer at Solihull, near Birmingham. In 1804 he was appointed an instructor of mathematics in the new military college at Great Marlow, which moved to Sandhurst in 1812. He remained in this post until failing eyesight forced him to resign on a pension in 1840.

Lowry was an early and frequent contributor to Leybourn'sMathematical Repository, which was published from 1799 to 1819. He authored an appendix on spherical trigonometry for the second volume of Dalby'sCourse of Mathematics, the textbook used at Sandhurst during the early nineteenth century. John Lowry died at Pimlico, London in 1850.

Paul Scott, at the University of Adelaide, tells me that M. J. Cohn is Michael J. Cohn. He came from Adelaide and earned an M.Sc. in math in 1971 at the University of Adelaide before moving to England to work on a Ph.D.

James T. Smith, a Professor Emeritus at San Francisco State University, has brought to my attention the early work of Alfred Tarski and a colleague Henryk Moese on bounding the minimum number of pieces required for various geometric dissections:Alfred Tarski, "O stopniu równoważności wielokątów,"Tarski defined theMłody matematyk1 (1931): 37-44. ("The degree of equivalence of polygons," translated by Izaak Wirszup, in Tarski and Moese 1952, 1-8.)

Henryk Moese, "Przyczynek do problemu A. Tarskiego: `O stopniu równoważności wielokątów,'"Parametr2 (1931-1932): 305-309. ("Contribution to A. Tarski's problem `On the degree of equivalence of polygons,'" translated by Izaak Wirszup, in Tarski and Moese 1952, 9-14.)

Alfred Tarski, "Uwagi o stopniu równoważności wielokątów,"Parametr2 (1931-1932): 310-314. ("Further remarks about the degree of equivalence of polygons," translated by Izaak Wirszup, in Tarski and Moese 1952, 15-20.)

Alfred Tarski and Henryk Moese,Concerning the Degree of Equivalence of Polygons,translated by Izaak Wirszup. Chicago: The College, University of Chicago, 1952.degree of equivalence of two equivalent polygons W and Vto be the least natural number n satisfying the condition: each of the polygons W and V can be divided into n polygons in such a way that the polygons obtained by the division of W are respectively congruent to the polygons obtained by the division of V.

Among other things, Tarski proved that if W and V are polygons with the same area, and W is convex, then a lower bound on the number of pieces in a dissection of W and V is δ(W)/δ(V), where δ(W) is the diameter of W. He used this to show that the minimum number of pieces to dissect a (3a×a/3)-rectangle to a square is precisely 3.

Considering the dissection of an (xa×a/x)-rectangle to a square, for any positive real number x, Tarsksi defined τ(x) to be the degree of equivalence. He claimed that:τ(x) ≤ ⌈(xMoese claimed that:^{2}-1)⌉+2 for any x ≥ 1.

τ(x) ≥ √ [(x^{4}+1)/(2x^{2})] for any x > 0.τ(x) ≤ ⌈x⌉+1 for any x ≥ 1.Tarski defined the "latitude" π of a polygon V to be the width of the narrowest strip that can cover V. The Tarski proved that if W is a parallelogram and V is a polygon of the same area, then the minimum number of pieces to dissect W into V is at least π(W)/π(V).

τ((n+1)/n) = τ(n/(n+1)) = 2 for any natural number n.

τ(n+1/k) ≤ n+1 where n and k are any natural numbers.

Evangelos Kranakis, Danny Krizanc, and Jorge Urrutia have bounded the minimum number of pieces from above and below for dissecting a regular polygon ofnsides to a square. They give the proofs in their paper, "Efficient Regular Polygon Dissections,"Discrete and Computational Geometry, Japanese Conference, JCDCG'98, Lecture Notes in Computer Science,vol. 1763, Tokai University, Tokyo, Japan, December 1998, pages 172-187. A revised version of their paper has appeared inGeometriae Dedicata80(2000), pp. 247-262.

Among other things, the authors prove that number of pieces needed to dissect an {n} to a square is at leastn/4. They also show that the number of pieces is at mostn/2 +o(n), whereo(n)is a function ofnsuch that the limit ofo(n)/nasngoes to infinity is zero. Their analysis is similar in spirit to that done by M. J. Cohn (1975).

I belatedly realized that the pieces in Hart's dissection of two similar circumscribed polygons to one can be hinged. In particular, the pieces on the left in Fig. 19.6 can be cyclicly hinged. This is not the only dissection in the book for which I have belatedly identified a hinging, but it is singular in that Hart's dissection of inscribed polygons cannot be so hinged and that the hinging reflects both the spirit and the hidden beauty of this dissection of irregular polygons. See my new bookHinged Dissections: Swinging & Twisting.

On page 226-27, I stated that either of Fig. 19.6 or Fig. 19.7 can be generalized to a two-into-two dissection. I gave an example for inscribed pentagons in Fig. 19.9, but did not give an example for circumscribed pentagons. I guess that I chose the wrong case to illustrate, because the two-into-two dissection of circumscribed polygons can be hinged. The idea follows from my observation in the paragraph above.

Harry Hart was instrumental in the introduction of a new mathematical syllabus at the Royal Military Academy in 1892. See "Mathematics in the Metropolis: A Survey of Victorian London," by Adrian Rice, inHistoria Mathematica 23(1996) pp. 376-417.

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