Updates to Chapter 19, "When Polygons Aren't Regular",
in Dissections: Plane & Fancy, by Greg N. Frederickson

More on William Wallace

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, in Historia Mathematica 26 (1999) pp. 239-267.

Short Biography for Mr. Lowry

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

Michael J. Cohn

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.

Tarski's work geometric dissections

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," Młody matematyk 1 (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,'" Parametr 2 (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," Parametr 2 (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.
Tarski defined the degree of equivalence of two equivalent polygons W and V to 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) ≤ ⌈(x2-1)⌉+2 for any x ≥ 1.
τ(x) ≥ √ [(x4+1)/(2x2)] for any x > 0.
Moese claimed that:
τ(x) ≤ ⌈x⌉+1 for any x ≥ 1.
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.
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).

General bounds on numbers of pieces

Evangelos Kranakis, Danny Krizanc, and Jorge Urrutia have bounded the minimum number of pieces from above and below for dissecting a regular polygon of n sides 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 in Geometriae Dedicata 80 (2000), pp. 247-262.
Among other things, the authors prove that number of pieces needed to dissect an  {n}  to a square is at least  n/4.  They also show that the number of pieces is at most  n/2 + o(n),  where  o(n)  is a function of  n  such that the limit of  o(n)/n  as  n  goes to infinity is zero. Their analysis is similar in spirit to that done by M. J. Cohn (1975).

Hinging one of Hart's dissections

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 book Hinged Dissections: Swinging & Twisting.

Hinging two similar circumscribed polygons to two

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.

More on Harry Hart

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, in Historia Mathematica 23 (1996) pp. 376-417.


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Last updated June 17, 2010.