Book reviews for Hinged Dissections: Swinging & Twisting

Review in Cubism For Fun

Rik van Grol wrote a short review in the November 2002 issue (#59) of Cubism For Fun, a newsletter in English published by the Nederlandse Kubus Club NKC (Dutch Cubists Club). Follow this link to the homepage of the newsletter. Excerpting from the review, which appears on page 32:
"In 23 chapters (over 250 pages) Frederickson leads us through an admittedly fascinating world of swing-hinged and twist-hinged dissections. A hinged dissection is a dissection in which the pieces are linked together and which can transform from one figure to another. In swing-hinged dissections the hinges allow the pieces to swing from one position to another, see Figure 1. In twist-hinged dissections the pieces can be turned around at the hinge point, see Figure 2. In 3D there are also piano-hinges."

Review in SciTech Book News

A short review appeared in the December 1, 2002 issue of SciTech Book News, published by Book News, Inc., of Portland, Oregon. Excerpting from the review:
"For anyone who has had a course in high school geometry and thought that regular hexagons were rather pretty, Frederickson (computer science, Purdue U.) explores how to cut a geometric figure such that rotating the pieces at their intersections can form another figure."

Review for NRK Radio

Tormod Guldvog wrote a very nice review (in Norwegian) that appeared on the website for the Norwegian Radio NRK on December 5, 2002. Follow this link to the homepage for the radio. The review appeared in the column "Verdt å vite" ("Worth to know"), and can be accessed from this link. Geir Ellingsrud, a professor in mathematics at the University of Oslo, was a guest on the radio program on December 5 and talked about the book. He was impressed with the book and recommended it to people who enjoy recreational puzzles etc, and also to teachers at all levels for use in the classroom.
Excerpting from the review:
"Hvordan kan en trekant bli til en firkant? Enkelt. Klipp trekanten i fire deler, riktignok etter en spesiell oppskrift, så kan du legge bitene slik at de danner en firkant - med nøyaktig samme areal. Utfordringen blir derimot ikke enklere dersom det settes som begrensning at bitene ikke skal tas fra hverandre, men roteres på en måte som gjør at minst et hjørne eller en side av hver bit fortsatt skal henge sammen med nabobiten."
Translation (with the assistance of Eirik Herskedal throughout): "How can a triangle become a square? Simple. Cut the triangle into four parts, following a special recipe, then you can lay the pieces so that they form a square - with precisely the same area. The challenge does not become easier under the limitation that the pieces shall not be separated from each other, but rotated in a manner that makes at least one corner or one side of each piece still hang together with neighboring pieces."
"Forvirret? Boka "Hinged Dissections: Swinging & Twisting" er full av nettopp sånne puslespill. Eksemplet med trekanten som blir firkant er nærmest for amatørmessig å regne når du har lært å løse oppgavene i boka. For her er det mangekanter i utalliger varianter, og gjerne et titalls biter, som på finurlig vis danner de flotteste rosetter og stjerner. Et av de mer ekstreme eksemplene er en femkant som er dannet av 27 mindre femkanter pluss åtte femkantede stjerner."
Translation: "Confused? The book "Hinged Dissections: Swinging & Twisting" is full of exactly such puzzles. The example with a triangle becoming a square is almost trivial after you have learned to solve the problems in the book. Because they include polygons in innumerable variations, also dozens of pieces that in an amazing way become the most extravagant rosettes and stars. One of the more extreme examples is a pentagon formed out of 27 smaller pentagons plus eight five-pointed stars."
"Dette er en bok som tar temaet på alvor, noe som spesielt vises i de utallige illustrasjonene. Her formerlig danser polygonene på hver eneste side. Forfatteren tar for seg de ulike måtene å lage puslespillene på, og hvilke formler og metoder som brukes for å få løst oppgavene. Frederickson er flink til å trekke på historiske ressurser, og det er en svært god bibliografi bak i boka, for dem som vil lære mer om dette fenomenet."
Translation: "This is a book that takes its subject seriously, as shown by those countless illustrations. Here the polygons literally dance on every single page. The author deals with the different ways to create puzzles and which formulas and methods are used to solve these problems. Frederickson is clever to draw on historical resources, and he has a very good bibliography in the back of the book, for those who want to learn more about these phenomena."
"For deg som har vært borte i dette før, er dette en svært godt gjennomført bok som vil gi deg mange timer med både lærerikt tankespinn og frustrerende oppgaveløsning."
Translation: "For those of you who have some experience in this area, this is a very good, thorough book that will give you many hours of both instructive mental exercise and frustrating problem solving."

Review in American Scientist

Mike Eisenberg, in the Computer Science department at the University of Colorado, Boulder, wrote a very nice review, which appeared in the May-June, 2003, issue (vol. 91, no. 3) of American Scientist, the magazine of Sigma Xi, The Scientific Research Society. Excerpting from the review, entitled "King of Swing" which appears on pages 269-270:
"Greg Frederickson's Hinged Dissections is emphatically a book for the geometers. It's an extended meditation on the fun of playing with pictures."
"Many of the examples in the book dissect one large shape into smaller ones by using separate "chunks" of hinged pieces, but the most jaw-dropping examples are those in which a single collection of hinged pieces can be rearranged into two remarkably different forms."
"In a succession of increasingly startling chapters, Frederickson describes hinged dissections created both by himself and by others. Since dissection fans paid little attention to "hingeability" until a few years ago, most of the examples are recent; in some cases, "classic" dissections also happen to be hingeable. (Frederickson shows two simple but lovely dissections of this sort created by Leonardo da Vinci.) Like a magician pulling a succession of stranger and wilder animals out of his hat, the author shows dissections from multiple polygons (squares, triangles, hexagons, octagons) to one; dissections involving crosses (Maltese, Latin, Greek); and dissections involving curved forms, stars and (in an extended aside) polyominoes. In the final chapters of the book, we encounter three-dimensional hinged dissections (one beautiful but very straightforward example rearranges two truncated octahedra into a cube), and two-dimensional dissections with more powerful hinges that allow pieces to "flip around" out of the plane, either around a vertex (flip hinges) or around the midpoint of an edge (twist hinges). The book is truly a feast, not so much for the eye as for the "inner eye": The real fun is in staring at the diagrams and imagining the transition between shape A and shape B via the "spread out" hinged forms that Frederickson depicts. The diagrams are clear and (as far as this reader can discern) accurate and beautifully edited."
"I would put Frederickson's dissection books, along with Alan Holden's Shapes, Space, and Symmetry (Columbia University Press, 1971), on the shortlist of beautiful, inspirational and accessible works in recreational geometry. I heartily recommend Hinged Dissections to anyone who loves geometry, or wants to."

Review in MAA Reviews

Mihaela Poplicher, a mathematics professor at the University of Cincinnati, wrote a very nice review for MAA Reviews. It first appeared May 2003.
Some excerpts from the review:
"the author includes detailed explanations of very ingenious new techniques, as well as puzzles (with solutions!) The book is a lot of fun to read, and the lively text includes hinged dissections for polygons (triangles, squares, stars, crosses, etc), as well as curved and three-dimensional figures. Since hinged dissections refer to the cutting of a geometric figure into hinged pieces that can be rearranged to form another figure, it is obvious that the book needed to include many illustrations. Of these, the author includes a wealth, and all of them are very good!"
"hinged dissections are not only fun and entertaining, but also useful outside mathematics --- one more reason for everybody interested in finding entertaining and challenging mathematical puzzles, from high school students to mathematicians, to read Frederickson's book."

Review in Mathematics Teaching

Paul Gailiunas, a chemistry teacher at Gosforth High School in Newcastle, England, wrote a review in the June 2003 issue (#183) of Mathematics Teaching, a quarterly journal published by the Association of Teachers of Mathematics, in England. Excerpting from the review, which appears on page 43:
"As in his earlier book, Frederickson has taken great care to credit solutions accurately, and place them in a historical perspective. In particular, the detailed presentation of techniques for solving these problems is interspersed with a series of short chapters reviewing the contribution of Henry Dudeney ..."
"Certainly any student who was motivated to work through even a few of the technical sections would begin to develop considerable geometric insight. It is very well illustrated, and many of the dissections have an immediate visual appeal ..."

Review in Choice

David Feldman, a professor of mathematics at the University of New Hampshire, wrote a very nice, short review in the June 2003 issue (vol. 40, no. 10) of CHOICE: Current Reviews for Academic Libraries, a monthly periodical published by the the Association of College & Research Libraries, which is a division of the American Library Association. Excerpting from the review (40-5849), which appears on pages 1781-1782:
"Sometimes recreational mathematics can seem merely frivolous, as when it involves entirely unnatural manipulations with numbers, but recreational geometry has a solid appeal. Dissection problems--which involve cutting up one figure and then reassembling the pieces to make another--even attract the attention of research mathematicians (e.g., Hilbert's second problem)."
"This book, like the author's first (Dissections: Plane and Fancy, 1998), displays the sort of mind-boggling ingenuity that will intrigue anyone interested in the limits of human imagination.... Highly recommended."

Review in Mathematical Reviews

Dr. Roswitha Blind, in Stuttgart, Germany, wrote a review that appeared in Mathematical Reviews, a publication of the American Mathematical Society. (Follow this link to the Mathematical Reviews on the web.) Excerpting from review 2003h:52018, on page 6135 of the August 2003 issue:
"This book is a fine contribution to recreational mathematics."
"The book is beautifully illustrated and easy to read---only a basic knowledge of high-school geometry is needed."

Review in BMS-NCM News

Adhemar Bultheel, a professor in the Department of Computer Science at the Katholieke Universiteit Leuven, wrote a nice review that appeared in the September 15, 2003 (no. 44) issue of the BMS-NCM News, the newsletter of the Belgian Mathematical Society and the National Committee for Mathematics. This newsletter is published five times a year by the Belgian Mathematical Society. The review covers my first two books and appears on pages 10-11, as well as online. An excerpt from the review:

"These books give many new perspectives for all those who love math puzzles. They will adore the second book as much as they loved the first, and they will devour it with great enthusiasm. The style is lively and pleasant to read. Practically no mathematical prerequisites are needed, so that everybody can be intrigued by this fascinating play of fancy pieces swinging and twisting around."

Review in Mathematics Teacher

Beth M. Schlesinger, a mathematics teacher in the Torah High Schools, San Diego, California, wrote a review that appeared in Mathematics Teacher, a periodical of the National Council of Teachers of Mathematics. Excerpting from the review, which appears on page 523 of the October 2003 issue (vol. 96, issue 7):
"The beautifully illustrated book includes problems, along with drawings of solutions. Some of the history of these dissections is interspersed with the mathematics, and the book includes a comprehensive bibliography."
"This book is not for the casual high school reader, because it requires sophisticated thought, reading ability, and visualization. However, it can be used as a source of enrichment for students and teachers or as inspiration for science fair or mathematics projects. Hinged Dissections gives a clear understanding of the complex and elegant games that mathematicians play and is a source of challenging and imaginative puzzles."

Review in American Mathematical Monthly

Judith N. Cederberg, in the math department at St. Olaf College, wrote a telegraphic review that appeared in the November 2003 (vol. 110, no. 9) issue of the American Mathematical Monthly, a publication of the Mathematical Association of America. Quoting from the review, which appears on page 866:
"A fascinating collection of figure dissections into hinged pieces that swing to form a second figure. Well illustrated, with careful explanations and interesting annotations."

Review in Mathematics in School

John Sykes, at Sedbergh School, in Sedbergh, Cumbria, England, wrote a review that appeared in the November 2003 (volume 32, number 5) issue of Mathematics in School, a periodical published by the Mathematical Association of Great Britain. (Follow this link to the homepage for the periodical.) Excerpting from the review, which appears on page 46:
"Frederickson is much more ambitious. ... There are swing hinges and triple hinges, piano hinges and socket hinges, Q steps and Q swings. One of the earlier chapters shows how the dissections and the hinges can be obtained from the superposition of tessellations of the object and image shapes. Quite fascinating."
"Some of Frederickson's book is very readable and the generous scattering of excruciating puns help.... if you want to know how to find hinged dissections to convert squares into crosses (whether they be Greek, Lorraine or Latin) or dodecagons to hexagons or hexagrams to triangles or four octagons to one or ... or ... or ... then this must be the definitive source of information."

Review in Elemente der Mathematik

Hans Walser, a Lecturer in the Mathematical Institute at the University of Basel, Switzerland, wrote a review that appeared in the February 2004 (vol. 59, no. 1) issue of Elemente der Mathematik, a publication of the Swiss Academy of Sciences. Quoting from the review, which appears on page 44:
"Das Buch enthält eine Fülle, fast eine Überfülle von Beispielen... Der Autor gibt theoretische Hinweise zur Konstruktion solcher Zerlegungen - sehr hilfreich sind dabei Überlagerungen geeigneter Rasterungen und Parkettierungen - wie auch einige Praktische Tips zur Modellherstellung."
Translation (with the assistance of Susanne Hambrusch): "The book contains an abundance, almost an overabundance of examples... The author gives theoretical hints for the construction of such dissections - the accompanying overlays of suitable tessellations and tilings are very helpful - as well some practical tips for producing models."
"Das Buch ist geeignet als Unterlage für geometrische Unterrichtssequenzen in formaler Hinsicht wie auch mit Blick auf `Geometrie zum Anfassen' auf allen Schulstufen, und nicht zuletzt für virtuelle Modelle mit dynamischer Geometrie Software."
Translation: "The book is suitable as a resource for a sequence of formal geometric studies in school as well as for `hands-on geometry' at all school levels, and last but not least for virtual models with dynamic geometry software."

Review in The Mathematical Gazette

A review by Peter Ruane, who was senior lecturer in mathematics education at Anglia Polytechnic University, England, appeared in the March 2004 issue (volume 88, no. 511) of The Mathematical Gazette, a publication of the Mathematical Association of Great Britain. (Follow this link to the homepage for the periodical.) Excerpting from the review, which appears on pages 183-184:
"The author says that this text can be read by anyone with a high-school maths background, but I suggest that it could be used at an earlier stage than that. For example, primary school teachers could use some of the material as the basis of suitable investigative activities..."
"this book is highly recommended. It is extremely readable, attractively presented and there is hardly a page devoid of eye-catching illustrations. The enthusiasm of the author for his subject is evident throughout..."

Review in Mathematical Spectrum

Will Donovan, a fourth-year physics student at Queens' College, Cambridge, wrote a nice review in the May 2004 issue (volume 36, number 3) of the Mathematical Spectrum, a magazine for students and teachers in schools, colleges, and universities, and published by the Applied Probability Trust of Great Britain. (Follow this link to the homepage for the periodical.) Excerpting from the review, which appears on page 70:
"The techniques are fascinating due to their surprising use of symmetry and tessellations. The sheer weight of examples means that this is a book aimed at the enthusiast; there is an enormous amount of material here, and it requires some passion to wade through it."
"Frederickson's book is full of gorgeous results, described in a lively style; it will reward serious study, and provide a massive store of puzzles."

Review in Optische Fenomenen

A nice review in the May 2004 issue (number 203) of Optische Fenomenen, (Optical Phenomena) the newsletter of the Nederlandse Stichting voor Waarneming & Holografie (Dutch Foundation for Perception and Holography). (Follow this link to the homepage for the periodical.) Excerpting from the review, which appears on page 4, which is entitled "Vlakken worden vlakverdelingen" (Areas become area dissections):
"Na het uitkomen van zijn eerste boek "Dissections: Plane & Fancy" (1997) kreeg samensteller, auteur en wiskundige Greg N. Frederickson zoveel lof en vragen naar een tweede boek met nog meer grappen en grollen met vlakken en vlakverdelingen, dat hij zijn tweede boek met veel plezier presenteerde." De inhoud geeft weer vele nieuwe vindingen en inspiraties voor eigen experimenten."
Rough translation: "After the publication of his first book Dissections: Plane & Fancy (1997), compiler, author, and mathematician Greg N. Frederickson got so much praise and questions about a second book about dissections with still more jokes, that with much pleasure he presented a second book. The contents reflect many new findings and inspiration from his own experiments."
"Een echt puzzelboek voor ontwerpers, wiskundigen en ieder ander die zich wil laten inspireren tot het tekenen van vlakken en vlakverdelingen."
Rough translation: "A genuine puzzle book for designers, mathematicians and anyone else who wants to be inspired to draw areas and dissections."

Review in Internationale Mathematische Nachrichten

Peter Schmitt, an ausserordentliche professor in the Department of Mathematics at the University of Vienna, in Vienna, Austria, wrote a nice review in the August 2005 issue (number 199) of the Internationale Mathematische Nachrichten, or International Mathematical News, a newsletter published by the Österreichische Mathematische Gesellschaft (Austrian Mathematical Society). (Follow this link to the homepage for the periodical.) Excerpting from the review, which appears on page 43:
"This book can be read as a research monograph in discrete geometry (it both presents much new material and is the first systematic treatment of a subject which, up to now, could be found only in a few scattered sources), but also as a leisure-time book for everybody interested in geometrical puzzles and patterns, in ingenious and surprising constructions, that is, for all who love the beauty and diversity of geometrical objects."

Review in Bulletin de l'Association Mathématique du Québec

Robert Bilinski, at the Collège Montmorency in Laval, Québec, Canada, wrote a very nice review in the December 2005 issue (volume 45, number 4) of the Bulletin de l'Association Mathématique du Québec, published by the Association Mathématique du Québec. (Follow this link to the homepage for the periodical.) Excerpting from the review (of both Dissections: Plane & Fancy and Hinged Dissections), which appears on pages 63-65:
"Je vais aussi être franc, je n'ai pas fini de lire les deux livres avant de faire la chronique. Mais, il est clair qu'ils ont été bien écrits, de manière consciencieuse et minutieuse, avec passion et surtout avec rigueur. De plus, je ne pense pas me tromper en parlant de la fascination et de l'émerveillement que ce livre a suscités chez moi et les autres (non-mathématiciens) à qui j'ai montré les livres. De tout manière, je n'ai aucune honte à le dire, parce que ce livre est à mon chevet depuis quelques mois de manière « active ». Au moins une fois par semaine, j'ouvre au hasard un des deux livres et je découvre quelque chose de nouveau, de beau et de fascinant. Ces livres sont pleins de constructions abracadabrantes."
Approximate translation (any additional assistance would be greatly appreciated!): "I will also be honest, I did not finish reading the two books before writing this column. But it is clear that they have been well written, in a conscientious and meticulous manner, with passion and especially with rigor. Moreover, I do not think I am mistaken while speaking about the fascination and amazement that this book caused in me and others (nonmathematicians) to whom I have shown the books. In any case, I am not ashamed to state this because this book has been by my bedside for a few months in an `active' way. At least once a week, I randomly open one of the two books and discover something new, beautiful and fascinating. These books are full of extravagant constructions."
"Voilà une lecture de longue haleine ou un livre à mettre sur la table de son salon pour faire la conversation à la manière d'un livre d'art. Bonne lecture!"
Approximate translation (with the assistance of Luc Mongeau): "Here is a seminal book, to keep on the coffee table to stimulate conversations, like an art book. Good Reading!"

Review in Nieuw Archief voor Wiskunde

Helena Verrill, an assistant professor of mathematics at Louisiana State University, in Baton Rouge, LA, wrote a nice review in the March 2007 issue (fifth series, volume 8, number 1) of the Nieuw Archief voor Wiskunde, the magazine for the members of the Koninklijk Wiskundig Genootschap, the Royal Dutch Mathematical Association. (Follow this link to the homepage for the periodical.) Excerpting from the review, which appears on page 63:
"Anyone interested in recreational mathematics, especially geometric puzzles, will enjoy this book."
"The book is richly illustrated, with several pictures on almost every page. There are a good number of puzzles throughout the text, with solutions at the end. Also many open problems are given."
"Physical models of hinged dissections can easily be constructed with paper, thread and tape, and teachers will be able to use these captivating puzzles to help their students understand and appreciate basic geometry."

Review in tangente: l'aventura mathématique

Michel Criton wrote a very nice review in the co-authored (with Élisabeth Busser) "notes de lecture" section entitled "Découpages: la trilogie de Greg Frederickson," of the French mathematics magazine tangente: l'aventura mathématique, Hors série no. 64 (Septembre 2017), p. 19. He and Élisabeth Busser identified my three books, Dissections: Plane and Fancy, Hinged Dissections: Swinging and Twisting, and Piano-Hinged Dissections: Time to Fold! as a trilogy! The subsection on Hinged Dissections: Swinging and Twisting was entitled "La bible sur les découpages articulés" which translates to "The bible on hinged dissections". Excerpting from his review:

"Le livre Dissections: Plane and Fancy par Greg Frederickson semblait être un aboutissement, une référence pour les vingt ans à venir. Peu après sa publication, l'auteur s'est pourtant intéressé aux découpages articulés, dans lequels toutes les pièces du puzzle peuvant être attachées les unes aux autres par des charnières qu'il suffit de faire jouer dans un sens ou dans l'autre pour obtenir les deux dispositions du jeu. Il pensait alors en faire un chapitre supplémentaire lors d'une réédition ultérieure de son livre. Mais plus il a creusé la question, plus les découvertes se sont faites nombreuses. Frederickson a commencé par reprendre tous les découpages connus et par se poser la question: est-il possible de relier toutes les pièces par des charnières de façon que le puzzle fonctionne? Contre toute attente, un nombre important de découpages connus étaient "articulables", c'est-à-dire que l'on pouvait effectivement les rendre articulés sans modification. Pour d'autres, il fallait adapter le découpage, au prix souvent d'une augmentation du nombre de pièces minimales... ce qui ouvre tout un champ de nouveaux records à battre! Bien qu'aucun théorème n'ait été démontré à ce sujet (qui semble difficile), Frederickson conjecture qu'étant donné deux polygones de même aire, il existe toujours un découpage articulé permettant de passer de l'un à l'autre. Il réunit finalement tant de découpages articulés qu'il fait paraître en 2002 un ouvrage qui leur est entièrement consacré. Il explore également des découpages de solides articulés (une bonne vision spatiale est recommandée) ainsi que les pavages du plan articulés."
(which is translated as:)
"The book Dissections: Plane and Fancy by Greg Frederickson seemed to become a reference for the next twenty years. Shortly after its publication, however, the author became interested in hinged dissections, in which all pieces of the puzzle could be attached to each other by hinges so that just swinging them one way or the other would give each of the two configurations of the puzzle. He thought then to make an additional chapter during a later edition of his book. But the more he examined the question, the more discoveries he made. Frederickson began by taking all the known dissections and asking himself the question: Is it possible to connect all the pieces of each by hinges so that the puzzle works? Unexpectedly, a large number of known dissections were "hingeable", that is to say that they could actually be hinged without modification. For others, it was necessary to adapt the dissection, often at the price of an increase in the number of pieces ... which opens up a whole new field of new records to beat! Although no theorem has been shown on this subject (which seems difficult), Frederickson conjectured that, given two polygons of the same area, there is always a hinged dissection allowing to move from one to the other. It finally brings together so many hinged dissections that in 2002 his book Hinged Dissections: Swinging and Twisting appeared which was entirely devoted to them. It also explored the dissection of solid objects (a good spatial vision is recommended) as well as tilings of the hinged plane."
(actually, such a theorem was proved afterwards and can be found in:)
Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott Duke Kominers, "Hinged dissections exist," Discrete & Computational Geometry, vol. 47, no. 1 (2012), pp. 150-186.

Last updated October 26, 2017.