Book reviews for Piano-Hinged Dissections: Time to Fold!

Review in MAA Online

Brian Borchers, a professor of Mathematics at the New Mexico Institute of Mining and Technology, wrote a review for "Read this!", the book review column in MAA Online. It first appeared February 22, 2007, and can be found online:
"Read This: Piano-Hinged Dissections".
Some excerpts from the review:
"Frederickson's latest book discusses another class of hinged dissections, piano-hinged dissections, in which edges are connected by hinges that allow the pieces to fold over each other or to fold out flat. In a piano-hinged dissection the pieces are required to overlap in two layers to form each of the geometric figures. Unlike with swinging and twisting hinged dissections it is particularly easy to build models of piano-hinged dissections out of card stock or wood. There are also obvious connections between piano-hinged dissections and origami. Thus piano-hinged dissection problems should be particularly attractive to fans of recreational mathematics."
"highly recommended to fans of geometric dissections who are looking for some new and challenging dissection problems"

Review in Cubism For Fun

Rik van Grol, the editor, wrote a very nice review in the March 2007 issue (#72) 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 pages 24-25:
"With this book Greg fanatically explores a world filled with piano-hinged dissections. Initially you would think it to be impossible to make more than a few dissections with piano-hinges (a piano-hinge is a long narrow hinge the runs the full length of a joint), but Greg show[s] that given time and effort many more dissection[s] can be realized with piano-hinges than you could ever imagine. In this book an amazing piano-hinged version of the Triangle to Square is presented."
"The book comes with a CD with a video in which Greg actually demonstrates his hinged dissections: almost an hour of wonderful dissections folding and unfolding before your eyes. Most of these are paper realisations, but Greg also shows a number of amazing wooden piano-hinged dissections; simply great!"
"Greg's books are valuable sources for the puzzle designers and an obvious must have for dissection lovers."

Review in SciTech Book News

There is a great review in the March 2007 issue (Vol. 31, No. 1) of SciTech Book News, published by Book News, Inc., of Portland Oregon. It's so enthusiastic (and not so long) that I couldn't resist reproducing the whole thing, which appears on page 34:
"Armed, as it were, with little more than an inquiring mind, moderate hand-eye coordination and the anticipation of a delightful outcome, you can follow Frederickson (computer science, Purdue U.) into the world of dissecting polygons so the resulting pieces, which are attached by hinges that must fold along an edge rather than swing or twist, form another polygon. The geometry is fascinating, and so are the illustrations Frederickson offers to show how he came up with the results, which range from the remarkably complex Theobald 11-piece dissection to a plethora of hexagrams that become hexagons. Working in some cases with the unique approach of Ernest Irving Freese, Frederickson produces both beautiful dissections and the concepts to back them up. His self-made video on the accompanying CD-ROM truly helps those of us who need more hands-on training and less apprehension that it cannot be done."

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 March 15, 2007 (no. 62) 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 appears on pages 8-9, as well as online. Two excerpts from the review:

"Because essentially all the moves discussed need a three-dimensional space, it is sometimes difficult to give a clear explanation of the operations to be performed. This gave Frederickson the idea of including a cdrom on which he demonstrates the folding, decomposing and recomposing the dissections. Some are really complicated, and even if you see him doing it, it is sometimes impossible to see how all the pieces fall into place. At least it is a visual proof that the method does indeed work. It is quite funny if you see him fighting with a wooden model of a hinged dissection that is a loop consisting of 30 triangles hinged together."
"And there are many other of these geometrical puzzles. Some of them are relatively simple and e.g., related to the way we fold a paper large map into some handy pocket sized format. Others, in fact most of them, are quite a challenge and are beautiful in their complexity."

Review in Choice

David V. Feldman, a professor of mathematics at the University of New Hampshire, wrote a very nice review in the July 2007 issue (vol. 44, no. 11) 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 (44-6281), which appears on pages 1944-1945:

"The search for elegant mutual dissections has formed a principal theme of recreational mathematics for several centuries. `Elegant' can simply mean dissection into an unexpectedly small number of pieces, or alternatively, a dissection subject to some interesting side condition.

"the author has now invented yet a new art form, two-ply models with polygonal pieces connected by easy-to-find and simple-to-work piano hinges. The body of this book works variations on the main theme that seem nothing short of ingenious. An accompanying video-CD-ROM offers readers the chance to see these remarkable models in action. Best of all, the author captures the infectious tone and constant excitement of an effectively theatrical lecture-demonstration."

Review in Mathematics Teacher

Ira Lee Riddle, a tutor in the Learning Center of Penn State University, Abington, wrote a review that appeared in the September, 2007 issue (vol. 101, no. 2) of the Mathematics Teacher. This journal is published nine times a year by the National Council of Teachers of Mathematics. The review appears on pages 159-160. Two excerpts from the review:

"When the term [dissection] is used in mathematics, it refers to demonstrating that the area of a figure is a constant, even when its pieces are rearranged. Shapes are cut apart, and the pieces are put together again into a new shape. Tangrams come to mind immediately, but this book goes a bit beyond tangrams."
"challenging for teachers as well as students, but it is a good bit of fun as well"

Review in EMS Newsletter

Martina Bečvářová, a member of the Department of Mathematics Education at Charles University, in Prague, Czech Republic, wrote a very nice review that appeared in the September, 2007 issue (no. 65) of the EMS Newsletter. The newsletter is published four times a year by the European Mathematical Society. The review appears on page 57. Excerpting from the review:

"This brilliant book can be recommended to students of geometry and teachers of mathematics, as well as students and all people who are interested in geometric dissections. Every creative reader will find new material for his own discoveries. The reader can easily experiment with the piano-hinge dissections because their mechanism can be simulated by folding a piece of paper without special mathematical knowledge, materials, computer programs, etc."

Review in Zentralblatt MATH

Mowaffaq Hajja, a member of the Department of Mathematics, Yarmouk University, Irbid, Jordan, wrote a very nice review that appeared in Zentralblatt MATH in November 2007. Zentralblatt MATH is published by the European Mathematical Society, FIZ Karlsruhe, and Heidelberger Akademie der Wissenschaften. Excerpting from the review, which is indexed as 1126.52014 and appears in volume 1126:

"This beautiful book is the author's third book on dissections.

"The book under review lays the mathematical foundation of the theory of piano-hinged dissections by examining the definition of such a dissection and suggesting ways for overcoming technical difficulties that arise from the thickness that two-dimensional pieces have to be assumed to have. Above all, it introduces so many beautiful and ingenious piano-hinged dissections that have never been known before and that are so non-trivial to discover, one such beautiful example being an ellipse-to-heart piano-hinged dissection.

"There are several asides of both mathematical and recreational interest. Notable among these is a fascinating section on what is usually referred to as the open box problem and for which the author had duly won the Polya Award for expository writing from the Mathematical Association of America.

"Beside having a lot of perspective diagrams that illustrate how the folds are to be made, the book also has a CD-ROM that contains a lot of videos that are extremely helpful for understanding the moves. Without this CD, many of the moves that are verbally described would be hard and time-consuming to understand and follow. If a perspective diagram is worth a thousand pictures, then an animation is probably worth a thousand perspective diagrams."

Faux Review in Monatshefte für Mathematik

The Monatshefte für Mathematik gave what was purported to be a review of my book in volume 155, number 1 (September 2008), page 102. Unfortunately, it appears that there was a mixup, and the review is actually of the book Fractal Geometry, Complex Dimensions and Zeta Functions, by Michel Lapidus and Machiel van Frankenhuyen. See the fifth review down from here for an actual review of my book that appeared in this journal in 2013.

Review in Journal of Recreational Mathematics

Charles Ashbacher, one of the two editors of the Journal of Recreational Mathematics, wrote a 5-star review, "Complex, yet fascinating dissections with a flip," which appeared on October 28, 2009 on amazon.com and which will appear in due course in the journal itself. Excerpting from the review:

"A piano hinge dissection is one where a hinge runs the full length of a joint. The analogy is to the hinge that allows the top of a grand piano to be opened although in this case the hinge can allow the two pieces to be folded in either of the two directions. Add in multiple hinges and the potential for the pieces to overlap and the additional degrees of freedom can make for a complicated structure. I often found myself wondering how the dissection had been discovered.

"Frederickson uses paper and wooden models to illustrate the folding that allows you to transform one figure into another. Seeing the transformation by executing one active fold after another makes it so much clearer. Quite honestly, I am not sure if I would have completely grasped some of the more complex transformations without the video.

"The use of one or more piano hinges in a dissection creates a significantly higher level of complexity. However, in complexity there is joy and while I had difficulty in the explanations, the wonder of seeing it work made the time of difficulty well spent."

Review in Mathematical Spectrum

A very nice review appeared in the Mathematical Spectrum, vol. 42 (2009/2010), no. 1, on page 50. Excerpting from the review:

"Traditional dissections involve pieces that are not attached to each other—add hinges between the pieces, and you are looking at a whole new set of rules and challenges. This book showcases a new type of hinged dissection that generates even more challenges."

"This mechanism can be simulated by folding a piece of paper, so you can test and experiment with piano-hinged dissections without needing special materials: just paper and scissors–and some intuition and creativity!"

Review in SIAM Review

Les Pook, a visiting professor in the Department of Mechanical Engineering at the University College London, wrote a book review that appeared in the SIAM Review, vol. 52, no. 1 (March 2010), on pages 208-213. The bulk of the review is a chapter-by-chapter summary of the contents of the book, along with complaints that the style is not appropriate for a monograph, that the text does not always establish that the methods produce mathematically exact dissections, that nets are not furnished for all of the dissections, and that the series of "Folderol" segments and "Manuscript" segments are tangential and would better have been moved to appendices, or removed altogether. However, this reviewer was nonetheless intrigued by the piano-hinged dissections, as revealed by this excerpt from his introduction:

"Piano-hinged dissections are motion structures that can only be fully appreciated by manipulating models or viewing videos. They are highly addictive. I made 54 paper models while preparing this review. Individually, the piano-hinged dissections described by Frederickson are usually not of great interest, apart from admiring his ingenuity in deriving them. Collectively, they are of interest in demonstrating the wide range of possibilities and the relationships between different dissections."

Excerpting from his summary:

"It is a useful and fascinating monograph on one of the many aspects of paper folding. It includes much original information. The book can be enjoyed at the recreational mathematics level by making and manipulating models of the piano-hinged dissections described. It can also be enjoyed at the serious mathematics level by studying methods used to derive piano-hinged dissections and also the relationships between different dissections. The book is well produced with only a few minor errors."

Review in CMS Notes de la SMC

Keith Johnson, a professor in the Department of Mathematics and Statistics at Dalhousie University, wrote a book review that appeared in the "Brief Book Reviews" section of CMS Notes de la SMC, vol. 43, no. 5 (October/November 2011), on page 6. Excerpting from the review:

"A special class of these, dissections in which the reassembly is by means of piano hinge joints i.e. folding only along a specified set of edges, is the topic of this book. It is fairly easy to think of simple examples of these, such as the dissection of a triangle using three piano hinge joints which bring the three vertices together at a point on the longest side to give a rectangle (and so illustrate that the sum of the three angles of the triangle is π) but the variety of more complicated examples the author exhibits is quite astonishing."

"The book has a couple of unusual features. One is that many early results on the topic were discovered in the 1930's by an American architect and amateur geometer, Ernest Irving Freese, and described in an almost lost manuscript, parts of which are reproduced here for the first time. The other is an included CD with videos of the author showing the operation of many of the dissections."

Review in Monatshefte für Mathematik

Prof. Dr. Harald Rindler, Dean of the Faculty of Mathematics and Head of the Department of Mathematics at the University of Vienna, wrote a great book review that appeared in the Book Reviews section of Monatshefte für Mathematik vol. 169, issue 2 (February 2013), on page 251. Quoting from the review:

"Greg Frederickson ist ein international anerkannter Großmeister beim Entdecken und Kreieren beeindruckender Zerlegungen (Zerschneidungen), Faltungen, Aufklappungen und überraschender Umwandlungen vorgegebener Formen in faszinierende Objekte und bereichert hier eine jahrhundertelange Tradition mit einer Fülle neuer Ergebnisse und Anregungen. Der Autor bietet auch interessante Resultate eines "verlorenen" Manuskripts von E. Irving, eines Architekten in LosAngeles, der gegen Ende seines Lebens eine Passion für Zerlegungen entwickelte. Zusätzlich gibt es eine CD mit Videoclips und Anleitungen, die die Konstruktion der Objekte leicht ermöglichen."
Rough translation: Greg Frederickson is an internationally recognized Grand Master in discovering and creating impressive decompositions (dissections), folds, flaps, and surprising transformations of given shapes into fascinating objects, enriching here a centuries-old tradition with a wealth of new results and inspiration. The author also provides interesting results of a "lost" manuscript by E. Irving, an architect in Los Angeles who towards the end his life developed a passion for dissections. There is also a CD with video clips and instructions that make possible the easy construction of the objects.
"Einem großen Interessentenkreis sehr zu empfehlendes Buch!"
Rough translation: A highly recommended book for a large prospective audience!

Last updated April 19, 2013.