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"
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
of the newsletter.
Excerpting from the review,
which appears on pages 24-25:
"With this book Greg fanatically explores a world filled with
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."
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."
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
the newsletter of the Belgian Mathematical Society and the National Committee for
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."
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."
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
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"
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
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."
Mowaffaq Hajja, a member of the Department of Mathematics, Yarmouk University, Irbid, Jordan,
wrote a very nice review that appeared in
in November 2007.
Zentralblatt MATH is published by the
European Mathematical Society,
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."
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.
Charles Ashbacher, one of the two editors of the
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."
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 otheradd 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 scissorsand some intuition and creativity!"
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:
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
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."
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.