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Piano-Hinged Dissections: Time to Fold!

Greg N. Frederickson
A K Peters
Publication Date: 
Number of Pages: 
Hardcover with CDROM
[Reviewed by
Brian Borchers
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A geometric dissection is a way of cutting up a geometric figure and rearranging its parts into another figure or several figures. For example, Plato showed how to dissect a square into four triangles that can be put together to form two squares each of one half the size of the original square. Another example, is the "proof without words" of the Pythagorean theorem in which a square of side c is dissected into two squares with sides a and b. Geometric dissection puzzles have become a staple of recreational mathematics.

Greg Frederickson is one of the foremost contemporary experts on geometric dissections. Frederickson's first book, Dissections: Plane & Fancy is a general introduction to geometric dissection problems. His second book, Hinged Dissection: Swinging & Twisting introduced hinged dissections in which the pieces of a dissection are connected by hinges so that both figures can be obtained by unfolding the dissection in the right way. Frederickson considered a number of different kinds of hinged joints, including swinging joints between the corners of pieces and twisting joints in which two pieces rotate around a point on the interior of their edges.

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.

Frederickson discussions a number of general strategies for developing piano-hinged dissections. These strategies are illustrated by many examples of hinged dissections of various polygons. In addition to figures, the book includes a CD containing videos of the author demonstrating many of the dissections with card stock and wooden models. These videos make clear many of the dissections that are very hard to understand from the figures alone. The book also contains a number of dissection puzzles with solutions. Solving these problems requires great skill in visualizing the dissections.

Although this book is not a good starting point for those new to geometric dissections, it is highly recommended to fans of geometric dissections who are looking for some new and challenging dissection problems.


Frederickson, Greg N. Dissections Plane & Fancy , Cambridge University Press, 1997.

Frederickson, Greg N. Hinged Dissections: Swinging & Twisting , Cambridge University Press, 2002.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems. 

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