Hinged Dissections: Swinging & Twisting

by Greg Frederickson, Cambridge University Press, 2002, ISBN 0-521-81192-9 (hardcover).

A geometric dissection is a cutting of a geometric figure into pieces that we can rearrange to form another figure. As visual demonstrations of relationships such as the Pythagorean theorem, dissections have had a surprisingly rich history, reaching back to Arabic-Islamic mathematicians a millennium ago and Greek mathematicians more than two millennia ago. As mathematical puzzles they enjoyed great popularity a century ago, in newspaper and magazine columns written by the American Sam Loyd and the Englishman Henry Ernest Dudeney. Loyd and Dudeney set as a goal the minimization of the number of pieces. Their puzzles charmed and challenged readers, especially when Dudeney introduced an intriguing variation in his 1907 book The Canterbury Puzzles. After presenting the remarkable 4-piece solution for the dissection of an equilateral triangle to a square, Dudeney wrote:
I add an illustration showing the puzzle in a rather curious practical form, as it was made in polished mahogany with brass hinges for use by certain audiences. It will be seen that the four pieces form a sort of chain, and that when they are closed up in one direction they form a triangle, and when closed in the other direction they form a square.

This hinged model, has captivated readers ever since. There is something irresistible about the idea of swinging hinged pieces one way to form one figure, and another way to form another figure. You do not really need a physical model to enjoy this property. Once you have examined this figure, you will be swinging mental images of the pieces around in your mind.
When I wrote my first book on geometric dissections, I became fascinated with hingeable dissections. I explicitly illustrated hingings in 18 figures and identified the hingeability of a number of others. I discovered that many hingeable dissections have been published without any indication that they are hingeable. There were enough that I wondered about the following question: Given any pair of figures that are of equal area and bounded by straight line segments, is it possible to find a hingeable dissection of them?
I haven't found an answer to this question, but as I worked on it, I found lots and lots of nifty hinged dissections. Most of them are two-dimensional, although I do have a chapter on three-dimensional hinged dissections. For three-dimensional dissections, I first considered the "piano hinge", which is the obvious analog of the "swing hinge" above. But I also thought about alternative types of hinges, which led me in turn to a basic hinge for two-dimensional dissections that no one in the field of dissections had thought about. I call this new hinge a "twist hinge". I have discovered a surprising wealth of twist-hinged dissections, which I explore in the last two chapters of my book. You can ask the same question about twist-hinged dissections as I have about swing-hinged dissections. And I don't know the answer to that question either.
Below is a twist-hinged dissection of an equilateral triangle to a square. A small circle on an edge indicates the position of a twist hinge. A piece that is labeled with an asterisk in the equilateral triangle is flipped over an odd number of times in converting to the square.

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News Flash — My third book is now published! :

Piano-hinged Dissections: Time to Fold!. A book about a new type of dissections — folding dissections!


Recent excitement (for me, anyway!):


Animations of twist-hinged ring benches! - August 2007
I have posted a webpage containing a number of twist-hinged dissections of ring benches.
I demonstrated these animations at the Bridges Donostia conference in San Sebastian Spain in July 2007. The corresponding paper is in the proceedings on pages 21-28.

Note: I handed out a souvenir/offer at the end of my talk, but unfortunately had not brought enough along. If you did not receive one then, and would like me to send you one now, please send me an email.
Twist-hinged animations! - June 2007
I have posted a webpage containing a number of wild twist-hinged dissections.
I first demonstrated these animations at the 7th Gathering for Gardner in 2006, and they are meant to be a supplement to an article that is to appear in the Mathematical Intelligencer.
A hinged dissection in a lesson plan! - January 2006
Pat Baggett and Andrzej Ehrenfeucht have posted a lesson plan based on the swing-hinged dissection of the equilateral triangle to the square: Triangle to square: A hinged dissection.
I especially like the way the hinges are incorporated into the design, using the thick foam rubber that makes up the pieces. Nifty! And the animations on the webpage are really effective, too.
Animations in wood and java! - October 2005
After seeing David Gunderson's wooden models of Rick Mabry created

Students swing! - June 2003
Middle school math teacher Kim Sharp got his pre-algebra class swinging in the classroom aisles!
He called it `Springtime Swingtime'!

Great review in American Scientist! - May 2003
Mike Eisenberg wrote an enthusiastic review for the May-June, 2003, issue of American Scientist. The review, entitled "King of Swing," concludes with:
"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."

Hinged dissections classified as Art! - April 2003
Joe Malkevitch, a professor of mathematics at York College (CUNY-The City University of New York), wrote a monthly essay, "What's New in Mathematics" for the American Mathematical Society. His column for April, 2003, was "Mathematics and Art." See Section 5, "Polyhedra, tilings, and dissections," which referred to some of my work, in particular, photos of models of my hinged dissections.

Chemists discovered (hinged) dissections! - January 2003
The chemists do their version of `Look, Ma - No hands'

Hinged dissections established a beachhead in France! - December 2002
Jean-Paul Delahaye wrote a column entitled "Logique et calcul" ("Logic and calculation"), for Pour La Science,the French edition of Scientific American. In the December 2002 issue (number 302) of Pour La Science, he wrote about the book and related material in "Découpages articulés" ("Hinged dissection"), pages 164-169.

AMS took notice of hinged dissections! - December 2002
The Notices of the AMS (American Mathematical Society) listed the book in its Book List on page 1408 of Volume 49, Number 11 (December 2002). As explained at the beginning of the section:
"The Book List highlights books that have mathematical themes and hold appeal for a wide audience, including mathematicians, students, and a significant portion of the general public."
The listing continued in succeeding issues, but was gone in December 2003.

More information:

Table of contents.

Comments, reviews, and news.

Updates and corrections.

Exact measurements for the swing-hinged triangle to square dissection.

A photo gallery.

A video gallery.

Past and future events.

Education.

Citations.

Animations of hinged dissections.
Have you seen my article in the summer 2001 issue of the Mathematical Intelligencer?
On the table of contents page, there is a picture of me manipulating a model of my twist-hinged dissection of an equilateral triangle to a square.
Related to this dissection, there is a (minor) mistake in one of the figures. Here is a correction.

Last updated September, 2007.