The Puzzling World of Polyhedral Dissections
By Stewart T. Coffin

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Chapter 1 - Two-Dimensional Dissections

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Most of the designs described in this book can be thought of as dissections of some sort. By way of introduction, we will first consider some simple two-dimensional geometrical dissections, which in their physical embodiment become assembly puzzles.

Jigsaw Puzzles

To dissect means literally to cut into pieces. Just about any chunk of material cut into pieces becomes a sort of dissection puzzle. If sawn freely perpendicular to the surface of a sheet of plywood (or die-cut of cardboard), the result is the familiar jigsaw puzzle. Most jigsaw puzzles are not designed to exercise or perplex the mind, at least in the sense that other types of puzzles do, and it is perhaps stretching the definition a bit to even call them puzzles. The definition given in the dictionary for the noun puzzle seems to have been purposely broadened so as to include what are really pastimes of pattern recognition, memory, and patience. The definition given for the verb to puzzle contains no such connotation.

Jigsaw puzzles have been around for over 200 years, longer than nearly any other type of puzzle. Although their relationship to burrs and polyhedral dissections may appear to be remote, they are probably an important historical root. The ancestry of inventions in general must be an incredibly complex web of ideas branching backward in time into just about every nook and cranny of human culture. Puzzles are certainly no exception, and jigsaw puzzles, by their sheer numbers and long history, must play at least a minor role in the evolution of many present-day geometrical puzzles and recreations. How many of us played with jigsaw puzzles at one time and then began to ponder, perhaps subconsciously, variations along logical and mathematical lines?

Various schemes have been employed to make jigsaw puzzles more clever, such as by sawing on two different faces of a rectangular block or along multiple axes of a sphere. Some of these are quite entertaining, but still they are essentially non-geometrical in principle.


Fig. 5


If, instead of cutting freely, the dissection is done according to some simple geometrical plan, an entirely different type of puzzle results. Many fewer pieces are required to create interesting puzzle problems. Three characteristics of such puzzles are that they nearly always use straight line cuts, they usually assemble into many different puzzle shapes, and the problem shapes often have more than one solution.

Of the types of puzzles covered in this book, the oldest known is the popular seven-piece dissection of the square known as Tangram. It was at one time thought to be thousands of years old, but is now known to have originated in China sometime before 1780. (A quite similar Japanese seven-piece square dissection has been dated back to 1742.) Tangram became popular throughout Europe and America in the 19th century and continues to be so to this day. It is made and sold in many different materials. Thousands of problem shapes have been published for it over the years, and it is mentioned in many books. For more background information on Tangram and many similar puzzles, the reader is referred to Puzzles Old and New by Botermans and Slocum. Here we will discuss some of the curious mathematical aspects of the puzzle not generally mentioned in the literature. The dissection is shown in Fig. 6a and 6b. Fig. 6a is a reproduction by the German firm Richter and Co. of a Tangram set originally produced by them in the 1890s. Richter was well known for their Anchor Stone Building Sets. They called it Der Kopfzerbrecher which translates to "The Head Cracker". Fig. 6b is a set of Chinese candy dishes made around 1860 in China.

fig006a fig006b
Fig. 6a Fig. 6b

In designing dissection puzzles of this type, the idea is to divide the whole according to some simple geometrical plan so that the pieces will fit together many different ways. The way this is accomplished in Tangram is shown in Fig. 7. A diagonal square grid is superimposed onto the square whole such that the diagonal of the square measures four units and the area is eight square units. The only lines of dissection allowed are those that follow the grid or diagonals of the grid. To put it another way, the basic structural unit is an isosceles right-angled triangle made by bisecting a grid square, and all larger puzzle pieces are composed of these unit triangles joined together different ways. In Tangram, there are two of the unit triangles alone, three pieces made up of two unit triangles joined all possible ways, and two large triangles made up of four unit triangles, for a total of 16 unit triangles. The relative lengths of all edges are thus powers of formula0.


Fig. 7

The first Tangram problem is to scatter the pieces and then reassemble the square. Note that it has only one solution, usually a mark of good design. (Rotations and reflections are not counted as separate solutions.) For the countless other problem shapes, you can try to solve the published ones found in many books and magazines or you can invent your own.

The easiest way to discover Tangram patterns is just by playing around with the pieces. Start by trying to make the simplest and most obvious geometrical shapes - triangle, rectangle, trapezoid parallelogram, and so on, always using all of the pieces. An alternate method is to draw some simple shape on graph paper following the rules already given and having an area of eight squares, and then try to solve it. Which of the examples shown in Fig. 8 are possible to construct?


Fig. 8

Published Tangram patterns range all the way from the geometrical shapes shown above to the other extreme of animated figures created by arranging the pieces artistically. This range is represented by the row of figures shown in Fig. 9, reading left to right. Only those solutions that conform to a regular grid can be considered true geometrical constructions. Careful inspection will show those to be the three on the left. The others may be very artistic and imaginative, but they are not within the province of this book.


Fig. 9

The theme of discrete rather than random or incommensurable ratios of dimensions is one that plays continuously in the background throughout this book. In the case of Tangram-like dissection puzzles, it is easy to see that they cannot be made to work properly any other way. Beyond that, though, there must be something inherently appealing to our aesthetic sensibilities in simple, discrete ratios. They are, after all, the foundation of all music, although probably no one understands exactly why.

Fig. 10 shows 13 convex Tangram pattern problems. A convex pattern is one that can be cut out with a paper cutter straight away, i.e. with no holes or inside corners. They are all possible to construct. Are any others possible?


Fig. 10

For a slight change of pace from the usual Tangram problem, consider the following puzzler, which by the way is based more or less on an actual happenstance: Karl Essley made two Tangram sets as gifts - one to be sent to his sister and the other to his brother. The instructions were simply to assemble all the pieces into a square. Karl's sister brought hers back and declared (correctly) that the solution was impossible. Examining her set, they discovered that Karl had made a mistake in packing and had accidentally put two pieces into the wrong box, so one person got a set of five pieces and the other got nine. Embarrassed, Karl suggested that they phone their brother and explain the mistake. But his sister reflected for a moment and then said, "No that won't be necessary - he can make a square with his set." Can you tell who got the two extra pieces and what shape or shapes they were? (Answer) Be careful - this puzzler contains a nasty trap.

In a similar vein to the above puzzler, note the pairs of figures shown in Fig. 11. In each pair, one figure appears to be complete and the other appears to have a piece missing; yet they both use all seven pieces, as all Tangram figures must. Can you discover the common characteristic that all such confusing pairs have? (Answer) What other such pairs can you discover?


Fig. 11

In order to be entirely satisfactory, especially considering the examples just given, even simple puzzles such as this one should be accurately made of stable materials. If sawn directly out of a square of plywood, there will be noticeable errors introduced by the saw kerf. A more accurate way is to lay it out on cardboard, cut the cardboard with scissors, and then use the cardboard pieces as patterns.

Throughout this book, unscaled drawings are given for puzzle constructions. There are always a few readers who will report being unable to use such drawings, having been indoctrinated in school with the notion that nothing can be made out of wood without standard workshop blueprints with dimensions. Dimensions are omitted for the following reasons:

1. They are unnecessary. It should be obvious for example that in Tangram all of the angles are 45 or 90 degrees.
2. They are not as accurate as geometrical constructions. If the overall Tangram square is integral, all of the diagonal measurements are irrational and can be expressed in sixteenths of an inch or whatever only by rounding off.
3. Adding practical dimensions would only tend to obscure the elegantly discrete mathematical essence of the problem with unessential detail.
4. You may scale the puzzle to any size you wish.

Other Tangram-Like Puzzles

The great popularity of Tangram has spawned many imitations. Most notable of these were the famous Anchor Stone puzzles produced by Richter and Co. of Germany starting in the 1800s and on into the early 1900s. In Puzzles Old and New, Botermans and Slocum show 36 different designs and some of these are worth examining. Six of them, including Tangram, are squares dissected according to the usual square grid with diagonals. Three of these however, are on a grid with a finer scale than Tangram, i.e. containing more grid squares and unit triangles. The diagrams in Fig. 12 should make this clear. The number below each one indicates the number of grid squares enclosed for the coarsest grid that will conform.


Fig. 12

For a given number of pieces, dissections with coarser grids are likely to have more mutually compatible edges - thus the three on the left in Fig. 12 are the better designs in this respect. A dissection that accomplishes its purpose with the fewest pieces is usually to be preferred - thus the two on the left in Fig. 12 emerge as the better designs. The final test is to see which of these two sets constructs more interesting puzzle figures, and this task is left to the reader. The one on the far left is of course Tangram, and the other one was sold under the name Pythagoras.

Incidentally, note that the next smaller possible grid would contain only four squares and eight unit triangles. Are these too few to make an interesting puzzle? The most obvious such set (see Fig. 13) would be Tangram with the two large triangles omitted. This simple little set of five pieces probably contains a treasure-trove of undiscovered recreation: For example, how many convex patterns will it form? (Answer)


Fig. 13

A square can be dissected into numbers of equal isosceles right-angled triangles given by the following series: 2, 4, 8, 16, ... What is the next number in this series? (This question is reminiscent of "IQ" tests school children used to be given, and probably still are. Example: given the series 4, 6, 8, ..., what is the next number? A precocious student interested in prime numbers might answer 9, while one intrigued by the Platonic solids might say 12. But of course, by the time the students are supposed to know that the way the systems works is to always give the answer that the teacher wants, no matter how uninspired!)

Next in the Richter series, we find eight puzzles similar to those in Fig. 12 except rectangular rather than square. These are shown in Fig. 14 without further comment, except to point out that puzzles with mostly dissimilar pieces are generally more interesting than those with many duplicates or triplicates.


Fig. 14

All of the Richter puzzles shown so far have used only 45-degree and 90-degree angles. Eight of the Richter puzzles are polygonal shapes dissected into pieces with 30-60-90-degree angles. These are shown in Fig. 15 arranged by increasing numbers of pieces.


Fig. 15

Most of the other Richter puzzles have curved outlines or other complications. For example, the two shown in Fig 16 have more complicated angles. In dissection puzzles of this type, if all of the angles and linear dimensions are not immediately obvious by inspection, then the design is probably not very well conceived.


Fig. 16

To digress slightly, a most curious dissection is the one shown in Fig. 17 on the left. This construction within a square appears in Curiosités Géométriques, by E. Fourrey, published in Paris in 1907. It is said to have been discovered in a 10th-century manuscript and is supposed to have been the work of Archimedes. At least three slightly different versions of it have appeared in modern puzzle books, all supposing it to be a geometrical dissection puzzle and calling it the "Loculus of Archimedes". One learns to be skeptical about such things, especially when they do not appear to make much sense and the original documents are reported lost. The mystery of its origin and its actual purpose is a challenging problem for recreational maths historians.


Fig. 17

It has been pointed out by some authors that the areas in the Loculus are cleverly devised to be in the ratios of whole numbers, as indicated. But there is nothing unusual about that. It is easily proven if not immediately obvious, that all polygons formed by connecting points on a regular square grid must have areas in the ratios of whole numbers. Less obvious but also provable is that polygons formed by intersections of such lines must also have this property, as in the example shown in Fig. 17 on the right. Exercise for the reader: compute the relative areas in this figure.

Note that none of the Richter puzzles has fewer than seven pieces, and several have more. One always tries to minimize the number of pieces without sacrificing other design objectives. Satisfactory dissection puzzles of this type with fewer than seven pieces are not as common, but possible. Consider the experience of another puzzle acquaintance of mine, Bill Trong. Bill made for himself a Tangram set from published plans, but he carelessly failed to make one cut, so he ended up with two of the pieces joined together and thus a set of six pieces. Surprisingly, he found he could construct all 13 of the convex patterns (Fig. 10) with this set. Which two pieces were joined together? Judge for yourself if this six-piece version is an improvement over the original Tangram.

Previously, the reader was asked if other convex Tangram solutions could be found. According to an article in American Mathematical Monthly, vol. 49, in 1942 Fu Traing Wang and Chuan-Chih Hsiung of the National University of Chekiang proved that no more than 13 different convex Tangrams can be formed. Their proof involved showing that there are only 20 possible ways of assembling the 16 unit triangles convexly, of which 13 were found to have Tangram solutions. An excellent discussion of this is given in Tangram, by Joost Elffers.

The point to be made here, before leaving the subject of Tangram, is that the simplest and most familiar puzzles often contain surprising recreational potential, much of which may have been overlooked. Some of the practical innovations may be quite clever too. Figure 18 shows an example of what one skilled and inspired woodcraftsman - Allan Boardman - has done with Tangram. The seven pieces fit with watchmaker's precision two layers deep into the tiny square box complete with sliding cover, all beautifully crafted of pearwood.


Fig. 18

A Five-Piece Square Dissection

Fig. 19 shows Sam Loyd's well known square-dissection puzzle. It is made by locating the midpoints of all four sides of the square, drawing the appropriate lines, and dissecting. The five pieces construct all of the puzzle patterns shown. Again note the interesting paradox of the two on the right - one being a solid rectangle and the other a rectangle with a corner missing, yet both use all five pieces.


Fig. 19

When one of Loyd's pieces is divided in two, the number of possible interesting puzzle patterns is approximately doubled. Some of these new patterns are shown in Fig. 20. The first problem for the reader is to discover the additional cut. It should be obvious which piece to divide. But which way?


Fig. 20

The reader is now encouraged to experiment with new and original dissection puzzles. Start with a simple shape such as a square or rectangle and dissect it according to some simple geometrical plan, the idea of which is to make pieces that fit together many different ways. Six or seven pieces is a good number. Try to avoid having many pieces alike, then create your own catalogue of pattern problems.

Geometrical Dissections

To mathematicians, the term geometrical dissection has a slightly different meaning from the one we have been using here. It usually refers to two different polygons being formed from the same set of pieces. This is essentially an analytical problem, and a minor branch of mathematics is devoted to it. It has been proven that any polygon can be dissected to form any other polygon of the same area. Most attention has been given to the regular polygons. Choose my two regular polygons, and cut one of them into as many pieces as you wish to form the other. It may sound easy until you actually try it!

The classic problem in geometrical dissections is to find the minimum number of pieces required to perform a dissection between various pairs of common polygons. An excellent book on the subject is Recreational Problems in Geometrical Dissections and How to Solve Them, by Harry Lindgren.

Famous puzzle inventor Henry Dudeney was a pioneer in geometrical dissections. His classic four-piece dissection between the square and equilateral triangle, first published in 1902, is shown in Fig. 21. This must be the simplest of all possible dissections between two regular polygons. Yet if the reader will try to construct the dissection, even after glancing at the drawing, it will immediately obvious that the methods described earlier in this chapter do not work!


Fig. 21

Start by constructing a square and equilateral triangle of equal area. Thus, if the square is 1 x 1, the sides of the triangle are formula1. Next, note that all points marked (*) are midpoints of sides. Therefore, triangle ABC is equilateral and point B on the square is located by measuring formula2 from point A, after which the rest is obvious.

In geometrical recreations of this sort the essence of the puzzle is discovering the dissection. Given the dissections, their physical embodiment in the form of actual puzzle pieces has never enjoyed much popularity as practical manipulative puzzles. Perhaps it is because the two solutions are quickly memorized, and then there are no more problems. But there are exceptions. The Sam Loyd dissection puzzle described in the previous section was most likely developed by dissecting the square into the cross, after which the other interesting problem shapes were probably discovered. Creative Puzzles of the World, by van Delft and Botermans, contains an excellent chapter on geometrical dissections as practical puzzles. Further investigation might uncover a dissection by which several polygons could be constructed from a neat set of pieces. For example, what are the fewest pieces required to construct three different regular polygons? (Answer unknown, at least to the author.)


Checkerboard puzzles consist of a dissected standard 8 x 8 checkerboard (draughtsboard). The object is not only to reassemble the pieces into an 8 x 8 square, but to do so with the proper checkering. A Compendium of Checkerboard Puzzles compiled by Jerry Slocum in 1983 lists 33 different versions, and it includes only those that have been manufactured, patented, or published. The numbers of pieces range from 8 to 15, with 12, 13, and 14 being the most common. The oldest is dated 1880. The commercial versions were usually made of die-cut cardboard printed on one side only, so the pieces may not be flipped. Some are printed on both sides, and the checkering may not be the same on both sides. Those made of light and dark wooden squares can of course be flipped. A typical 12-piece dissection taken from Slocum's Compendium is shown in Fig. 22. The pieces may not be flipped. It is known to have at least two solutions.


Fig. 22

Taken as a whole, checkerboard dissections tend not to be the most inspired of puzzle designs. All that can be said for most of them is that they differ slightly from each other. Any reader wishing to make a checkerboard dissection puzzle might just as well create an original design rather than copy someone else's. Here are some design suggestions:

1. As the number of pieces is increased, the difficulty increases, reaches a maximum, and then diminishes. For the checkerboard, maximum difficulty occurs around 11 or 12 pieces.
2. Difficulty of finding one solution varies inversely with the number of solutions possible. Designs with only one solution are considered especially clever (but how do you know?)
3. Pieces with compact shapes approximating square or rectangular, such as those containing a 2 x 2 square lend themselves more easily to solutions and increase the number of solutions. Contrarily, skinny, angular, complicated shapes do just the opposite, especially those that refuse to fit into corners.
4. To be avoided are pieces having rotational symmetry, and especially pieces identical to each other. (There will be more on this later. For a simple explanation here, imagine a checkerboard dissection in which this rule is grossly violated and see how exceedingly uninteresting it would be.)

It is interesting to note that the additional constraint imposed by the checkering may make the solution (or solutions) easier or harder, depending upon the circumstances. If only one mechanical solution exists to begin with, obviously the checkering makes it much easier to find. On the other hand, if hundreds of solutions exist, but only one with the correct checkering, then the addition of the checkering has turned it into a real puzzler!

©1990-2012 by Stewart T. Coffin
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