# Mathematics on the Chopping Block

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Mathematics on the chopping block

Author: Theodore Kolokolnkov

ABSTRACT.

The production of end-grain wooden cutting boards involves repeated operations that lead to interesting mathematical questions. I explore some of the mathematical issues that arise in the process; they pose some interesting mathematical puzzles. I also show how mathematics can be used to create intricate artistic designs for cutting boards that are amenable to woodworking. These designs have been tested in real wood. Some of them belong to a class of iterated-function-system fractals but others do not. The overall goal is to produce an aesthetically pleasing design suggested by mathematics and implementable in wood.

## Introduction

Suppose you want to make a wooden chess board. You have some dark wood (such as walnut) and some light wood (such as maple) to make black and white squares. You also have a table saw and some wood glue. One possibility is to cut 32 white squares and 32 black squares, and then glue them up. However, arranging all of the squares and trying to glue them all at once will inadvertently introduce many imperfections. A much better way — and the way this is done in practice — is illustrated in Figure 1. First, cut eight strips of wood — four white and four black — and then glue them together into a board, iterating black and white. Let the glue dry; then make seven cuts across, resulting in 8 stripes as in Figure 1. Flip every second stripe and re-glue. Voila: a chess board.

The above procedure requires two glue-ups and seven cuts after the initial eight strips are cut. Each glue-up is along one direction only, which facilitates the use of clamps to fix the glue while it dries. A similar procedure, using only two glue-ups, is used to make surprisingly complex and artistic end-grain cutting boards, some of which are illustrated in Figure 2. These boards were produced by Andrey Muntian (MTMWood) who is a master of the craft [1]. It is a nice mathematical puzzle for readers to try to deduce the process that generates these patterns. (See Andrey’s website referenced in the figure for answers.)

In this paper, I discuss several techniques for producing a large variety of cutting board designs. Some examples — constructed in wood — are illustrated in Figures 2 and 3. I explore connections to various branches of mathematics, including group theory, combinatorics, and fractals. I also propose several open mathematical questions that are motivated by making various designs. Some of the designs provide interesting challenges for woodworkers as well.

There is a fine balance between what an artist or computer can do, what is aestethically pleasing, and what is implementable in wood in a workshop without fancy tools such as a CNC router and within a reasonable time and budget. For the most part, I emphasize realistically obtainable designs, as tested by actually building them. The primary goal is to produce an artistic design using mathematical techniques as a guide and to motivate readers to try out some of these techniques to create and build their own designs.

## Basic square designs

As mentioned in the introduction, the basic technique is to glue some strips, cut them across, rearrange, and reglue. After the second set of cuts, the wood should be turned end-grain up to create an end-grain cutting board. There are several reasons for doing this [1]. End-grain boards are best for stability. They keep the knives relatively sharp since the knife actually goes through the wood fibers instead of across them. This also extends the lifespan of the board, since the wood fibres close up after each cut, rather than the knife cutting through the fibres. Moreover, the glue holds much better when applied along the fibres, and turning the wood end-grain up guarantees that all of the glueing-up is done along the fibers.

How many different designs can be produced using two glue-ups? To formulate this question mathematically, assume we start with $n$ black-and-white stripes. Glue them together horizontally, and then make $m-1$ cuts to make $m$ strips vertically, rearrange, and glue again. Let $N\left(n,m\right)$ be the number of resulting patterns.

Assuming two choices for each stripe (either black or white), there are $$2^n$$ possible stripe selections before the first glue-up, so that $$N(n,1)=2^{n}$$. After making $$m$$ vertical strips, each strip can be either left alone or flipped vertically. Permuting resulting strips is allowed, but it does not alter the pattern. This leads to following recursion relationship: $$N(n,m+1)=2\left( N(n,m)-x(n)\right) +x(n). \label{recurse} \tag{2.1}$$

Here, $$x(n)$$ is the total possible number among the $$n$$ black-and-white squares in a column that remains invariant under flipping it. To count $$x(n)$$, suppose $$n$$ is even. Then the first $$n/2$$ squares are chosen arbitrarily, and the last $$n/2$$ squares are a reflection of the first. This yields $$x\left( n\right) =2^{n/2}$$ if $$n$$ is even. However, if $$n$$ is odd, the central square can be chosen arbitrary, and the last $$(n-1)/2$$ squares is the reflection of the first $$(n-1)/2$$ squares. Hence $$x(n)=2\times 2^{(n-1)/2}$$ if $$n$$ is odd. In summary, $$x(n)=2^{\left\lfloor (n+1)/2\right\rfloor }$$, where $$\left\lfloor {}\right\rfloor$$ denotes the floor function. The linear recursion (\ref{recurse}) is readily solved to obtain $N(n,m)=2^{\left\lfloor (n+1)/2\right\rfloor }(1-2^{m-1})+2^{n+m-1}.$

For instance, this gives 30736 “basic” designs for an 8 x 8 board. Among these, there are some that are isomorphic up rotations and flips. This leads to the following open question.

Challenge: How many unique designs up to rotations and flips can be made starting with $$n$$ black and white strips followed by $$m$$ cross-cuts?

## Connections to group theory

For practical purposes, one rarely uses more than two sets of cut-and-glue operations. Each cut turns some wood into sawdust (the width of a saw blade is about 3mm), so one can quickly eat through a lot of wood when making excessively many cuts. Making two sets of cuts ensures that the board is uniform in either the vertical or horizontal direction, which increases its stability and prevents the board from bending over time. But further glue-ups do not improve the board’s stability.

Nonetheless, for the purposes of making artistic decorative designs as well as writing a math paper, we are free to repeat the cutting and glue-up procedure. This leads naturally to the notion of what I will call a “cutting board group”, a kind of a Rubic’s cube-type game. This group is generated by the following operations on an $n×m$ array of tiles:

• flip any row or column of tiles;

• swap any two rows;

• swap any two columns.

A natural question is to ask what kind of patterns can be obtained using these group operations, starting with black and white strips? If enough cuts can be made, any pattern with the same number of tiles as the initial configuration is producible. More precisely, I show the following [2]:

Theorem 3.1. Starting with $$n_1$$ black and $$n_2$$ white rows of an $$n\times m$$ array of tiles, where $$n=n_{1}+n_{2}$$, it is possible to obtain any array that consists of $$n_{1}m$$ black and $$n_{2}m$$ white tiles in any position using only row/column flips and swaps.

A much more difficult question is the following: what is the minimum number of moves required to produce a given pattern? Recently, a similar question was answered for a Rubic’s cube: it was shown in [3] that 20 moves is the “god’s number” of moves; that is, any position can be solved using 20 moves or less, and moreover there are positions that require 20 moves. Although upper and lower bounds existed for a long time, it took about 30 years to find the optimal solution.

Define “god's number” $g$ to be the minimum number of flips and swaps required to obtain any pattern, starting with ${n}_{1}$ rows of black and ${n}_{2}=n-{n}_{1}$ rows of white tiles. While it appears to be very hard to determine god’s number, it is relatively easy to come up with upper and lower bounds for it. For example, in [2], I show that if $n=m$ and ${n}_{1}=n/2,$ then the god’s number is bounded by ${C}_{1}\frac{{n}^{2}}{log\left(n\right)}\le g\le 32{n}^{2}$where as $n\to \infty .$ If, however, ${n}_{1}=1,$ then $\phantom{\rule{4pt}{0ex}}{C}_{1}n\le g\le 32n$, where as In other words, $g$ scales linearly with $n$ when ${n}_{1}=1.$ However, it scales somewhere between $O\left({n}^{2}/logn\right)$ and $O\left({n}^{2}\right)$ when ${n}_{1}=n/2.$ It is an interesting open question to determine the exact scaling of $g$ in this case.

## Hexagonal patterns

What if we want a hexagonal lattice instead of squares? This requires miter cuts (i.e. cuts at an angle). The procedure is illustrated in Figure 4. Start by cutting, say, 5 black and 5 white strips (see Figure 4(a)). Rip (i.e. cut along a grain) a 60${}^{0}$ corner from each, and then glue to a strip of opposite colour as in Figure 4(b). Rubber bands can be used instead of clamps to glue at this angle. Next, glue the ten strips together as in Figure 4(c). I will refer to this as the generator of the pattern. Once dry, make bunch of cross cuts, resulting in identical stripes. Turn them end-grain up, and arrange them as in Figure 4(d).

A huge variety of different boards can be made by using a different order and orientation of the mixed-colour stripes. Figure 5 shows a star-lattice design, a 3D-effect design, and several others. All of them are produced in the same way and require the same amount of wood to make. Readers are invited to play with various rules using a Javascript applet that I wrote [4].

## Fractal designs

Several standard fractals are amenable to woodworking. Figure 6 shows how to build a Sierpinski triangle without having a CNC router.

Start with a black square strip. Cross-cut into three equal strips. Also produce a white strip of the same dimensions. Glue them together as shown in Figure 6. You obtain a square strip that is 1/3 the length of the original and twice the other dimensions. Rinse and repeat. Each iteration requires two glue-ups. (Remember that each glue-up should be done along one direction only.)

More generally, there are a total of 8 different rigid transformations of each strip possible that preserve its dimensions: rotation by 0, 90, 180, or 270 degrees; or flipping a strip either horizontally, vertically, or along two diagonals. Labelling these transformations 0 to 7, they are: \begin{align*} T_{k}(z) & =\exp\left( i\frac{\pi}{2}k\right) z,\ k=0,1,2,3;\ \\ T_{k}(z) & =T_{k-4}(\bar{z}),\quad k=4,5,6,7. \end{align*} Thus, there are a total of ${8}^{3}=512$ possible fractals that can be produced with this method, assuming the rules for each iteration are the same. (Of course, many more “fractals” are possible if successive iterations use different rules.) Figure 6 shows some of them after 10 iterations. These are easily produced with a few lines of computer code. However, even building four iterations poses significant challenge for woodworking, and more than four is very difficult to do.

One practical complication is that each iteration requires an additional “blank board” to be produced in a separate process. Also, in practice, one should start with several black strips separately, so that the first iteration should be done three times in parallel, rather than starting with a single strip. This is because the width of a strip decreases by ${3}^{k}$ after $k$ iterations, so in practice it is difficult to achieve more than three iterations (27 cross cuts) out of a single strip of reasonable length.

The limit set of these fractals can be described by an iterated function system (IFS) using three functions that move and transform the three boards that make up each iteration. These functions are: $$f_1=(T_a(z)-1+i)/2$$, $$f_2=(T_a(z)+1+i)/2$$, $$f_1=(T_a(z)-1-i)/2$$. This rescales the board so that its corners have coordinates $$\left( \pm 1,\pm1 \right)$$. The usual procedure for an IFS can be used to produce the bottom row of Figure 6: start with a random point, choose one ${f}_{1},{f}_{2},{f}_{3}$ at random, apply it to the point, and then repeat a billion times.

To avoid lots of white space, one can vary the colour of the “blank board” inserted at each iteration. An example of this is illustrated in Figure 6 (bottom right).

Duplicating fractal. A more interesting fractal design can obtained by making four copies at each iteration, instead of making three copies and a blank slate; this also avoids the difficulty of making a new blank slate at each iteration. Start with a square board $P.$ Cut across to make four copies of $P$. Then reassemble and glue up the four boards into a new square board ${P}^{\text{'}}$ whose sides are twice the dimension of the original square. Before assembly, each of the four pieces is transformed in one of eight possible ways (rotations and flips) according to a predetermined rule.

The resulting design depends on the initial board. Figure 7 shows various patterns that are obtained by starting from an initial square that is half light and half dark: . Readers are invited to try various designs on their own with the help of a javascript applet [4]. There are  ${8}^{4}=4064$ different rules for a given initial board, although many of the rules lead to the same patterns. Overall, it tends to produce more artistic boards — especially at a lower resolution necessary for actual production (e.g. 3 iterations) — than the standard iterated function fractal. In part, this is because there is a better mixing of two colours than the iterated map fractal, which is dominated by one colour after a few iterations.

Most of the resulting designs are non-repeating. For example, consider the design shown in Figure 7 (top left). Take any any row and substitute 0 for light colour and 1 for dark colour to obtain a sequence of 0s and 1s. For instance,         the first row of iteration 1 then reads 0110, the first row of iteration 2 reads 01101001, and so on. The resulting sequence is the well-known and ubiquitous Thue–Morse sequence [5]. This sequence is obtained iteratively as follows. Start with ${s}_{1}=0.$ Then negate ${s}_{1}$ (i.e. replace any 0 by 1 and any 1 by 0) and append it to the end of ${s}_{1}$ to obtain ${s}_{2}=01;$ similarly, ${s}_{3}=0110,$ and so on. It is an easy exercise to show that the Thue–Morse sequence is aperiodic. (In fact, it is transcendental [6, 7], though that is much harder to show.)

Almost all duplicating fractals are non-repeating. Computer experimentation suggests that there are only four distinct periodic patterns that appear with this method. These are: stripe pattern, checker-board pattern, and the two “swastika”-type designs shown on bottom right of Figure 7. All other patterns do not have any discernible periodicity. This leads to the following open question:

Open question: Starting with initial configuration , which duplicating fractals have periodic patterns? What about more general initial configuration? The conjecture is that the only periodic patterns are the four mentioned above.

From an artistic point of view, some of the more interesting patterns can be designed by combining the duplicating fractal with an IFS fractal as an initial board. An example is shown in Figure 8. It uses two iterations of an IFS fractal as a initial board configuration for the duplicating fractal, and then two iterations of the duplicating fractal.

## Double fractal

Finally, we discuss a fractal construction that generalizes both the duplicating fractal and the IFS fractals. It creates two boards at the same time. Start with two initial boards $P$ and $Q.$ They need to be very thick. For example, the board $P$ can be taken all white and the board $P$ can be taken all dark (and more complex initial conditions can also be considered). For each iteration, a total of eight copies (some $P$, some $Q$) are assembled. For example, cut across to make four copies of $P$ and four copies of $Q.$ Then assemble and glue up two new boards with twice the dimension from these eight pieces according to a rule ${P}^{\text{'}}=\left[\begin{array}{cc}P& P\\ P& Q\end{array}\right]$ and ${Q}^{\text{'}}=\left[\begin{array}{cc}P& Q\\ Q& Q\end{array}\right].$ Rinse and repeat. The resulting pattern is shown in Figure 10 (top left) after four iterations. As before, we also can rotate or flip the various pieces to create intricate designs. I constructed an example of such a cutting board; see Figure 9. See Figure 10 for further examples.

A duplicating fractal is a special case of a double fractal, obtained by setting ${Q}^{\text{'}}={P}^{\text{'}}$. An IFS fractal is also a special case and is obtained by taking ${Q}^{\text{'}}=\left[\begin{array}{cc}Q& Q\\ Q& Q\end{array}\right].$

There are eight possible positions for each piece; in addition to ${the 2}^{8}$ possible piece arrangements, this gives a total of ${8}^{8}{2}^{8}\approx 4.3$ billion possible rules, although of course some rules will generate the same (up to rotations and reflections) patterns. Readers are invited to try various designs on their own with the help of a Javascript applet [4].

All of these fractals are special examples of Lindenmayer systems. The board in Figure 10 (bottom left) appears in photonics literature (see, for example, [8, 9]). It is a two-dimensional analogue of the Thue–Morse Sequence: its rows and columns form a one-dimensional Thue–Morse sequence.

## Discussion

One of my main motivations was to produce artistically beautiful designs in wood. Of course, beauty is very subjective. In my opinion, appealing designs often appear at a boundary between order and chaos. This is natural: the design should have some pattern to it in order to spark interest, but not too much lest it becomes boring. This is why fractal designs can be appealing: they have infinitely many patterns, but none of it is repeating.

Complex mosaics with repeated geometric motifs appear in many cultures throughout the world. Figure Figure 11 shows two such examples made from wood: Hakone Yosegi Marquetry craft from Japan [10] and the art of Khatam from Iran [11]. Hakone marquetry dates back to the 18th century. It combines various basic strips of wood into ever-larger blocks, which are eventually planed into paper-thin veneer slices. These slices are, in turn, used for decorative boxes and furniture. The initial strips typically have a triangular profile and multiple colours are used, leading to very complex patterns. Khatam art uses a similar principle, starting with very thin triangular strips of various colours.

Woodworking provides some natural constraints on the kind of patterns that can be made easily. These constraints raise some interesting and novel questions for mathematicians. I also hope that some of the patterns discussed will spark interest among woodworkers (or inspire mathematicians to try woodworking!).

## References

1. MTMWood, The Basics of Making End Grain Cutting Boards; available from mtmwood.com, 2015.
2. T. Kolokolnikov, Mathematics on the chopping block, submitted, Math. Intelligencer.
3. T. Rokicki, H. Kociemba, M. Davidson, J. Dethridge, The Diameter of the Rubiks Cube Group is Twenty, SIAM Rev. 56(4) (2014) 645–670.
4. Readers are invited to play around with applets that generate the designs in this paper and many others. These can be found on my website: http://www.mathstat.dal.ca/~tkolokol/board.
5. J.-P. Allouche, J. Shallit, The ubiquitous Prouhet–Thue–Morse sequence, in: Sequences and their Applications, Springer, 1999, pp. 1–16.
6. F. Dekking, Transcendance du Nombre de Thue–Morse, CR Acad. Sci. Paris 285 (1977) 157–160.
7. B. Adamczewski, Y. Bugeaud, A Short Proof of the Transcendence of Thue–Morse Continued Fractions, The American Mathematical Monthly 114(6) (2007) 536--540.
8. L. Moretti, V. Mocella, Two-dimensional Photonic Aperiodic Crystals Based on Thue–Morse Sequence, Optics Express 15(23) (2007) 15314–15323.
9. V. Matarazzo, S. De Nicola, G. Zito, P. Mormile, M. Rippa, G. Abbate, J. Zhou, L. Petti, Spectral Characterization of Two-Dimensional Thue–Morse Quasicrystals Realized with High Resolution Lithography, Journal of Optics 13(1) (2010) 015602.
11. N. Shojanoori, A Background of Khatam Art, European Online Journal of Natural and Social Sciences 3(4(s)) (2014) 330.