Proofs without Words III

The concept of the book is clear from the title. Except that "without words" does not mean that there are only formulas, but on the contrary formulas are mostly avoided and the proofs are represented by one or a few graphics. Moreover, it is the third book in a row collecting this kind of proofs, so that the concept may be familiar already to many mathematicians, especially to teachers. The previous volumes were also published by the MAA in 1993 and 2000. This kind of mathematical proofs keeps showing up in magazines and journals so that bundling a new collection in this booklet became a natural thing to do.

Most of the 176 proofs that this collection contains are harvested from publications of the MAA. Few are found in other books or on the web, and only five never appeared in print before. Each item appears on one page and gives a formulation of the theorem (in words or formulas), but the proof consists of one or more graphics which should make the argument clear. Occasionally there are a few formulas to help understand the reasoning. The graphics are not reproductions of the originals, but they are redrawn so that the book gets a uniform appearance. For some theorems many different graphical proofs exist. In such cases, the numbering is indicated by roman numerals and the numbering is a continuation of the previous volumes. For example this book opens with The Pythagorean Theorem XIII, since 12 other versions were published in previous collections.

The 3 proofs of the Pythagorean theorem given in this collection are like puzzles. They could be directly turned into a set of wooden polygonal jigsaw shapes that can be arranged into some form showing the two squares $a^2$ and $b^2$, and then are rearranged so as to form the square $c^2$. The ways in which the squares can be cut into pieces is endless, as long as the sides $a$, $b$, and $c$ are recognized. This can lead to beautiful and pleasing arrangements. In fact this geometric approach to proving mathematical theorems was the standard way since the ancient Greek, which was only replaced by algebraic methods introduced by the Arabs, in particular al-Kwarizmi around 800.

To catch a proof in one graphic (or few graphics) requires to reduce it to its essential elements, which is often a matter of deep insight similar to catching an idea in a haiku or a picture in a few strokes of a brush on the canvas. It is a matter of elegance and esthetics. It does not always mean that the proof is obvious at first sight. The subtitle of these books refer to exercises in visual thinking, which is an accurate description because some of these "proofs" do need reasoning and mental exercise. Printed graphics are static. That is why sometimes more than one drawing is needed to represent the successive steps in the argument. Nowadays, animated graphics are easily made available on the internet, which in some cases, might be a better medium to visualize the arguments used.

The proofs in this book are arranged in 5 groups. The first is geometry and algebra. This involves plane geometry with triangles, squares and circles. Sometimes this leads to algebraic identities. There is one exception about the volume of a triangular pyramid. The second part is called trigonometry, calculus, and analytic geometry and relies mainly on constructions with right triangles. In the relatively short part on inequalities (only 18 proofs) there are four most pleasing proofs of the Cauchy-Schwarz inequality (only in $\mathbb{R}^2$). The part on integers and integer sums often has proofs that show dots, circles or square blocks arranged in a triangle or square grid. Arranged in a triangle, it can immediately be seen that for example the sum of the first $n$ odd numbers equals $n^2$. In other cases, like for summing squares, it is better to arrange items as Minecraft-like compilations of unit cubes that can grow outside the plane. A natural thing to do since sums of squares will involve cubes, which is three-dimensional counting. The last part deals with infinite series and other topics. Graphically, this may be the least satisfying one because the constructions inevitably involve ellipses (in the sense of three dots) because subdivisions or constructions have to continue forever, leading to infinitesimal areas representing infinitesimal quantities. Nevertheless we find proofs of the irrationality of $\sqrt{2}$,  the fixed point theorem, and summations involving the Riemann zeta function. Although the latter is just a matter of rearranging the order of summation, which means that one just has to arrange terms in an infinite square matrix that can be summed either columns first or rows first. Not really graphical but it does not require words to "see" the proof. So, in my view it is a bit of an outsider in this collection. It is perhaps less visually pleasing but it still fits under the title of the book. But if one accepts that this type of proof can be part of this collection of truly graphical proofs, then I can imagine that there are many more examples that one can come up with.

The author never claims to be exhaustive, and as said above, the collection is mainly restricted to items that appeared in print. Mostly in MAA publications. This immediately sets boundaries of what can be included here. There are of course many fancy applets and websites with graphical animation, which also can be categorized as proofs without words, but these are by the conceptual constraints set out from the beginning, excluded from this type of collection. But still, what is in this booklet is a very pleasing collection. Without the formulation of the theorem, the graphics often do not make sense, but given the theorem, it is fun to analyze the figure and mentally reconstruct the argument that it suggests. With the abstraction of modern mathematics, this kind of geometric arguments have somewhat lost interest in research and therefore to some extent also in education. Even though modern society is dominated by a visual culture of pictures, images, and videos, graphical proofs are not really trendy. In that sense, it is a good thing this booklet helps saving an endangered skill of graphical reasoning

Adhemar Bultheel
Book details

This is the third collection of graphical proofs of classical and less classical mathematical properties and theorems. Most of the proofs are reprints from MAA journals and magazines. Each theorem fits on one page that consists of its formulation and one or a few graphics in grayscale, possibly completed with a few formulas.

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9780883857908 (pbk)
£27.99 (pbk)