# Curves for the Mathematically Curious

Julian Havil has already produced several popular math books. Some of them have been reviewed here: Impossible? (2008), The Irrationals (2012) and John Napier (2014). In this book, containing an anthology of ten iconic curves, he takes another angle of approach to tell more stories about mathematics. Havil's popular math books are more of the recreational kind. I mean that while telling his story, he is not hiding away the mathematics. There can be many formulas and derivations that are however easy to follow with some background in basic calculus.

The curves selected have names. That is because they are in some sense important. If it is the name of a mathematician, it is, like often in mathematics, not always the name of the one who first studied the object. This is again illustrated by Havil in this book when he explores the history underlying the origin of the curve. There is one chapter for every curve. Sometimes it is just one particular curve described by a unique formula like the catenary, but often these curves have parameters or it is just a whole collection of curves with a special property like space filling curves. "Why these ten?" is an obvious question to ask, and Havil has anticipated this because he opens every chapter with a section that explains why he has chosen this curve. Whatever reason he gives, what is important for the reader is that there is always a story or stories worth telling that can be connected to that curve and in some cases these also have a very long history. Let me illustrate the yeast of the book by a telegraphic survey of the ten chapters.

1. *The Euler spiral*. Its parametrizations are analyzed and the connection with elastic curves and Fresnel integrals. It is also known under other names (e.g. Coru spiral and clothoid), and Havil also explains the history of how and why this has happened.

2. *The Weierstrass curve*. This is defined as an infinite sum and it is probably the first fractal ever described: a continuous function that is nowhere differentiable. The proof of Weierstrass for these properties is included.

3. *Bézier curves*. This is an introduction to the characterization of these curves and how they are constructed by the Casteljau algorithm. There are two fun stories connected to these curves. One is about a Bézier curve called Lump which is the name of a dachshund as it was sketched by Picasso caught in one smooth Bézier curve. Havil provides its control points. Another story on the side is about how these curves are used to design letter fonts.

4. *The rectangular hyperbola*. This is an excellent occasion to tell the history of how logarithms were invented. This is of course described in much more detail in Havil's book about John Napier.

5. *The quadratrix of Hippia*. The history of this curve is connected to the classic Greek problem of trisecting an angle using only compass and straight edge, but the story would not be complete if one did not recall also the other "impossible" problems of squaring the circle and doubling the cube. The quadratrix is formed by the intersection of two moving lines one translating and another rotating at constant speed. If one could construct that curve, then trisecting an angle and squaring the circle became possible as well as constructing segments whose length is a unit fraction or a square root. The latter are examples of how Havil manages to add some extra mathematics of his own to a well known story.

6. *Two space-filling curves*. Cantor, Hilbert, and Peano, are three names connected with these curves. The construction of these curves is of course related to the study of cardinality. The Peano curve is a continuous map from a unit interval to a unit square but it is not surjective.

7. *Curves of constant width*. These are curves like the Reuleaux triangle that looks like a triangle that is slightly inflated, and yet shares many properties with a circle. If it is used as a drill, it will produce square holes (with slightly rounded corners). But there are several generalizations to study. Again, the latter are typical examples of Havil's mathematical extras.

8. *The normal curve*. This bell shaped curve is probably best known since it represents the normal probability distribution and it is related to the accumulation of rounding errors in long computations. No introduction to probability or statistics is possible without it. There are a few less known names of mathematicians that show up in the birth history of this curve.

9. *The catenary*. This is the curve formed by a chain loosely hanging from its fixed extremes. It looks deceptively like a parabola, but it isn't and that has fooled some mathematicians of the past. It is of course a place to discuss also the other hyperbolic functions. This is one of the curves that has been used to shape bridges and arches. It is also the shape of the road on which one can smoothly drive with square wheels.

10. *Elliptic curves*. These are the most complex curves of the book. They are related to Diophantine equations and they are most famous for their use in cryptography.

It is clear that the variety of topics is very broad: form constructions with compass and straight edge to cryptography and from the foundations of mathematics to the design on fonts with Bézier curves and the Casteljau algorithm. There are also seven short appendices explaining some preliminaries or expanding on some topics. However the first appendix is a surprise. On one of the very first pages of the book (page ii, before the title page) there are two 13 × 41 blocks of decimal digits or a number *N* of over 500 digits spread over 13 lines. No reference, no explanation. The explanation comes in the first appendix. It shows a complicated formula whose main ingredient is a modulo 2 formula for an expression depending on *x* and *y*. It thus gives a 0 or 1 depending on *x* and *y* which are assumed within certain bounds. The bounds for *y* depend on a number *N*, It turns out that it describes the pixel values within a rectangle of a page that will reproduce a pixelated image of the formula on a 106 × 17 pixel grid. Thus the *N* is the decimal representation of the binary number with 106 × 17 = 1802 bits giving the bit pattern of the pixmap one wants to generate. The two blocks at the beginning of the book give the two *N* values needed to reproduce the title of this book in pixel-form. The idea is from a 2001 paper by computer scientist Jeff Tupper.

A few pages further at the beginning of the book on page vi shortly after the title page, there is a mathematical doodle with nine wild curves symmetrically arranged in a 3 x 3 matrix, and a trigonometric formula. No further explanation, hence leaving it as a puzzle and a challenge to tease the reader.

There is more serious mathematics to be found in some other relatively long excursions in the chapters. Many of them are following some historical evolution of the problem. For example in the chapter on the normal distribution there is a lot of formula manipulation to move from a binomial distribution, via summing binomial coefficients and Bernoulli numbers, to finally arrive at the exponential expression. The discussion that a bijective map from the unit interval to the unit square cannot be continuous is illustrated by following the steps of the proof of continuity and non-differentiability as given for the Peano curve. The move from an parametrization of the Euler spiral to a simple one, parametrized by arc length, is fully explained and variations in the parametrization can produce very frivolous curves. And there are more not-so-trivial derivations in other chapters that can set the reader on a DIY path for further exploration. The fun items on page ii and vi will certainly trigger the interest of the mathematical puzzlers to find explanations or variations. The conclusion of the book is that $x^2+(\frac{5}{4}y-\sqrt{|x|})^2=1$ is the most important curve of all and it is indeed a lovely one.

**Submitted by Adhemar Bultheel |

**2 / Dec / 2019