About 12 years elapsed since Johan Gielis wrote his book *Inventing the circle: The geometry of nature* in which he describes his superformula. This formula is a generalisation of the equation for the ellipse. In its simplest form it is $(x/a)^n+(y/b)^n=R^n$ with the Euclidean circle for $n=2$ and $a=b$ as a special case and more generally it represents an ellipse in the $\ell^n$-metric of $\mathbb{R}^2$. These graphs are also known as Lamé curves. The Danish artist and mathematician Piet Hein declared that $n=2.5$ resulted in the most aesthetic shape and made the super ellipse and its 3D analog, the super egg, an icon of the 1970's Swedish design. Martin Gardner picked it up in his Scientific American column popularising it worldwide. In the superellipse, the four points $(\pm a,0)$ and $(0,\pm b)$ are fixed for all $n$. For $n\gt2$ the ellipses are convex like a rectangle with rounded corners in between the ellipse and the circumscribed rectangle to which they converge as $n\to\infty$. For $1\lt n\lt 2$ they are in between the circle and the inscribed rhombus and for $n\lt 1$ they become concave like 4 pointed stars and converge to the coordinate axes as $n\to0$.

Gielis was inspired by the shape of bamboo stems whose cross section had this rounded square shape. As a bio-engineer he got interested in describing also other shapes like flowers and leaves and 3D botanical components. He transformed the equation to polar coordinates and introduced more parameters to arrive at $r(\theta)=[|x(\theta)|^{n_2}+|y(\theta)|^{n_3}]^{−1/n_1}$ with $x(\theta)=\frac{\cos(m_1\theta/4)}{a}R$ and $y(\theta)=\frac{\sin(m_2\theta/4)}{b}R$. In a second stage also $R$ can depend on $\theta$ which allows for example to generate spirals. The result is that almost any appealing shape from nature can be generated. The wikipedia page on the superformula shows several examples, and provides matlab and octave scripts to generate them as well as links to interactive websites where one may experiment with the parameters to generate different shapes.

In the present book the adjective "super" is replaced by "Gielis". So there are Gielis transforms, Gielis curves, Gielis surfaces, etc. Gielis is announced as an acronym on page 5, but the explanation comes only in chapter 7 on page 118: '**G**eneralised (or **G**eometric) **I**ntrinsic and **E**xtrinsic **L**engths **I**n **S**ubmanifolds'. The idea is still the same as in the previous book: namely to illustrate that the Gielis transform (of the basic circle and spiral) can describe almost any 2D or 3D form appearing spontaneously in nature (in particular in botany). Some slightly further generalised variants are considered which for example allow Möbius band-like objects, which combine symmetry in certain coordinate directions and twists in others. But Gielis has explored the material much further and he has collaborated with mathematicians which resulted in several published papers relating the transform to several other, not always plant-related, applications. Besides giving more historical, mathematical and even philosophical background on the formula, this book is also a summary of some of these papers.

The book has 6 parts as described in the comments by Proclus (412-485 CE) on Euclid's Elements. These are the 6 steps that have to be followed to prove a theorem in the Platonic tradition. In this way a bridge is spanned between the classical Greek geometric approach to mathematics (including the Pythagorean theorem and the commensurability problem) on the one hand and on the other hand the proof that when we use an elastic notion of length, that is if the unit used to measure the length $r$ of the radius depends on the angle $\theta$, then the Gielis transform just describes a circle or a spiral in this flexible metric, and it can be applied in particular to describe many natural shapes in 2D, or, with obvious adaptations, curves and surfaces in 3D.

The first part, the *Propositio*, Gielis explores the idea and provides some elements that will be used later. In this case it concerns the algebraic, geometric, and harmonic means of two numbers; some historical background on the problem; and it gives a discussion about the relation between mathematical shapes and how these shapes appear in nature.

In the *Expositio* he then generalises how these different notions of means can be generalised giving different weights to each of the two elements. But there are also reflections on Newton's fluxions, derivatives and multiplications to conclude that polynomials are in fact transforms of monomials. These prepare the reader to accept flexibility (via parameterisation) in how we should look at mathematical definitions. This fundamental idea is applied when finally the 2D and 3D Gielis transforms are introduced to generate far reaching variants of the circle in 2D and the ball in 3D.

The more advanced mathematical elements are introduced in the *Determinatio*. Some slight generalisations for the Gielis transform can be generated or the transform can consist of combinations of Gielis transforms for different values of the parameters. Other excursions into the mathematics are dealing with Pythagorean trees, Lindemayer-systems, fractals, and R-functions. The latter are functions named after Rvachev who introduced them in 2D and that later were generalised by Fichera for 3D. These are multivariate functions whose sign only changes when one of its arguments changes sign. When the sign is interpreted as true or false, these functions can be applied in logic and define cobordisms. An object can be defined as for example the set of points $x\in\mathbb{R}^d$ for which $R(x)>0$. Highly complex objects can be described in this way.

The Gielis formula can transform the circle or the spiral into almost anything. That may lead to a complexity theory for topology: the oligomials (oligo is Greek for few). The complexity of a curve can be expressed by the degree of its polynomial equation. If it is rational, the polynomial degree is infinite, but the degree of the rational expression is a finite alternative. Similarly the (topological) complexity of an object can be expressed in terms of the Gielis transform (or transforms) required to represent it. In that sense, the circle is the simplest among all Lamé curves (its radius does not depend on the angle) but all other curves in that class have essentially the same complexity.

Furthermore the concepts of intrinsic and extrinsic measure are introduced. These notions point us to the idea of an elastic, anisotropic measure of length that can be applied on manifolds. This gives rise to all kinds of geometries (Minkowski, semirigid, Riemann, ...), curves on manifolds (with applications in phyllotaxis), etc. The Gielis transform which so far is only described in polar coordinates can also be described in Euclidean coordinates thanks to the connection with Chebyshev polynomials: $T_n(x)=\cos(n\theta)$ if $x=\cos\theta$. That completes the picture showing that Gielis is the natural extension of the historical line that connects Pythagoras and Lamé.

The *Constructio* lifts the machinery developed from botanical observations to a higher level, showing that it can me employed in a much broader (mathematical) context. While the rest of the text is more descriptive, here we find definitions and proper mathematical theorems. All the elements of the previous step are set to work in studying solutions of differential equations (boundary value problems) which can be applied to explain for example colour patterns in flowers. Differential geometry and definitions of curvature are also employed to explain natural shapes of manifolds i.e., surfaces with constant mean *anisotropic* curvature, where anisotropic refers to this notion of elastic length.

In the *Demonstratio* the actual demonstration is given that plant and other shapes in nature can be described with a Gielis transform of the circle and the spiral and they can be explained by satisfying some natural optimality condition. The keyword is that all this is possible thanks to an elastic anisotropic concept of what a unit (of length) is.

In the last step, the *Conclusio*, some reflections are attached to what has eventually been obtained. Is everything just based on a generalization of the Pythagorean theorem? Should plants be at the core of our world view, rather than physics or humans?

The book has many beautiful colour pictures of natural forms that illustrate the mathematical shapes generated by the relatively elementary superformula. In general the book is rather accessible, at least for those who can understand and appreciate the Gielis transform. Not much more than elementary mathematics is needed. Some chapters (in particular those of the *Expositio* and the *Constructio*) where text reaches out to the more mathematical aspects of some connections and applications, are more demanding. For the interested readers, a more in-depth analysis of the references is probably needed since not everything is fully explained. But since it often concerns summaries of published papers, it should be easy for the interested reader to look up the originals. The Gielis transform is clearly the North Star that shows the way through the book, but it is not always very clear where the excursions taken along the winding road will lead the reader to. For example the discussion of arithmetic, geometric, harmonic, and Gaussian means show up at different places but it is not very clear how they contribute to the understanding of the Gielis transform, except perhaps in the chapter on curvature. Finally I could spot some typos like on page 114 where $M(\sqrt{2)},1)$ has an extra bracket and $n'''=\sqrt{m''n'}$ is missing a prime in the second $n$, and on page 37, it is written "Sine and cosine are examples of simple polynomials...", but otherwise it is a well-groomed text.