Matilde Marcolli is currently a mathematics professor at CalTech, working in the division of physics, mathematics, and astronomy. She is also interested in information theory and computational linguistics. Therefore, it is not a surprise to find a lot of mathematics, (particle) physics, and cosmology in this book, which she discusses in some detail. Less obvious is that this is also a book about art (mostly modern and often abstract paintings) and that she makes a remarkable (and rather successful) effort to link art and science and to point to parallels between both, sometimes by exploring the common underlying philosophical ideas.

However, in the introduction, she warns the reader that she is not an art critic or an art historian. And yet, it is a book about art, and it is also a book about mathematics and physics, but it is not a science book either. This book project grew out of a series of lectures that she has given in which she has explored the ideas that are bundled and extended in this book project. Its purpose is to explain modern science to the artists and to enlighten the art for scientists. It should prevent utterances from ignorant colleagues scientists like "a child can do this" when judging an abstract painting. It is unwise to give such an opinion if one is ignorant about the artistic background.

The title of the book Lumen Naturae captures that common idea in art and science where insight is obtained from exploring nature, its physics and the structure of matter. A form of Enlightenment that can be captured by both the artist and the scientist. It is the elegance of the structures, the beauty of physical laws, the understanding of space and time that is revealed to us by progressive insight. All that is there for us to explore, and that does not come from a superior being or transcendence. Artists and scientists are both spearheads pulling society forward to the next level.

To convey these ideas to the reader, Marcolli does not shy away from explaining detailing mathematical concepts. Her defence is that if desired, the reader can jump over the details and still catch the main idea, while the more savvy reader will only be stimulated to explore the quite extensive literature through which she guides the interested reader who is willing to dig somewhat into the concepts or to do some extra reading to understand the formulas or concepts that are mentioned here. On the other hand, she links the mathematics being discussed to many paintings that are reproduced in colour but on a relatively small scale (it is not a picture book). This link is obviously her (personal) interpretation which may differ from the reader's. However, unlike the scientific elements of this book, these paintings and the artists are not discussed in great detail. There is little or no historical context and there are no biographies of the artists. Here again she refers to the literature for the details.

So what exactly does Marcolli discuss to illustrate her objectives outlined above? Besides an introductory chapter, there are ten more chapters of variable lengths in which she brings a coherent story about some topic. The first one is somewhat unexpectedly about still life paintings. She chose this topic to illustrate the spacetime model. I will discuss this in somewhat more detail although it is a chapter with almost no mathematical discussion. Still life painting has survived from the Flemish painters of the 17th century till the cubist and dadaist artists of the 20th century. Time can be introduced in the scene by putting together fruits or flowers from different seasons on the same scene just like cubists tried to catch different viewpoints and different instances in the same picture or a music instrument, placed amidst food is an indication of time, since music can only be experienced as time passes. And there is of course the symbolic vanitas that places time in perspective of a human lifetime, and seashells on a table have rings that refer to the time it took to grow into their particular shape. Books and scientific instruments may refer to the flow of knowledge as time evolves. In some paintings, the objects depicted become abstractions, detached from their environment just like science is more and more relying on abstraction. With some imagination, the elegant curves in paintings by Cézanne can be seen as spacetime curved by gravity, while surrealists like for example Dalí depict the relativity and flexibility of time more explicitly as a melting clock.

As I said, there is little mathematics in that first chapter, but the theme of space and time is a recurrent one, repeated in several of the subsequent chapters. For example chapter three, discusses space, form and structure. Structure is illustrated with the mathematical number system that starts with integers, then rationals, reals, complex numbers,... and with Klein's Erlangen geometry project. There are many meanings in mathematics connected to "space": from vector space, to topological space, and from projective geometry to the piecewise linear space of computer graphics, there are knots, graphs and networks, metric and normed spaces, fractals, measure theory and tilings. In other words, this chapter is like a selected survey of mathematics where art is mentioned, but it is not the main topic.

In the next two chapters she discusses entropy, randomness, information, and complexity. This gives her the occasion to give some critique on the pamphlet by Rudolf Arnheim from 1971 in which he considers the notions of order and disorder in modern art. Indeed some abstract paintings seem to place objects randomly in space or they look rather chaotic (think of Jackson Pollock), but there are many kinds of randomness, and even in chaos there is some information to be found. The seemingly random cosmic background radiation gives important information about how our universe was formed, and stochastic processes like random walks follow precise rules.

The void may be represented by painting a black square or even a white square on a white background. In any case, 20th century avant-garde postwar artists were looking for a way to represent it, just like physicists tried to understand empty space, which turned out to be not empty at all since they are facing phenomena like dark matter and dark energy. Empty space definitely has some shape because gravity is defining its geometry. It is also very dynamical with particles popping in and out of existence as quantum mechanics shows (this is a pretext for Marcolli to tell us more about Feynman diagrams, quantum field theory and quantum gravity, and the different interpretations of quantum theory).

Then there are some shorter chapters of a more theoretical nature. One on the geometry of numbers (Von Neumann definition of natural numbers, primes, their randomness, Ulam spirals and the zeta function); one on laws of physics (the standard model of particle physics); and one on the shape of our universe (from the Kepler's laws of the solar system to cosmic topology). The book ends with two chapters that are somewhat different from the previous ones. There is a larger chapter about a 20th century avant-garde futurist movement with an anarchist and socialist undertone, and there is a shorter chapter about illustrations in books, which leads to her personal watercolor paintings over pages in her mathematical notebook.

The futurist chapter starts with the train as the power engine bringing society to a new destination, Similarly networks of roads connect people over larger distances, while networks of electrification and other services and provisions make that people clump together in cities. Mechanisation and specialisation should give individuals more time for leisure and entertainment. The importance of science is realised and being a scientist becomes a respected profession. Gradually the human body is also conceived as a mechanical machine or factory that can be studied by physics and chemistry. Science rules and there is no limit to what can be achieved, including the conquest of space and understanding the cosmos.

It may be clear from this review that in this book Marcolli can (and does) connect mathematical and physical laws to many paintings but the main content of the book and most detailed information is given about the former. I can imagine that the reader-scientist will enjoy to see how many paintings can be linked to abstract theory and that the same kind of philosophy, the same ideas, and even the same emotional sensation can be associated with a theory or a set of formulas as with some visualisation produced by a creative artist. An artist (or any other reader) who is not skilled in the theoretical aspects that are forwarded by Marcolli, may have a harder time to understand all the formulas, but he or she can well be inspired to look up some literature, and here Marcolli does an excellent job in guiding all kind of readers through the extensive list of references after every chapter. The book is carefully edited, but the glossy pages, necessary to give numerous colour reproductions of the paintings, makes it heavy to hold while reading. I spotted a few minor typos. For example page 261: "modifi ed" and "defi ned" and p. 316: "lass" instead of "less", but these are really minor and do not harm an otherwise excellent typography. The pictures are always close to the reference in the text, which saves a lot of paging back and forth. Also the index is extensive and carefully compiled to easily retrieve the many subjects that were discussed in the text.