Mathematicians consider some mathematics to be beautiful, and there has indeed been scientific research measuring that mathematicians showed increasing brain activity in the frontal cortex when seeing mathematical formulas. This brain activity is similar to what is observed when people see a beautiful painting or listen to music. So there must be some link between mathematics and art. Several mathematicians are known to be also great musicians of produce visual art and most mathematicians have a soft spot for a certain kind of visual works of art that have some mathematical flavour. The Bridges Organization promotes the interaction between mathematics and visual art, music, architecture, and it organizes an annual conference around these ideas.

Several books are available in which mathematics is linked to art. Some are coffee table books, mainly consisting of pictures, others are philosophical essays with some illustrations. This beautifully illustrated book by Ornes is about the "mathematical art" of nineteen contemporary visual artists. Its format is a careful balance between a sketch of the artist, a short discussion of his or her work, and an easily accessible introduction to the "mathematics behind the art". Some of the artists were once represented at these Bridges conferences and some are at the MoMath museum. For obvious reasons Ornes is mainly interested in visually pleasing work, with a strong mathematical background. Most of the artists are still living and Ornes is quoting some of them, which shows that he has interviewed or at least spoken with these artists.

Divided into four parts, Ornes discusses 19 artists and their work. A telegraphic survey of the contents and the list of artists is given at the end of this review. Some of the artists are professional mathematicians, but others are just inspired by mathematics. The selection of the art is quite diverse. It can be monumental sculptures, weaving, computer generated curves, quilts, 3D-printed objects, wood carving, or crocheting. The work presented is selected to serve the purpose of this book. The artists can have other work of a quite different nature, or it can be early work and they may have moved more recently to a different kind of work. The appendices about the mathematics that served as an inspiration is diverse as well. There is pi and phi (the golden ratio), the Fibonacci numbers and primes, the Pythagoras theorem, set theory and infinity, geometry with classical Platonic and Archimedian solids, fractals and non-Euclidean geometry, topology and the Moebius band, space filling curves and tilings, computer science with complexity theory, algebra with symmetries and groups, and more. An appendix is linked to one artist, but there are cross references to other artists as well. Clearly the selection of topics and artists is very diverse, but this is only a very small section from a vast domain showing a growing interest for this kind of interaction between mathematics and art.

The size of the book is nearly square (9 x 9.5 inches) and it is printed on glossy paper. So it can serve as a coffee table book but it has more to read than it has to see. The cover is black with a white design by Bathsheba Grossman. I could not find the reference in the book for the cover picture (although all other pictures are properly credited). The picture is actually a dodecahedron based design for a lamp that is 3D printed by Materialise. It is also an illustration on Grossman's Wikipedia page (3 Sept 2019). Grossman has also a Klein bottle opener, i.e., an operational bottle opener in the shape of a Klein bottle.

I like the book very much. Unlike some other popularizing math books, it literally illustrates the beauty of mathematics, and makes this beauty accessible to non-mathematicians. Hopefully they will be triggered by the beauty of the pictures, to also read the mathematical appendices, which are written at a level that can be read and understood by anyone.

To conclude, a quick summary of the 19 cases that are collected in four parts.

• Part 1: Making sense of the universe.
  • The art of pi - John Sims, who among other work, produces quilts like coloured QR codes where colours are defined by the digits of pi.
  • Geometry in motion - John Edmark designs objects that require dynamics, and here one should consult his website to understand and appreciate his work. The mathematics here deals with the Fibonacci numbers and the golden section.
  • The proof is in the painting - Crockett Johnson has paintings that are inspired by graphical proofs of the Pythagoras theorem.
  • One to one to infinity - Dorothea Rockburne produces abstract art, sculptures and installations, that draw inspiration from set theory. The mathematical appendix discusses set theory and different orders of infinity and gives a proof that there are infinitely many primes.
  • The many faces of geometry - George Hart makes sculptures by weaving several identical components together that shapes Platonic solids. The mathematics is about regular and classical polyhedra and their stellations.
• Part 2: Stranger shapes
  • Space and beyond - Bathsheba Grossman makes sculptures that are periodic minimal surfaces or projections of the 120-cell in 3D space.
  • The consequences of never choosing - Helaman Ferguson has monumental sculptures like an umbilic torus decorated with a Peano space filling curve. This and other space filling curves are discussed in the appendix.
  • The tangled, torturous universe of fractals - Robert Fathauer produces fractal organic sculptures. Fractals are introduced in the appendix but is continued in the next case.
  • The mystical and the mathematical - Melina Green focusses on the Mandelbrot set and generates an image of the set that suggests the shape of a Buddha.
  • The equations of nature - David Bachman is a topologist and his art was originally the result of describing nature by equations and then generate artificial objects that look very natural. More recently his work visualizes more abstract ideas. The appendix is discussing topology.
• Part 3: Journeys
  • The wandering mathematician - Robert Bosch produces a piecewise linear Jordan curve that is denser at some places which, from a distance, gives the impression of a grey-scale reproduction of for example the Mona Lisa of whatever other image one cares to choose. The construction of the curve is based on a traveling salesman algorithm which is discussed in the appendix together with the P versus NP problem.
  • The curves in the machine - Anita Chowdry is inspired by the Lissajous curves and produces some steampunk instrument to generate such complex curves.
  • The algorithms of art - Roman Verostko (born in 1929) has embraced the first computers and designed algorithms to produce graphical art. The appendix discusses some elements from complexity theory and quantum computing.
  • Projections - Henry Segerman has work inspired by stereographic projection, producing a Riemann sphere that is the projection on the sphere of for example a regular grid in the plane.
• Part 4: (near) Impossibilities
  • Following yarn beyond Euclid - Daina Taimina is known for her crochet work representing hyperbolic geometry. The appendix explains and illustrates rather well hyperbolic geometry with the Poincaré disk or half plane models.
  • Bounding infinities - Frank Farris produces symmetric images and transitions in wall paper groups using deformed photographs as a stamp. Some of his work is discussed in Creating symmetry.
  • Connections - Carlo Séquin is a computer scientist who produces complex large mathematically inspired sculptures. The appendix discusses symmetry and group theory.
  • Math and the woodcarver's magic - Bjarne Jespersen is a wood carver who produces a wooden object where a sphere is capture inside a polyhedral structure that, unable to take it out or put it in. The appendix is about tessellations that cover the plane.
  • The possibilities - Eva Knoll uses many different media to express herself among which weaving where some relative prime repetition of patterns creates some extra pattern on top of the underlying one. The appendix gives a discussion of algebra and all its different meanings in mathematics.