## Book review: Beautiful Symmetry

[Disclosure of Material Connection: MIT Press kindly provided me with a free review copy of this book. I was not required to write a positive review. The opinions expressed are my own.]

Beautiful Symmetry: A Coloring Book about Math
Alex Berke
The MIT Press, 2020

Alex Berke’s new book, Beautiful Symmetry, is an introduction to basic concepts of group theory (which I’ve written about before) through symmetries of geometric designs. But it’s not the kind of book in which you just read definitions and theorems! First of all, it is actually a coloring book: the whole book is printed in black and white on thick matte paper, and the reader is invited to color geometric designs in various ways (more on this later). Second, it also comes with a web page of interactive animations! So the book actually comes with two different modes in which to interactively experience the concepts of group theory. This is fantastic, and exactly the kind of thing you absolutely need to really build a good intuition for groups.

The book is not, nor does it claim to be, a comprehensive introduction to group theory; it focuses exclusively on groups that arise as physical symmetries in two dimensions. It first motivates and introduces the definitions of groups and subgroups, using 2D point groups (cyclic groups $C_n$ and dihedral groups $D_n$) and then going on to catalogue all frieze and wallpaper groups (all the possible types of symmetry in 2D), which I very much enjoyed learning about. I had heard of them before but never really learned much about them.

One thing I really like is the way Berke characterizes subgroups by means of breaking symmetry via coloring; I had never really thought about subgroups in this way before. For example, consider a simple octagon:

An octagon has the symmetry group $D_8$, meaning that it has rotational symmetry (by $1/8$ of a turn, or any multiple thereof) and also reflection symmetry (there are $8$ different mirrors across which we could reflect it).

However, if we color it like this, we break some of the symmetry:

$1/8$ turns would no longer leave the colored octagon looking the same (it would switch the blue and white triangles). We can now only do $1/4$ turns, and there are only $4$ mirrors, so it has $D_4$ symmetry, the same as a square. In particular, the fact that we can color something with $D_8$ symmetry in such a way that it turns into $D_4$ symmetry tells us that $D_4$ is a subgroup of $D_8$. Likewise we could color it so it only has $D_2$ symmetry (we can rotate by $1/2$ turn, or reflect across two different mirrors; left image below) or $D_1$ symmetry (there is only a single mirror and no turns; right below). Hence $D_2$ and $D_1$ are also subgroups of $D_8$.

Along different lines, we could color it like this, so we can still turn it by $1/8$ but we can no longer reflect it across any mirrors (the reflections now switch blue and white):

This symmetry group (8 rotations only) is called $C_8$; we have learned that $C_8$ is a subgroup of $D_8$. Likewise we could color it in one of the ways below:

yielding the subgroups $C_4$, $C_2$, and $C_1$. Note $D_1$ and $C_2$ are abstractly the same: both feature a single symmetry which is its own inverse (a mirror reflection in the case of $D_1$ and a $180^\circ$ rotation in the case of $C_2$), although geometrically they are two different kinds of symmetry. $C_1$ is also known as the “trivial group”: the colored octagon on the right has no symmetry.

Anyway, I really like this way of thinking about subgroups as “breaking” some symmetry and seeing what symmetry is left. If you like coloring, and/or you’d like to learn a bit about group theory, or read a nice presentation and explanation of all the frieze and wallpaper groups, you should definitely check it out!