Matt Parker is a "standup-mathematician", i.e., stand-up comedian and a maths communicator and he loves to mix both. So you may expect this book to be fun as well. In fact, that is a major point he wants to make with this book. Many people experience mathematics as annoyingly difficult, extremely boring, and exclusively for nerds. However, what Parker takes as a starting point is that mathematics is fun to do and that people should start doing mathematics for the fun of it. One should not necessarily necessarily do mathematics to solve particular problems. That mathematics is useful and turns out to have practical applications is a fortunate side effect. As Richard Hamming formulated it in 1980: Mathematics is unreasonably effective.
That being said and given the title of the book, Parker promises a lot, but to better understand everything that is needed to work in the fourth dimension, one has to start from the beginning and build things up step by step. So he starts on a long journey meandering through a very diverse set of math related topics that leads to sphere packing in hypercubes and other objects in the fourth and higher dimensions. But it does not stop there. Parker goes on to other advanced topics such as the zeta function, Cantor's aleph numbers and Gödel's incompleteness theorems.
Of course there are many things to make and do, and it is often recreational mathematics, but there are also surprisingly many serious mathematical topics embedded in this seemingly relaxing narrative and, most important for the objective of the book, they are very well explained for the layman. The many illustrations certainly help. Some of these are pictures or computer generated (no colour plates though), but many of them, and also many of the displayed formulas are handwritten. Not out of necessity, because there are also (not many) perfectly typset mathematical formulas. But, as I experienced it as a reader, it is as if the author is writing them down for you. It creates some "intimacy" or "nearness" between the author and the reader, like he is personally writing it down, trying very hard to make you personally understand. I think it is a smart idea.
Inpossible to enumerate all the topics touched upon in a review. It starts with number systems, slicing pizzas in unusual ways, plane geometry, origami and flexagons, platonic solids, hyperbolic knitting, sphere packing, prime numbers, knots, graph theory, and colouring problems in the plane and on 3d surfaces. Here, halfway the book, we meet the fourth dimension for the first time with the hypercube. I think, there is also some historical timeline underneath the subsequent topics discussed following to some extent the development of mathematics throughout the centuries (although not very strict and with many excursions). So: enter the algorithmic and computer issues, somewhat halfway the book, and the festival of topics continues with algorithms and recursion applied in card tricks, towers of Hanoi, building a computer (i.e., circuit logic) with tumbling domino sequences, the zeta function, geometry in higher dimensions, coding theory, number systems (rational, irrational, computable, and normal), to end up with the mathematics of Cantor, Hilbert and Gödel. With this enumeration (which is offensively incomplete) I try to reflect the mixture of the recreational stuff, the things that need to be constructed, tricks you can surprise your friends with, puzzles that have to be solved (some answers are provided in an appendix), but also how these often have an historical origin and how they are rooted in some mathematical theory that turns out to be "unreasonably effective" to solve the poblem.
The number of books with recreational and popular mathematics, explaining "everyday mathematics" for a broad public is booming. However, seldon did I come accross a book of this volume that is so entertaining, yet touching on many of the nice mathematics as well, without ever being pedantic. With his many lectures in classrooms and his stage performances, Parker certainly got a lot of experience, so that it knows all too well how to fascinate and entertain his public. Even if some classical subjects are treated, it is almost always with a twist, just going beyond where other books stop. For example Reuleaux polygons and volumes get a lot of attention besides classic Platonic solids. Of course the reader will not learn all of the underying mathematics itself, but he or she will at least know what some mathematicians are doing, and what it means when they talk about graphs, codes, higher dimensional spaces, the zeta function, etc.
Unfortunately, there is no subject index. There are so many topics that are covered, and the narrating style makes some unexpected bridges between subjects, which makes it difficult to locate a specific topic with only the chapter titles available. So after you finished reading the book, it might be difficult to retrieve a subject you remember, but that was only mentioned in a few lines somewhere. For example Édouard Lucas, a French mathematician from the 19th century is mentioned at several places, but it would be difficult to find out all these instances, and to which results or puzzles he was related.