We all take for granted that mathematics can be used to describe the world, but when you think about it this fact is rather stunning. This article explores what the applicability of maths says about the various branches of mathematical philosophy.

The sense of going on a journey with a brilliant and entertaining companion, of feeling like you are never sure why the conversation is veering in this new direction, yet being confident that there is a good reason for it, is the strongest sense which I got from this lovely new book by Marcus du Sautoy.

If you like mathematics because things are either true or false, then you'll be worried to hear that in some quarters this basic concept is hotly disputed. In this article Phil Wilson looks at constructivist mathematics, which holds that some things are neither true, nor false, nor anything in between.

Symmetry abounds: the wallpaper, your chair, even your own body. Familiar types of symmetry include reflection in a line and rotation about a point. Creating a repeating pattern by translating a core segment to a new place, common in wallpaper, also counts as a symmetry, as does switching without the use of a mirror from an anticlockwise segment to one otherwise identical but oriented clockwise.

How do you do it? Horizontally from side to side, or perhaps criss-cross, producing a series of Xs running up your feet? Towards the end of The shoelace book, its author Burkard Polster raises a troubling question. Despite all the here-today, gone-tomorrow vagaries of fashion, and in spite of the huge variety of shoe styles available to us in this golden age of footwear, why does almost everyone lace their shoes in one of these two ways?

It turns out your mum was wrong after all: you can judge a book by its cover. This book has a colourful, detailed, and tantalising cover adorned with portraits of people you may or may not know. Who are they, and what do they have to do with numbers at work, and the culture in which we live?

Leonhard Euler was one of the greatest and most prolific mathematicians of all time. His work was of vital importance to a bewildering variety of fields, many of which he himself created