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The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere,  consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).

A homotopy from a circle around a sphere down to a single point.

The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from an n-sphere in to a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology.

Picture of the month

Missing square puzzle animation.

The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures. It depicts two arrangements of shapes, each of which apparently forms a 13×5 right-angled triangle, but one of which has a 1×1 hole in it.

The puzzle works as the blue and red triangles are not similar so their hypotenuses are not parallel. In one arrangement the line bends one way, in one arrangement it bends the other way, and the gap between these bent lines is exactly one unit.

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Did you know...

• ...that the line separating the numerator and denominator of a fraction is called a solidus if written as a diagonal line or a vinculum if written as a horizontal line?
• ...that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type the complete works of William Shakespeare?
• ... that there are 115,200 solutions to the ménage problem of permuting six couples at a twelve-person table so that men and women alternate and are seated away from their partners?
• ... that mathematician Paul Erdős called the Hadwiger conjecture, a still-open generalization of the four-color problem, "one of the deepest unsolved problems in graph theory"?
• ...that the six permutations of the vector (1,2,3) form a hexagon in 3d space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
• ...that Ostomachion is a mathematical treatise attributed to Archimedes on a 14-piece tiling puzzle similar to tangram?
• ...that some functions can be written as an infinite sum of trigonometric polynomials? This sum is called the Fourier series of that function.
• ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
• ...that a regular heptagon is the regular polygon with the fewest number of sides which is not constructible with a compass and straightedge?

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