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The socle of a group G is the subgroup generated by its minimal normal subgroups. For example, the symmetric group S_4 has two nontrivial normal subgroups: A_4 and ...

The geometry of the Lie group R semidirect product with R^2, where R acts on R^2 by (t,(x,y))->(e^tx,e^(-t)y).

The space join of a topological space X and a point P, C(X)=X*P.

A Jordan algebra which is isomorphic to a subalgebra.

Johnson solid J_(86).

Johnson solid J_(88).

Let a, b, and c be the sides of a spherical triangle, then the spherical defect is defined as D=2pi-(a+b+c).

The image on the Riemann sphere of any circle under a complex rational mapping with numerator and denominator having degrees no more than n has length no longer than 2npi.

A three-dimensional knot spun about a plane in four dimensions. Unlike suspended knots, spun knots are smoothly embedded at the poles.

A square grid is a grid formed by tiling the plane regularly with squares.