This paper has two aspects, one mathematical, the other in mathematics education. Mathematically, it investigates the bifurcation set K of the linear differential equation x = Ax on the plane, when the real 2x2 matrix A is regarded as a point in four‐dimensional space R4. The relation with mathematics education is to provide an exposition showing in detail how the material combines ideas from analysis, topology, geometry and algebra in an elementary and meaningful setting that is often not found in undergraduate courses, when these subjects are treated separately. In the hope of cultivating spatial awareness, we replace R4without essential loss of information by the three‐sphere S3, which we then project stereographically into three‐space R3, to draw two‐dimensional sketches of the projection of the set K?S3. This projection divides R3into regions, in each of which the differential equation displays a structurally stable phase‐portrait, and we emphasise the role of ‘degenerate’ cases in allowing each such portrait to change to another qualitatively different one. Use is made of the Jordan canonical form, which is derived for 2x2 matrices through a method avoiding the conceptual difficulties of the usual n X n treatment. The associated changes of variable give a good example of a ‘concrete’ group action on R4, and the isotropy subgroups are calculated; the orbits are the levels of a Morse‐theoretic function and their topology as three‐manifolds is described.