# What is Poincare's "Fourth Geometry"?

In Science and Hypothesis, Poincare cryptically describes a "Fourth Geometry." Can anyone clarify what he is talking about? Is there a standard name for this geometry?

The Fourth Geometry.—Among these explicit axioms there is one which seems to me to deserve some attention, because when we abandon it we can construct a fourth geometry as coherent as those of Euclid, Lobatschewsky, and Riemann. To prove that we can always draw a per- pendicular at a point A to a straight line AB, we consider a straight line AC movable about the point A, and initially identical with the fixed straight line AB. We then can make it turn about the point A until it lies in AB produced. Thus we assume two propositions—first, that such a rotation is possible, and then that it may continue until the two lines lie the one in the other produced. If the first point is conceded and the second rejected, we are led to a series of theorems even stranger than those of Lobatschewsky and Riemann, but equally free from contradiction. I shall give only one of these theorems, and I shall not choose the least remarkable of them. A real straight line may be perpendicular to itself.

It is the Minkowski plane, the lightlike lines are “perpendicular to themselves”, see e.g. Stachel, Poincaré and the Origins of Special Relativity. To get this geometry, one needs to expand what “geometry” means. The classical geometries of Euclid, Lobachevsky, and Riemann have what is now called sign-definite metrics, which define real distances. In contrast, the Minkowski plane has an indefinite metric with signature $$1,1$$, so the “distances” can be imaginary (along the timelike lines). It does have constant curvature, like the classical geometries, but is only homogeneous, not isotropic, spacelike and timelike directions can not be interchanged by Lorentz transformations (which preserve the metric). It can also be realized as the Lorentz geometry on the hyperboloid of one sheet, and is sometimes denoted $$H^{1,1}$$.