Geometry of the Hyperbolic Plane

Abstract

The Fifth Postulate of Euclid is equivalent to the statement that the sum of the angles in any triangle is equal to 180º (Hypothesis 5; see Section 2.1). From this we are able to deduce the possible regular and semiregular tilings of the plane (See Section 2.2). Now we are going to make the contrary assumption, that the sum of the angles in any triangle is less than 180º. (Recall Legendre’s and Saccheri’s Theorem 1.3.2, which says that either the sum of the angles in every triangle equals 180º or else the sum is always less than 180º.) This will lead to a very different conclusion about possible tilings. For example, four squares no longer fit together at their corners without leaving a gap. As we will see, however, it is possible for five squares to do so!