Department of Mathematics
Server

Many geometry books claim that the validity of the Pythagorean theorem in the absolute plane (i.e. without assuming the parallel postulate) implies the parallel postulate and hence makes the geometry euclidean (see e.g. [1], [3]). However, the euclidean distance is defined in terms of the additive group law + and the Pythagorean theorem itself is stated using this group law: (AB)² + (BC)² = (AC)². Hence the question is whether there is another group law f(x, y) definable in the hyperbolic plane such that the modified version of Pythagorean theorem is now valid with this as the underlying addition. Abraham Ungar has given such an example of a loop with this property. This operation is non-associative. In this talk we will present his loop operation and its connection with the geometry of the Poincare Model. It is not yet known whether there is a group operation in which the analog of Pythagorean theorem is still valid in the hyperbolic plane.

REFERENCES
[1] Dobbs, David E., A single instance of the Pythagorean theorem implies the parallel postulate. Internat. J. Math. Ed. Sci. Tech. 33 (2002), no. 4, 596--600.
[2] Ungar, A. A., Hyperbolic Pythagorean Theorem, Proc. A.M.S. 106, Issue 38, (1999) 759-763.
[3] E.C. Wallace and S.F. West, Roads to Geometry, Prentice Hall, N.J. 1992. (see pp 312-314).