Department of Mathematics
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David Hilbert proved that the axiom system defining the classical Euclidean plane is categorical i.e. any two models of Euclidean Plane are, indeed, isomorphic). However, this does not mean that the usual RxR-model (where straight lines are given by the solution sets of linear equations) is the only model of the plane. In this talk, we show some "new" models of the Euclidean plane where the straight lines are given by the solution sets of non-linear algebraic curves (or portions of higher degree curves, as in, say, the Poincare model for the hyperbolic plane). These models were extensively used by Branko Grunbaum, Jan Mycielski and others (see e.g. [1], [2]). Because the lines do not appear 'straight', the usual intuition fails here and we need to use the power of formal algebra to verify the 'seemingly obvious' Euclidean properties of the models. Thus these non-linear models demonstrate:

• the difference between the logical content of an axiom system and its physical interpretations
• the need of algebra (an abstract process?) to demonstrate the geometric truths (a concrete reality?)
• help to destroy the faith in a preordained single model of an ordered Euclidean plane
• make the models of other unfamiliar geometries more palatable.

REFERENCES
[1] Adolf Mader, An Euclidean model for Euclidean geometry, Amer. Math. Monthly, Vol. 96,1989, pp. 43-49
[2] R. S. Millman and G. D. Parker, Geometry: A Metric Approach With Models, Springer-Verlag, New York, 1991.