synesthetics-in-motion

NON - EUCLIDEAN GEOMETRIES

The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. In Euclidean geometry, if we start with a line l and a point A, which is not on l, then we can only draw one line through A that is parallel to l. In hyperbolic geometry, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometry, parallel lines do not exist.

In 1829, the Russian Nikolai Ivanovich Lobachevsky published a treatise of hyperbolic geometry. For this reason, hyperbolic geometry is sometimes called Lobachevskian geometry. He was the first to unabashedly claim that Euclidean geometry was not the only geometry, nor the only conceivable geometric structure for the universe. Lobachevsky termed Euclidean geometry, "ordinary geometry," and this new hyperbolic geometry, "imaginary geometry." However, the possibility still remained that the axioms for hyperbolic geometry were logically inconsistent.
Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same).

Many of the drawings of M. C. Escher embody concepts of non-Euclidean geometry, as expressed in the Poincarémodel of Lobaczewski's plane to Escher's "Circle Limit"







These woodcuts elaborate on illustrations that one can find in various monographs about non-Euclidean geometry.
A casual inspection of Escher's Circle Limit woodcuts suggests a relationship to plane symmetric patterns. Like the latter, a finite motif undergoes an infinite repetition which displays at least combinatorial regularity. Nevertheless, one finds few reflections, glide reflections an gyrations, at least in the Euclidean sense.

It turns out that non-Euclidean geometry requires us to measure distance in a different way. One may compare Poincare''s model, the hyperbolic plane, to a map of the earth. In a typical Mercator projection one finds small distances stretched profoundly near the poles (indeed such maps must omit the poles lest they stretch infinitely up and down). In Poincare''s model one has just the opposite: large hyperbolic distances appear compressed as one travels toward the boundary.

The richness of Poincare''s geometry, the hyperbolic plane, arguably exceeds Euclidean geometry itself. The theory of symmetric patterns gives one indication of this. While symmetric patterns on the Euclidean plane all possess one of 17 symmetry times, symmetric patterns on the hyperbolic plane show inexhaustable variety. Escher's Circle Limit woodcuts show just 4 examples.

posted by Mikela @ 1:37 PM