INTRODUCTIONLet no one ignorant of

INTRODUCTION
Let no one ignorant of geometry enter this door.
ENTRANCE TO PLATO'S ACADEMY
Most people are unaware that around a century and a half ago a
revolution took place in the field of geometry that was as scientifically
profound as the Copernican revolution in astronomy and, in its impact,
as philosophically important as the Darwinian theory of evolution.
"The effect of the discovery of hyperbolic geometry on our ideas
of truth and reality has been so profound," writes the great Canadian
geometer H. S. M. Coxeter, "that we can hardly imagine how shocking
the possibility of a geometry different from Euclid's must have
seemed in 1820." Today, however, we have all heard of the spacetime
geometry in Einstein's theory of relativity. "In fact, the geometry
of the space-time continuum is so closely related to the non-Euclidean
geometries that some knowledge of [these geometries] is an essential
prerequisite for a proper understanding of relativistic cosmology."
Euclidean geometry is the kind of geometry you learned in high
school, the geometry most of us use to visualize the physical universe.
It comes from the text by the Greek mathematician Euclid, the Elements,
written around 300 B.C. Our picture of the physical universe
based on this geometry was painted largely by Isaac Newton in the late
seventeenth century.
Geometries that differ from Euclid's own arose out of a deeper
study ofparallelism. Consider this diagram oftwo rays perpendicular to