  In this work (PartI) the qualitative analysis of statics and dynamics of
defects and textures in liquid crystals is performed with help of meanders and
train tracks. It is argued that similar analysis can be applied to 2+1 gravity.
More rigorous justifications are presentedin the companion paper (PartII).
Meanders were recently introduced by V.Arnold (Siberian J.of Math.
Vol.29,36(1988)). Train tracks were originally introduced by W.Thurston in 1979
in his Princeton U. Lecture Notes (http://www.msri.org/gt3m/) in connection
with description of self-homeomorphisms of 2 dimensional surfaces. Using train
tracks the master equation is obtained which could be used alternatively to the
Wheeler-DeWitt equation for 2+1 gravity. Since solution of this equation
requires a large scale numerical work, in this paper we resort to the
approximation of train tracks by the meandritic labyrinths. This allows us to
analyse possible phases (and phase transitions)in both liquid crystals and
gravity using Peierls- like arguments.
