  A set of basic vectors locally describing metric properties of an arbitrary
2-dimensional (2D) surface is used for construction of fundamental algebraic
objects having nilpotent and idempotent properties. It is shown that all
possible linear combinations of the objects when multiplied behave as a set of
hypercomples (in particular, quaternion) units; thus interior structure of the
3D space dimensions pointed by the vector units is exposed. Geometric
representations of elementary surfaces (2D-sells) structuring the dimensions
are studied in detail. Established mathematical link between a vector
quaternion triad treated as a frame in 3D space and elementary 2D-sells prompts
to raise an idea of "world screen" having 1/2 of a space dimension but
adequately reflecting kinematical properties of an ensemble of 3D frames.
