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A set of basic vectors locally describing metric properties of an arbitrary2-dimensional (2D) surface is used for construction of fundamental algebraicobjects having nilpotent and idempotent properties. It is shown that allpossible linear combinations of the objects when multiplied behave as a set ofhypercomples (in particular, quaternion) units; thus interior structure of the3D space dimensions pointed by the vector units is exposed. Geometricrepresentations of elementary surfaces (2D-sells) structuring the dimensionsare studied in detail. Established mathematical link between a vectorquaternion triad treated as a frame in 3D space and elementary 2D-sells promptsto raise an idea of "world screen" having 1/2 of a space dimension butadequately reflecting kinematical properties of an ensemble of 3D frames.

Splitting of 3d quaternion dimensions into 2d-sells and a "world screen technology"