Institute for Problems in Mechanics, Russian Academy of Sciences,
101-1, Vernadskii Ave., Moscow, 119526, Russia.
email: rylov@ipmnet.ru
Web site: http://rsfq1.physics.sunysb.edu/~rylov/yrylov.htm
or mirror Web site: http://www.ipmnet.ru/~rylov/yrylov.htm
Updated December 8, 2011
Metric approach to geometry
admits one to construct a physical geometry, i.e. geometry which is described
completely in terms and only in terms of the distance function \rho , or in
terms of the world function \sigma =0.5\rho
^{2}. A discrete geometry is a special case of physical geometry, when there
exist an elementary length \lambda _{0} and all distances in the discrete
geometry are larger, than the elementary length. A discrete geometry is
obtained as a generalization of the proper Euclidean geometry. To produce such
a generalization, a logical reloading in the description of the Euclidean
geometry is to be produced. It means that the Euclidean geometry begins to be described in terms of
the world function and only in terms of the world function. Such a description
contains general geometric relations, which are valid in all physical
geometries, and special relations, describing properties of the Euclidean world
function. Replacing the Euclidean world function \sigma _{\mathrm{E}} by the
world function \sigma _{\mathrm{d}} of the discrete geometry in all general
geometric relations and ignoring special relations, one obtains general
relations for the discrete geometry. As far as the form of \sigma _{\mathrm{d}} is supposed to be known,
the special relations for the discrete geometry are not needed
.Text of paper in English (pdf, ps) and in Russian (pdf, ps)