UDC 531.55+514.85 Tachyon Gas as a Candidate for Dark Matter Y. A. Rylov Institute for Problems in Mechanics, Russian Academy of Sciences 101-1, Vernadskii Ave., Moscow, 119526, Russia In the physical geometry (i.e. in geometry, described completely by its world function) identical geometric objects have identical description in terms of the world function. <...> As a result spacelike straight segment is a three-dimensional surface even in the space-time geometry of Minkowski. <...> Tachyons have two unexpected properties: (1) a single tachyon cannot be detected and (2) the tachyon gas can be detected by its gravitational influence. <...> The pressure of the tachyon gas depends on the gravitational potential and does not depend on temperature. <...> Key words and phrases: discrete geometry, tachyon, dark matter, dark energy, rotation curves. 1. <...> Introduction geometry is a monistic construction, which is described completely by the only quantity (world function), the only possibility may take place g1(σ) = g2(σ) = g(σ). (1) The conventional (Riemannian) space-time geometry is pluralistic. <...> This problem is not considered in the general relativity, which uses different geometries for different regions of the space-time. <...> The only geometric object which is considered in the general relativity is the world line of a free pointlike body. <...> It is supposed that the world line of a free body is a geodesic. <...> Let a physical of the world function and only in terms of the world function. <...> This conventional definition of the world line of a free body agrees with the definition (1) for timelike world lines. <...> However it disagrees with (1) for spacelike world lines, because in the physical geometry a spacelike straight segment is not a one-dimensional line. <...> Our conceptual logical consideration disagrees with the general opinion that the |v| (3) For spacelike distances the triangle axiom (3) is not fulfilled, and the set of points R satisfying equation (2) is 3-dimensional. 2 > c2 segment of straight is a one-dimensional set in any geometry. <...> Degenerated ellipsoid, where the point P3 on its surface coincides with one of focuses is by definition segment T[P0P1] = ELP0P1P1 of straight between focuses P0, P1. ellipsoid is <...>