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On the possibility of the realization of a tringle in a 3D space with a scalar product
2009jaz | D.G. Pavlov, G.I. Garas  //  , Bauman Moscow State Technical University, Moscow, Russia, Electrotechnical Institute of Russia, Moscow, Russia

The isometric and conform symmetry groups are of exceptional importance in mathematics and physics that can scarcely be overestimated. The former class of symmetry relates to the invariant of the element of length of the metric space, but the latter class of symmetry relates to the angle invariant. If there exists a continuation of this chain of the symmetry groups, isometric, conform etc, then there should exist objects tightly connected with this more generic class of symmetry group, which are common to call as tringles or, without any relation to the dimension, as ingles, and, to show the dimension m exceeding 3 -- as m-ingles. It is not possible to have ingles in the Euclidian or pseudo-Euclidian spaces, but, in contrast, it is possible to have ingles in the space with the dimension exceeding 2 and having scalar polyproducts, with the number of the vector arguments also above 2. In the present work, we build a real tringle accurate within a function of one real variable, and we derived its relation to the coordinates of the vectors in the space with a scalar triproduct, where the space is tightly connected with the Bervald-Moor 3D space, which is justified to be called as 3D-time. So, the existence of the tringles, which have been supposed to exist, is rigorously proven that implies a real possibility for m-ingles, with $m3$, to exist.


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