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In this paper, we introduce a new class of discrete orthogonal transforms (DОT) defined on lattices of integer elements of quadratic fields. The method of synthesis of such transforms essentially uses the specifics of the representation of integer quadratic elements in the so-called quasi-canonical number systems. This article, which presents the results of the first part of the author's research, deals exclusively with problems related to binary number systems in quadratic fields. We also consider the issues of synthesis of fast algorithms of the introduced and the possibility of their application to the analysis of fractal (or self-similar) objects. We also consider the issues of synthesis of fast algorithms of the introduced methods and the possibility of their application for the analysis of fractal (or self-similar) objects.

Eugene Wigner’s much-discussed notion of the “unreasonable effectiveness of mathematics” as applied to describing the physics of empirical reality is simultaneously both trivial and profound. After all, the relevant mathematics was (in the first instance) originally developed in order to be useful in describing empirical reality. On the other hand, certain aspects of the mathematical superstructure have by now taken on a life of their own, with at least some features of the mathematical superstructure greatly exceeding anything that can be directly probed or verified, or even justified, by empirical experiment. Specifically, I wish to raise the possibility that the real number system (with its nevertheless pragmatically very useful tools of real analysis and mathematically rigorous notions of differentiation and integration) may nevertheless constitute a “wrong turn” (a “sub-optimal” choice) when it comes to modelling empirical reality. Without making any definitive recommendation, I shall discuss several reasonably well-developed alternatives.

This article reviewed and analyzed the complexes for numerical artifacts from archaeological cultures the first Millennium of our era from the Northern Kama region, the Urals and Western Siberia. Identified system for accurate counting and accounting for calendar time, based on the numbers six and seven are reflected on the decoration of these products. For comparison, was analysied of Ob-Ugric, Altai Turks, Kets and Nenets folklore texts, for identify a numeric complexes in them. It turned out that the most common case is a numeric complex "seven - six". This complex in epics of the Altai Turks ("Maaday Kara" for example) dominates its frequency. The context in which it used, most often associated with the sky and annual way of the Sun. As the prayer of Torum, the count is seven-month cycle started with the birth of the "new sun" at the time of the winter solstice, compared with the birth of the "calf of the elk-sun" (by Patkanov), which ended in Elijah's day (beginning of August), a fracture of the weather on "winter" and a sharp reduction in the light of the day. The mapping of these features, numerical algorithms on the artifacts (in fact the ring calendars) and in the texts, allows to say that the reference system on the basis of which was constructed a numerical model on the products and the texts were natural realities.