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The chapter includes the summary of unified theory of space–time transformations of fundamental laws of nature in the form of three “great theorems” by Galileo, Lorentz and Poincare. This theory, in light of the initially existing inherent contradictions between three “great geometries” by Euclid, Lobachevsky and Riemann, reflects the dual features of three-dimensional, perspective and projective perception of the world around us through human eyes. Three “great theorems” related to the velocity of light propagation, gravitational radii and Planck’s quantities, reveal the essence of unified theory of space–time transformations. Its significance is considered for describing the total energy of oscillatory and wave processes as well as for formulating the fundamental laws of its development in the Universe. In light of unified theory of space–time transformations, these three “great theorems” have the dual form of space–time laws. The last ones unite Newtonian physics and thermodynamics with electromagnetic, gravitational, weak and strong fundamental interactions in regard to special nodal and bifurcation, perspective and projective reference points, which serve as the origin of coordinates of unified metric system.

In the chapter, the prerequisites and basic concepts of unified theory of space–time transformations of fundamental laws of nature in the form of three “great theorems” by Galileo, Lorentz and Poincare are considered in detail in the context of three “great geometries” by Euclid, Lobachevsky and Riemann using simple and obvious cases of description of free oscillations of string and the Earth. The author provides formulations and solutions of wave equation for free oscillations of string and radial, spheroidal, torsional free oscillations of the Earth as well. Considering these formulations and solutions from the point of view of new visible event horizons associated with the velocity of light propagation, gravitational radii and Planck’s quantities, the author comes up with the formulation of unified theory of space–time transformations of fundamental laws of nature. From the point of view of exact sciences—mathematics, geometry and physics, the unified theory of space–time transformations of fundamental laws of nature in the form of three theorems by Galileo, Lorentz and Poincare is supposed to be related to further development of infinitely large and infinitesimal in integral and differential calculus in the transition to new visible event horizons, determined by velocity of waves and light propagation, gravitational radii and Planck’s quantities. The chapter also includes the consideration of applied aspects of unified theory of space–time transforms of fundamental laws of nature, in particular, the relation of the Earth’s free oscillations to standing internal gravitational waves that determine the fractal hierarchy of dissipative structures of the Earth, which is associated with many aspects of its geological structure and evolution.

The principal undertaking of the methods described in this work is to reveal multi-ordinal structures of gravitational instability—or dissipative structures of the Earth—and identify their typomorphic elements. The range of methodological approaches aimed at solving this issue is rather wide and includes: the analysis of seismic tomography data; maps of global geoid anomalies and gravitational potential; graphical and analytical methods for the decomposition of relief, figures representing the Earth’s gravitational field; the deciphering of space and aerial images; morphometric, structural-geomorphological and structural-geological investigations.

The manifestation of the wave properties of matter in the processes of spontaneous structuring of density-unstable masses, originally described as a natural phenomenon characteristic of both the nonequilibrium density-unstable laboratory physical models, and the geological planetary layers and upper horizons of the Earth’s crust, is the basis for new approaches to geological mapping and to the interpretation of geological and geological-geophysical data at all stages of their processing.

The invariance of the revealed regularity in the spatial distribution of different-type hierarchies of the Earth’s gravitational instability structures throughout its geological history, along with the data discussed above on the distribution of mantle plumes and the sialic protocrust with respect to global structures such as G-hexagons in the eastern hemisphere and the L-hexagons in the western, confirms the emergence of these global structures, which determined the fundamental asymmetry of the Earth and divided it into the Indo-Atlantic and Pacific tectonic segments during the early Archean, simultaneously with the isolation of the liquid core, i.e. more than 3500 Ma. At the same time, paleogeodynamic reconstructions show that generational structures emerge during each subsequent stage of the Earth’s evolution, along with disordered instabilities. Moreover, these increasingly massive generational structures are presented predominantly by G-hexagon type in the eastern hemisphere and the L-hexagon type in the western. With the consideration of the relationship between the size and the stratification depth of the structures of gravitational instability, this means that with the course of geological time, the increasingly deep layers of the Earth’s crust and mantle became gravitationally unstable with respect to the underlying shells. From a physical point of view, the increase in the thickness of gravitationally unstable layers over time can easily be explained by the nonequilibrium cooling of the Earth through its surface, i.e., by external sinks of heat. With the increase in the size of the Earth’s gravitational instability structures, the associated tectonic regimes also change, ranging from fine-cellular intraplate (greenstone and granulite, protoaulacogenic and protogeosynclinal) to marginal-plate and global riftogenic regimes. In the terms of geodynamics, the horizontal component in the movement of the Earth’s crust becomes especially critical when the magnitude of its gravitational instability increases. The scale of its destruction increases, and a gradation from small-plate and intraplate tectonics to the tectonics of lithospheric plates can be observed.

Among the many structural features appearing throughout the Universe, including the Earth and its surrounding layers, the attention of contemporary scientists is drawn to the spontaneous structuring of these natural systems and the geometric commonality of the emerging structural forms, i.e., in terms of their cellular (square, hexagonal, polygonal) geometry. The structural similarity of these forms is revealed to be independent of the nature and scale of their manifestation, a factor having already been emphasised by many researchers from a philosophical perspective (V. V. Piotrovsky, M. A. Sadovsky, Yu. M. Pushcharovsky, V. E. Khain, L. I. Krasny, etc.). This has led the scientific community towards the view that the external manifestations of the spontaneous structuring and redistribution of masses in nonequilibrium density distribution in natural systems may be explained in terms of a transphenomenal physical mechanism. This perception is strengthened still further by turning to the applied natural scientific fields of fluid dynamics and thermodynamics, oceanology, meteorology and geophysics, where the same geometric commonality of spontaneous structuring can be seen as underlying otherwise apparently scattered examples.

The natural science of geology is in the first place a descriptive science; nevertheless, in this field a large body of accumulated empirical material is unified at the level of the fundamental laws of nature in the language of modern physics and mathematics. In this connection, we can state that with the development of various physical and mathematical methods for describing the processes of spontaneous structuring of density-unstable media, their wave nature becomes more and more evident regardless of the causes (whether thermodynamic or dynamic) of this instability.

From the standpoint of the material manifestation of wave properties in the processes of spontaneous structuring of unstable masses, the essence of the new general geological paradigm follows from the discovery of the colossal global structures of the Earth’s gravitational instability of G-hexagon type, exhibiting immersion in the centre and emersion in the marginal parts of the deep mantle substrate in the eastern hemisphere, and L-hexagon type, with emersion in the centre and immersion in the marginal parts of the deep mantle substrate in the western hemisphere, the basis of which development is explained in terms of wave processes.