Turkish Journal of Physics, 2012, Vol. 36, No 3, P. 461 – 464.
Sergey G. Fedosin
Perm, Perm Region, Russia
The scale dimension discovered in the theory of infinite nesting of matter is studied from the perspective of physical implementation of well-studied four-and n-dimensional geometric objects. Adding the scale dimension to Minkowski four-dimensional space means the necessity to use the five-dimensional spacetime.
Keywords: scale dimension; multi-dimensional space; theory of infinite nesting of matter.
Intensively developed in recent years the theory of infinite nesting of matter, which studies the entire hierarchy of space systems, faced the necessity of introducing scale dimension into the everyday life of science. The purpose of this article is to analyze the properties of the new dimension and its relationship with geometric theory of n-dimensional spaces.
The number of basic dimensions in physics is determined by the number of degrees of freedom or independent variables that set the location of a physical body and its elements, considered as points in a given frame of reference. The number of dimensions or degrees of freedom gives the dimensionality of the spacetime used. By adding the scale dimension to the four-dimensional spacetime, we obtain the five-dimensional manifold, which includes the usual spacetime. According to the order of historical understanding this can be written as (3 +1 +1)-space, where in the first place the spatial dimensions are reflected, and then the dimensions of time and scale. In terms of geometry it is convenient to consider the axes of the all dimensions perpendicular to each other.
In contrast to geometry, physics does not deal with mathematical points, but with specific material objects. Infinite division of these objects in parts does not lead to the emergence of mathematical points, but instead new objects of the smaller size appear. Although these objects seem more and more similar to the points, they contain an infinite number of matter carriers and the smallest field quanta. If the motion inside of material objects is regarded as motion along the scale axis, then this axis is a special dimension of space. During the motion along the scale axis transition between the scale levels of matter takes place and we can see that the objects of lower levels become parts of the objects of higher levels of matter. As a result the scale dimension can be represented as the dimension characterizing the complexity of space systems, in the sense of their composition of lower order systems.
The theory of infinite nesting of matter and the idea of the scale dimension acquired their modern form owing to the works by Robert Oldershaw  (self-similar cosmological model), Sergey Sukhonos [2-3] (scale similarity of space systems) and Sergey Fedosin [4-5] (similarity of matter levels).
Spatial relations are perceived by the observer through the shape of objects and their configuration relative to each other, and temporal relations are determined as the changes of shapes and configurations, the angular and linear shifts. Owing to the fact that objects at different levels of matter evolve and change relatively independently from each other and the speed of processes and the characteristic sizes and masses of objects differ significantly, the scale dimension is considered as a new dimension of spacetime. At the same time shapes and configurations of many three-dimensional objects at different levels of matter in many respects are similar to each other. This is fixed by relations of similarity for various physical quantities and SPФ-symmetry, which states the similarity of physical laws at different levels of matter .
In the field of research that uses large-scale transformations and spatial dimensions, the scale dimension can be studied by geometrical methods, and thus it is in the scope of geometry. In four-dimensional spacetime as the fourth dimension not the time t itself is taken, but the product , where c is the speed of light (since for the exact spacetime measurements electromagnetic waves are commonly used). Section of such spacetime that is perpendicular to the time axis at some time point, defines the hyperplane in the form of three-dimensional geometrical space at a given time. If in the three-dimensional space an object moves to the future upcoming events through the point of the present time, in the four-dimensional spacetime the movement from time 1 to time 2 is seen as a trace increasing in size from point 1 to point 2 that is associated with the object. The possibility of observing such a trace in the coordinate system means that the motion can be described as a set of three-dimensional images of an object at different times.
The introduction of the scale dimension takes into account that at different levels of matter the rates of time flow, regarded as the speeds of typical processes for similar objects, differ from each other. However, it is possible to introduce the total coordinate time, based for example on periodic processes in an electromagnetic wave. In this case, the hyperplanes which are perpendicular to the scale axis will determine the single four-dimensional spacetime at different levels of matter. By analogy with the four-dimensional Minkowski spacetime, now we can introduce the five-dimensional coordinate system and write down for each world event its vector as follows: , where is a three-dimensional radius-vector, − a coordinate along the scale axis.
Scale dimension allows us to understand and prove the possibility of existence of real four-dimensional bodies in five-dimensional spacetime. For constructing such bodies the method of induction may be used, based on the following reasonings:
– One edge is a line segment bounded by two points (vertices);
– Three edges in a plane give a triangle with three vertices, in the form of a face;
– Six edges with four vertices in three-dimensional space give a tetrahedron (triangular pyramid) with four faces;
– Ten edges with five vertices, with ten faces and the five tetrahedrons in four-dimensional space represent an object called pentachoron (4-simplex).
For a n-dimensional tetrahedron or a simplex with increasing n-dimensionality of space to one, the number of vertices increases, too, the number of vertices equals k = n + 1. Each edge is a connection between two arbitrary vertices of the tetrahedron, so that the number of edges depends on the number of vertices according to the formula for combinations of two with the k vertices. To determine the number of faces of n-dimensional tetrahedron in the form of triangles we need to find combinations of three with the k vertices. In general, the formula for r combinations of k elements is expressed by means of factorials and has the form:
Equation (1) allows us to find the number of edges, faces, r-polyhedrons as part of n-dimensional tetrahedron.
For a n-dimensional cube (hypercube), we can assume that its number of vertices, depends on the dimensionality of space n according to the formula: , where dimensionality of the point space is zero, the linear space has dimensionality 1, the flat space has dimensionality 2 etc. Consequently, a tesseract (a tetracube) as a hypercube in four-dimensional space must have 16 vertices, and according to (1) 120 edges. If we remove from this number the internal edges between the vertices of different faces, as well as the diagonal edges lying in the plane of faces, then only 32 external edges would remain, which are responsible for the convexity of a tesseract as a geometric figure. Besides a tesseract has 8 external three-dimensional cubic faces and 24 external two-dimensional square faces which limit it in the four-dimensional space. The common name for n-dimensional bodies with flat sides is the polytope, for four-dimensional bodies the term polychoron is accepted, whereas for three-dimensional bodies the term polyhedron is used, and for two-dimensional bodies the term polygon is used.
In space of n dimensions it is possible to construct a variety of figures, and find their properties, including the n-dimensional angles, areas and volumes. In particular, the volume of the n-dimensional sphere can be expressed by means of its radius R and the gamma function, according to :
This formula was known already in 19th century in the writings of the Swiss mathematician Ludwig Schläfli .
Currently, the branch of mathematics which studies geometric objects in space of n dimensions is developed so well that all possible objects are divided into classes according to symmetry properties . There are online resources, demonstrating the projection of multidimensional bodies on a plane or in three-dimensional space [9-10], obtained by computer modeling.
From the mathematical point of view, any space of smaller dimensionality, which is part of the space of higher dimensionality, is a hyperplane on which it is possible to make projections of points of n-dimensional body. If a simplex or a tesseract is oriented strictly along the fourth spatial dimension, then at each point of a three-dimensional hyperplane perpendicular to the axis of the fourth dimension, there will be a tetrahedron, or a cube, respectively. Similarly, if we dissect a tetrahedron or a cube perpendicular to their bases, then on the section plane (two-dimensional hyperplane) there will be a triangle, the size of which depends on the cut-point, or a square. While the four-dimensional body is passing through the three-dimensional space one would expect the sudden appearance and disappearance of the projections of this body and changing of their size. As in everyday life such examples are not commonly found, it is seen that the fourth spatial dimension does not reveal itself directly.
From the stated above it follows that the scale axis, discovered in the hierarchy of space systems is a physical embodiment of the fourth spatial dimension . A set of three-dimensional objects at different levels of matter represents a four-dimensional body oriented along the scale axis, so that each cross section of this axis represents a three-dimensional hyperplane, which may contain corresponding three-dimensional objects.
In contrast to the mathematical idealization, the objects of a natural system because of their limited total number are located on the scale axis with unequal probability, but unevenly, grouped into separate levels of matter. In addition, physical properties of objects’ substance are changed at different levels of matter what is usually ignored in geometry. At those levels of matter, where gravitational forces prevail, the dominating space objects have spherical shape.
Any set of objects, consisting of spherical galaxies, globular star clusters, individual stars, planets, nucleons, etc., forms a definite single four-dimensional spherical body, stretching along the scale axis as along the fourth spatial coordinate. This four-dimensional body is discovered not by its projections as the result of its passing through our three-dimensional world, but in the opposite situation, when the observer himself is moving relative to this body and examining its components in various hyperplanes. Thus, the scale dimension confirms the possibility of real existence of four-dimensional geometric objects in nature. This enables the application of mathematical methods for n-dimensional spaces for the study of composite physical objects at different scale levels of matter.
1. Oldershaw Robert L. Self-Similar Cosmological Model: Introduction and Empirical Tests // International Journal of Theoretical Physics . – 1989. – Vol. 28 . – No. 6. – P. 669-694. http://www.amherst.edu/~rlolders/OBS.HTM .
2. Сухонос С.И. Взгляд издали // Знание-сила. – 1981. – № 9. – С. 31-33.
3. Сухонос С. И. Масштабная гармония Вселенной. – М.: Новый центр. – 2002. – 312 с. ISBN 5-89117-096-5. http://www.trinitas.ru/rus/002/a0209004.htm .
4. Fedosin S.G. Fizika i filosofiia podobiia: ot preonov do metagalaktik. – Perm: Style-MG, 1999. – 544 p. ISBN 5-8131-0012-1.
5. Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009. – 844 p. ISBN 978-5-9901951-1-0. http://sergf.ru/knen.htm .
6. Араманович И.Г., Гутер Р.С., Люстерник Л.А. Математический анализ. Дифференцирование и интегрирование. – М.: ГИФМЛ, 1961. – 309 c.
7. Schläfli, Ludwig (1901) , Graf, J. H., ed. (in German), Theorie der vielfachen Kontinuität, Republished by Cornell University Library historical math monographs 2010, Zürich, Basel: Georg & Co., ISBN 978-1-4297-0481-6.
8. Coxeter H. S. M. Regular Polytopes. – 3-rd ed. – NY: Dover Publications, 1973. – 321 p. ISBN 0-486-61480-8.
9. The fourth dimension. Comments on behalf of the mathematician Ludwig Schläfli. – http://dimensions-math.org/ (Retrieved 2011-01-02).
10. Newbold Mark. HyperSpace Polytope Slicer. – http://dogfeathers.com/java/hyperslice.html (Retrieved 2010-12-30).
11. Wikiversity, Scale dimension. – http://en.wikiversity.org/wiki/Scale_dimension .