In the study of the structure of substances in recent decades,
phenomena in the higher dimension was discovered that was
previously unknown. These include spontaneous zooming (scaling
processes), discovery of crystals with the absence of translational
symmetry in three-dimensional space, detection of the fractal
nature of matter, hierarchical filling of space with polytopes of
higher dimension, and the highest dimension of most molecules of
chemical compounds. This forces research to expand the formulation
of the question of constructing n-dimensional spaces, posed by
David Hilbert in 1900, and to abandon the methods of considering
the construction of spaces by geometric figures that do not take
into account the accumulated discoveries in the physics of the
structure of substances. There is a need for research that accounts
for the new paradigm of the discrete world and provides a solution
to Hilbert's 18th problem of constructing spaces of higher
dimension using congruent figures. Normal Partitions and
Hierarchical Fillings of N-Dimensional Spaces aims to consider the
construction of spaces of various dimensions from two to any finite
dimension n, taking into account the indicated conditions,
including zooming in on shapes, properties of geometric figures of
higher dimensions, which have no analogue in three-dimensional
space. This book considers the conditions of existence of polytopes
of higher dimension, clusters of chemical compounds as polytopes of
the highest dimension, higher dimensions in the theory of heredity,
the geometric structure of the product of polytopes, the products
of polytopes on clusters and molecules, parallelohedron and
stereohedron of Delaunay, parallelohedron of higher dimension and
partition of n-dimensional spaces, hierarchical filling of
n-dimensional spaces, joint normal partitions, and hierarchical
fillings of n-dimensional spaces. In addition, it pays considerable
attention to biological problems. This book is a valuable reference
tool for practitioners, stakeholders, researchers, academicians,
and students who are interested in learning more about the latest
research on normal partitions and hierarchical fillings of
n-dimensional spaces.
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