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Embedding parallelohedra into primitive cubic networks and structural automata description.

Acta crystallographica. Section A, Foundations and advances·2020

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Tilings by hexagonal prisms and embeddings into primitive cubic networks.

Mikhail M Bouniaev¹, Nikolay P Dolbilin², Mikhail I Shtogrin²

¹School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, One University Boulevard, Brownsville, TX 78520, USA.

Acta Crystallographica. Section A, Foundations and Advances

|September 2, 2020

Summary

This study classifies how hexagonal prisms tile 3D space within cubic networks. It explores all combinatorial embeddings for these regular prism tilings.

Keywords:

embeddings hexagonal prisms parallelohedra primitive cubic networks tilings

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Area of Science:

Crystallography and Materials Science
Geometric Topology

Background:

Understanding space-filling polyhedra is crucial in materials science and crystallography.
Regular hexagonal prisms offer unique geometric properties for tiling three-dimensional space.

Purpose of the Study:

To systematically discuss and classify all possible combinatorial embeddings.
To analyze the integration of regular hexagonal prisms into primitive cubic networks.

Main Methods:

Combinatorial analysis of geometric structures.
Classification of spatial embeddings within cubic lattices.
Exploration of tilings using congruent and parallel regular hexagonal prisms.

Main Results:

A comprehensive catalog of combinatorial embeddings has been developed.
The study identifies and categorizes distinct ways hexagonal prisms can form cubic networks.
All possible arrangements are discussed and classified.

Conclusions:

The classification provides a fundamental framework for understanding 3D space-filling with hexagonal prisms.
This research contributes to the geometric and crystallographic understanding of complex network structures.