Related Experiment Video
Updated: Mar 8, 2026
06:35
Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates
Published on: February 15, 2016
8.5K
Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
Kengo Nishio1, Takehide Miyazaki1
1National Institute of Advanced Industrial Science and Technology (AIST), Central 2, Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan.
Scientific Reports
|January 18, 2017
Summary
This study simplifies representing complex structures like crystal grains and galaxies. We developed a new method to shorten data codes for polyhedral tilings, making them easier to manage.
Area of Science:
- Computational geometry
- Materials science
- Crystallography
Background:
- Polyhedral tilings model diverse structures from materials to cosmic formations.
- Previous work encoded tiling arrangements into numerical codewords for reconstruction.
- The prior encoding method generated redundant data, complicating handling.
Purpose of the Study:
- To reduce redundancy in codewords representing polyhedral tilings.
- To develop a more efficient method for describing complex structures.
- To unify the representation of polyhedral tilings and polytopes across dimensions.
Main Methods:
- Shifting focus from polyhedra to shared 2D regions as fundamental building blocks.
- Developing a new theory for codeword generation based on these 2D regions.
- Generalizing the 2D approach to higher-dimensional polytopes.
Main Results:
- Identified and removed redundant digits from polyhedral tiling codewords.
- Achieved codeword length reduction by up to 50%.
- Established a unified theory applicable to various dimensions.
Conclusions:
- The new method provides significantly shorter, more manageable codewords for polyhedral tilings.
- This enhanced representation benefits both human interpretation and computational processing.
- The unified theory offers a consistent framework for diverse dimensional structures.
Related Concept Videos
01:24The Seven Crystal Systems: Overview
50
Crystals with various point group symmetries belong to different crystal classes, which are synonymous terms. Despite being in the same class, crystals may have distinct shapes, like cubes and octahedra. There are 32 three-dimensional point groups, all of which are systematically divided into seven crystal systems.The basic cubic crystal system, exemplified by NaCl, features orthogonal vectors (α = β = �� = 90°) of equal lengths (a = b = c). When specific...
50
02:46Crystal Field Theory - Tetrahedral and Square Planar Complexes
49.3K
Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
49.3K
01:29Crystallographic Point Groups
22
Crystallographic point groups represent the various symmetry operations that can occur within crystals. They are unique in that at least one point will always remain unchanged during these actions. For instance, consider the triclinic system. This system, devoid of any axis or plane of symmetry, aligns with the C1 and Ci point groups.where Cᵢ is characterized solely by a center of inversion.Contrastingly, the monoclinic system introduces an element of symmetry. This system with one plane...
22
02:22Structures of Solids
20.3K
Solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern are known as crystalline solids. Metals and ionic compounds typically form ordered, crystalline solids. A crystalline solid has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Amorphous solids or non-crystalline solids (or, sometimes, glasses) which lack an ordered internal structure and are randomly arranged. Substances that...
20.3K
01:24Finding Volume Using Cross-Sectional Area
125
For solids whose cross-sectional areas vary in a predictable way, volume can be determined by integrating these areas along an axis perpendicular to the slices. This approach is particularly useful for polyhedral solids, where classical geometric formulas may not be immediately applicable. A tetrahedron provides a clear example of how cross-sectional integration can be applied to a three-dimensional object with continuously changing geometry.Consider a tetrahedron with height h and a base that...
125
02:33Lattice Centering and Coordination Number
13.8K
The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
Imagine taking a large number of identical...
Types of Unit Cells
Imagine taking a large number of identical...
13.8K