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The Seven Crystal Systems: Overview01:24

The Seven Crystal Systems: Overview

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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...
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Gauss's Law: Spherical Symmetry01:26

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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Gauss's Law: Planar Symmetry01:27

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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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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...
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Gauss's Law: Cylindrical Symmetry01:20

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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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.
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Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials
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A five-dimensional geometric uniformity framework for spherical diamond grids.

YuanZheng Duan1, JiangMeng Li1, Lei Shi1

  • 1Institute of Software Chinese Academy of Sciences, Beijing, China.

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|March 13, 2026
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Summary
This summary is machine-generated.

This study introduces a new framework to evaluate geometric uniformity in Discrete Global Grid Systems (DGGS). The icosahedron-based grid demonstrates superior uniformity, crucial for accurate digital Earth data analysis.

Keywords:
Diamond gridDiscrete global grid systemGeometric quality evaluationGrid uniformitySpherical convolutional neural network

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Area of Science:

  • Geographic Information Science
  • Geomatics Engineering
  • Computational Geometry

Background:

  • Discrete Global Grid Systems (DGGS) are essential for digital Earth but face geometric non-uniformity challenges.
  • Existing quality assessments are insufficient for diamond-based grids, especially regarding angular and distance uniformity.
  • Accurate DGGS are vital for reliable geospatial data representation and analysis.

Purpose of the Study:

  • To propose a comprehensive evaluation framework for spherical diamond grids.
  • To extend Goodchild's criteria with angular and distance uniformity metrics.
  • To compare the uniformity of cube, octahedron, and icosahedron-based diamond DGGS.

Main Methods:

  • Developed an integrated five-dimensional evaluation system (shape, topology, size, distance, angle).
  • Systematically compared three diamond DGGS derived from cube, octahedron, and icosahedron.
  • Constructed a Spherical Residual Network for Diamond Grids (SResNet-DG) for validation.

Main Results:

  • The icosahedron-based grid showed optimal uniformity across all five dimensions.
  • The octahedron-based grid exhibited significant angular distortion, proving less uniform than the cube-based grid.
  • Grid uniformity strongly correlated with SResNet-DG classification performance.

Conclusions:

  • The proposed framework effectively evaluates spherical diamond DGGS uniformity.
  • The icosahedron-based grid is the most suitable for DGGS applications requiring high geometric fidelity.
  • Enhanced grid uniformity directly improves the performance of machine learning models in geospatial tasks.