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Structures of Solids02:22

Structures of Solids

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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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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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Ionic Crystal Structures02:42

Ionic Crystal Structures

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Ionic crystals consist of two or more different kinds of ions that usually have different sizes. The packing of these ions into a crystal structure is more complex than the packing of metal atoms that are the same size.
Most monatomic ions behave as charged spheres, and their attraction for ions of opposite charge is the same in every direction. Consequently, stable structures for ionic compounds result (1) when ions of one charge are surrounded by as many ions as possible of the opposite...
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Crystallographic Point Groups01:29

Crystallographic Point Groups

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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...
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Theorems of Pappus and Guldinus: Problem Solving01:12

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Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
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Metallic Solids02:37

Metallic Solids

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Metallic solids such as crystals of copper, aluminum, and iron are formed by metal atoms. The structure of metallic crystals is often described as a uniform distribution of atomic nuclei within a “sea” of delocalized electrons. The atoms within such a metallic solid are held together by a unique force known as metallic bonding that gives rise to many useful and varied bulk properties.
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Fabrication of Three-Dimensional Graphene-Based Polyhedrons via Origami-Like Self-Folding
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Platonic solids generate their four-dimensional analogues.

Pierre Philippe Dechant1

  • 1Institute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, England.

Acta Crystallographica. Section A, Foundations of Crystallography
|October 18, 2013
PubMed
Summary
This summary is machine-generated.

This study constructs four-dimensional regular polytopes from 3D Platonic solids using spinorial geometry. This reveals novel links between 3D and 4D symmetries with potential applications in physics and materials science.

Keywords:
Clifford algebrasCoxeter groupsMcKay correspondencePlatonic solidsfour-dimensional geometrypolytopesquaternionsrepresentationsroot systemsspinorssymmetriestrinities

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Area of Science:

  • Geometry
  • Mathematical Physics
  • Group Theory

Background:

  • Regular convex 4-polytopes are four-dimensional analogues of Platonic solids.
  • Understanding their symmetries is crucial in various scientific fields.
  • Previous constructions often lacked a unified geometric framework.

Purpose of the Study:

  • To demonstrate a novel construction of regular convex 4-polytopes from 3D Platonic solids.
  • To elucidate the 'mysterious' symmetries of these 4D polytopes through a 3D spinorial perspective.
  • To establish a link between 3D and 4D geometries and their underlying symmetries.

Main Methods:

  • Utilizing the Cartan-Dieudonné theorem to generate rotations from reflective symmetries of 3D Platonic solids.
  • Employing a Clifford algebra framework to interpret 3D spinors within a 4D Euclidean structure.
  • Interpreting 3D spinors as vertices in 4D space for polytope construction.

Main Results:

  • A simple construction of 4D polytopes (16-cell, 24-cell, F4 root system, 600-cell) is presented.
  • The construction reveals the symmetries of these polytopes as almost trivial from a 3D spinorial viewpoint.
  • The spinor construction naturally generates rank-4 Coxeter groups and applies to other root systems, explaining structures like the grand antiprism and snub 24-cell.

Conclusions:

  • A novel link between 3D and 4D geometries is established via spinorial constructions.
  • The findings offer insights into symmetries relevant to (quasi)crystals, viruses, and high-energy physics theories.
  • This approach provides a unified framework for understanding complex polyhedral symmetries across dimensions.