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AI Overview: Polyhedral skeletal electron pair theory refers to a conceptual framework that discusses the arrangement of electron pairs in polyhedral structures, particularly emphasizing how these arrangements determine molecular geometries. This theory builds upon the Valence Shell Electron-Pair Repulsion (VSEPR) Theory, asserting that electron pairs, including both bonding and lone pairs, influence the shape of molecules by minimizing repulsion. In this context, the spatial arrangement of electron pairs around a central atom in polyhedral molecules can be analyzed, leading to an understanding of the resulting geometric structures.

  • Dual Polyhedron

    In geometry, every polyhedron is associated with a dual polyhedron, where the vertices of one correspond to the faces of the other and vice versa. The dual of a dual polyhedron returns the original polyhedron. Among the Platonic solids, the tetrahedron is self-dual, while the cube and octahedron, as well as the dodecahedron and icosahedron, are dual pairs.

  • Electron Configuration

    Electron configuration is the arrangement of electrons within an atom, crucial for understanding the structure of the Periodic Table. There are four types of orbitals: s, p, d, and f, each capable of holding a maximum of 2 electrons. The shape and energy levels of these orbitals vary, with s orbitals being spherical, p orbitals resembling dumbbells, d orbitals shaped like four-leaf clovers, and f orbitals having complex shapes. The filling order of these configurations is consistent and influences electron behavior around the nucleus.

  • Lone Pair

    Lone pairs are unbonded pairs of electrons that reside in the valence shell of an atom, distinct from bonding pairs. These high-energy electron pairs can participate in bond formation, particularly in nucleophilic attacks on electrophiles. Furthermore, lone pairs influence the molecular shape, as they occupy more spatial area compared to bonding electrons, leading to increased repulsion between them.

  • Polyhedroid

    The term refers to polychoron, a four-dimensional analogue of a polyhedron.

  • VSEPR Theory

    The Valence Shell Electron-Pair Repulsion (VSEPR) Theory explains the shape of molecules and ions based on the repulsion between electron pairs around the central atom. It emphasizes that only electron pairs around the central atom need to be considered, and they arrange themselves to minimize repulsion. Lone pairs exert greater repulsion than bonding pairs, influencing the overall molecular shape, regardless of whether the bonds are single, double, or triple.

  • Covalent Bonding

    This page redirects to the topic of Covalent Bond, which involves the sharing of electron pairs between atoms.

  • Archimedean Solids

    Archimedean solids are a category of convex polyhedra that are characterized by having regular polygonal faces and vertex arrangements that are identical for each vertex. These solids exhibit a high degree of symmetry and are discovered in various geometric contexts, making them significant in the study of polyhedral geometry.

  • Electron Crystallography

    Electron crystallography is a technique used to determine the arrangement of atoms in solids utilizing a transmission electron microscope (TEM). It is particularly effective for studying 2D crystals, viral capsids, and dispersed proteins, where X-ray crystallography is less successful due to its requirement for large 3D crystals. This method, invented by Aaron Klug, who received the Nobel Prize in Chemistry for his work in 1982, has been used to resolve high-resolution structures such as bacteriorhodopsin and the light-harvesting complex.

  • Convex Regular 4-Polytope

    A convex regular 4-polytope, or polychoron, is a four-dimensional convex shape that is the 4D counterpart of Platonic solids in three dimensions and regular polygons in two dimensions. First described by Ludwig Schläfli in the 19th century, there are six distinct convex regular polychora, each formed by regular Platonic solid cells. These polytopes possess properties that can be summarized in terms of vertices, edges, faces, cells, and symmetry groups, all represented by Coxeter groups. Key examples include the 5-cell (Pentachoron), Tesseract (8-cell), and 600-cell (Hexacosichoron). Their boundaries are topologically equivalent to a 3-sphere, which relates to Euler's polyhedral formula in four dimensions.

  • Archimedean Solids

    The Archimedean solids are a group of polyhedra that possess identical vertices and are composed of two or more types of regular polygons. They are known for their geometric properties and symmetrical structures, making them a significant topic in the study of mathematics and polyhedra.