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Related Concept Videos

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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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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Coordination Number and Geometry02:57

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For transition metal complexes, the coordination number determines the geometry around the central metal ion. Table 1 compares coordination numbers to molecular geometry. The most common structures of the complexes in coordination compounds are octahedral, tetrahedral, and square planar.
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Crystal Field Theory - Tetrahedral and Square Planar Complexes02:46

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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...
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Chirality02:25

Chirality

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Chirality is a term that describes the lack of mirror symmetry in an object. In other words, chiral objects cannot be superposed on their mirror images. For example, our feet are chiral, as the mirror image of the left foot, the right foot, cannot be superposed on the left foot.
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Valence Bond Theory02:42

Valence Bond Theory

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Coordination compounds and complexes exhibit different colors, geometries, and magnetic behavior, depending on the metal atom/ion and ligands from which they are composed. In an attempt to explain the bonding and structure of coordination complexes, Linus Pauling proposed the valence bond theory, or VBT, using the concepts of hybridization and the overlapping of the atomic orbitals. According to VBT, the central metal atom or ion (Lewis acid) hybridizes to provide empty orbitals of suitable...
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Isotopy classification of three-dimensional embedded nets.

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Related Experiment Video

Updated: Apr 30, 2026

Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates
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Polyhedra, complexes, nets and symmetry.

Egon Schulte1

  • 1Northeastern University, Department of Mathematics, Boston, MA 02115, USA.

Acta Crystallographica. Section A, Foundations and Advances
|May 13, 2014
PubMed
Summary

This study classifies regular polyhedra and polygonal complexes in 3D space. It identifies 48 regular polyhedra and 25 regular polygonal complexes, detailing their structures and symmetry properties.

Area of Science:

  • Geometry
  • Crystallography
  • Combinatorics

Background:

  • Skeletal polyhedra and polygonal complexes are 3-periodic structures in Euclidean 3-space.
  • These structures possess unique geometric, combinatorial, and algebraic characteristics.
  • They can be conceptualized as 3-periodic graphs (nets) with added face structure, allowing for skew, zigzag, or helical configurations.

Purpose of the Study:

  • To systematically classify regular polyhedra and polygonal complexes in 3-dimensional Euclidean space.
  • To enumerate all possible regular polyhedra and polygonal complexes based on symmetry and structure.
  • To identify the edge graphs (nets) of these structures and their relationship to known crystallographic nets.

Main Methods:

  • Definition of regularity based on symmetry group transitivity on flags (incident vertex-edge-face triples).
Keywords:
apeirohedracrystal netspolygonal complexesregular polyhedra

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  • Enumeration of structures satisfying the regularity criteria.
  • Explicit identification of the edge graphs (nets) for all classified structures.

    • Main Results:

    • Identification of 48 regular polyhedra, comprising 18 finite polyhedra and 30 infinite apeirohedra.
    • Discovery of 25 infinite regular polygonal complexes that are not polyhedra.
    • Characterization of six infinite families of chiral apeirohedra with distinct flag orbits.

    Conclusions:

    • The classification provides a comprehensive catalog of regular polyhedra and polygonal complexes.
    • The identified nets offer insights into the structural diversity of periodic frameworks.
    • The study contributes to understanding the geometry and symmetry of complex polyhedral structures.