Generally speaking, and taking into account that pure translational operations are not strictly considered as symmetry operations, we can say that finite objects can contain themselves, or may be repeated (excluding translation) by the following symmetry elements:

• The identity operation is the simplest symmetry element of all -- it does nothing! But it is important because all objects at the very least have the identity element, and there are many objects that have no other symmetry elements.

• The reflection is the symmetry operation that occurs when we put an object in front of a mirror. The image is found perpendicular to the reflection plane and equidistant from that plane, on the opposite side of the plane. The resulting object can be distinguishable or indistinguishable from the original, normally distinguishable, as they cannot be superimposed. If the resulting object is indistinguishable from the original, is because the reflection plane is passing through the object.

• The inversion operation occurs through a single point called the inversion center. Each part of the object is moved along a straight line through the inversion center to a point at an equal distance from the inversion center. The resulting object can be distinguishable or indistinguishable from the original, normally distinguishable, as they cannot be superimposed. If the resulting object is indistinguishable from the original, is because the inversion center is inside the object.

• The rotation operations (both proper and improper) occur with respect to a line called rotation axis. a) A proper rotation is performed by rotating the object 360°/n, where n is the order of the axis. The resulting rotated object is always indistinguishable from the original. b) An improper rotation is performed by rotating the object 360°/n followed by a reflection through a plane perpendicular to the rotation axis. The resulting object can be distinguishable or indistinguishable from the original, normally distinguishable, as they cannot be superimposed. If the resulting object is indistinguishable from the original, is because the improper rotation axis is passing through the object.

In addition to the name of the symmetry elements, we use graphical and numerical symbols to represent them. For example, a rotation axis of order 2 (a binary axis) is represented by the number 2, and a reflection plane is represented by the letter m.

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The association of elements of rotation with centers or planes of symmetry generates new elements of symmetry called improper rotations.

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Combining the rotation axes and the mirror planes with the characteristic translations of the crystals (which are shown below), new symmetry elements appear, with some "sliding" components: screw axes (or helicoidal axes) and glide planes.
 

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The symmetry elements of types center or mirror plane relate objects in a peculiar way; the same way that our two hands are related one to the other: they are not superimposable. Objects which in themselves do not contain any of these symmetry elements (center or plane) are called chiral and their repetition through these elements (center or plane) produce objects that are called enantiomers with respect to the original ones. The mirror image of one of our hands is the enantiomer of the one we put in front of the mirror.

The mirror image of either of our hands is the enantiomer of the other hand. They are objects not superimposable and as they do not contain (in themselves) symmetry centers or symmetry planes, are called chiral objects.

Chiral molecules have different properties than their enantiomers and so it is important that we are able to differentiate them. The correct determination of the absolute configuration or absolute structure of a molecule (differentiation between enantiomers) can be done in a secure manner through X-ray diffraction only, but this will be explained in another chapter 




Molecular chirality is critically important in both chemistry and biology for several reasons, particularly due to how it influences molecular interactions and the behavior of substances. Here's why it's so significant:  

1. Biological Interactions:

• Enzyme-Substrate Specificity, Many biological processes rely on the specific recognition of molecules by enzymes, receptors, and other biomolecules. Enzymes, for example, often recognize and catalyze reactions involving only one enantiomer (one of the two mirror-image forms of a chiral molecule). If the enantiomer is incorrect, the reaction may not proceed or could lead to harmful products.

• Chirality in Metabolism. Our bodies are highly selective in how they use chiral compounds. For example, the amino acids in proteins are almost exclusively left-handed (L-amino acids), while sugars in nucleic acids are right-handed (D-sugars). This specificity ensures the correct folding and function of proteins, enzymes, and nucleic acids.

2. Pharmacology and Drug Design:

• Different Biological Effects. A chiral molecule's two enantiomers can have drastically different effects in the body. One enantiomer of a drug might be effective in treating a disease, while the other could be less effective, inactive, or even toxic. A well-known example is thalidomide, a drug that caused birth defects because one enantiomer was beneficial, but the other caused harm.

• Pharmacokinetics and Pharmacodynamics. The body's absorption, distribution, metabolism, and excretion (ADME) of drugs can differ for each enantiomer. In some cases, one enantiomer might be metabolized more quickly, or might interact with a different set of receptors, which affects the overall efficacy and safety of a drug.

3. Optical Activity:

• Light Polarization: Chiral molecules can rotate the plane of polarized light. This property is important in many analytical techniques, including polarimetry, where the degree of rotation is used to determine the concentration of a chiral substance in a solution. The direction of rotation (clockwise or counterclockwise) can also distinguish between the two enantiomers of a compound.

4. Chemical Synthesis:

• Stereoselectivity in Synthesis. In chemical synthesis, controlling chirality is essential for creating molecules with the desired activity. Often, one enantiomer of a molecule is needed for a particular function, and selective synthesis methods must be employed to ensure that only one enantiomer is produced in high yield.

• Chiral Catalysis. Certain catalysts are used in the synthesis of chiral molecules to control the formation of one enantiomer over the other. This is important in producing pure pharmaceuticals, agrochemicals, and other specialty chemicals.

5. Molecular Recognition and Binding:

• Receptor Binding. In both chemical and biological systems, chiral molecules can bind to specific receptors or proteins in a manner similar to a key fitting into a lock. If the chirality is incorrect, the molecule may not fit the receptor properly, preventing it from having the desired effect. This selectivity is crucial for things like neurotransmitter function, immune responses, and enzyme catalysis.

6. Environmental and Ecological Impact:

• Chirality in Nature: Many natural products, such as amino acids, sugars, and alkaloids, are chiral. The chirality of these compounds is often essential for their biological roles. For instance, the chirality of the molecules in the cell membrane affects how cells communicate and interact with the environment, which can influence cellular processes and organismal behavior.

In conclusion, molecular chirality is important because it governs how molecules interact with biological systems, affects drug design and efficacy, and plays a crucial role in many chemical reactions and synthetic processes. In the world of living organisms, chirality is essential for proper biochemical function, and in synthetic chemistry, it's crucial for creating molecules with the desired properties and activities.

The interested reader should not forget to read the article written by Istvan Hargittai & Magdolna Hargittai, which can be found at this link.

Also regarding the chirality of crystals and of their building units (molecular or not), advanced readers should also consult the article by Howard D. Flack to be found through this link.
 

Thus, any finite object (such as a quartz crystal, a chair or a flower) shows that certain parts of it are repeated by symmetry operations that go through a point of the object. This set of symmetry operations is known as a symmetry point group. The advanced reader has also the opportunity to visit the nice work on point group symmetry elements offered through these links:

• Dean H. Johnston from the Otterbein University, and
• Margaret Kastner, Timathy Medlock and Kristy Brown from the University of Bucknell

Additionally, the reader can download (totally virus free!!!) and run on his own computer this Java application that, as an introduction to the symmetry of the polyhedra, that was developed by Gervais Chapuis and Nicolas Schöni (École Polytechnique Fédérale de Lausanne, Switzerland).

Symmetry in crystals

In crystals, the symmetry axes (rotation axes) can only be two-fold (2), three-fold (3), four-fold (4) or six-fold (6), depending on the number of times (order of rotation) that a motif can be repeated by a rotation operation, being transformed into a new state indistinguishable from its starting state. Thus, a rotation axis of order 3 (3-fold) produces 3 repetitions (copies) of the motif, one every 120 degrees (= 360/3) of rotation. If the reader wonders why only symmetry axes of order 2, 3, 4 and 6  can occur in crystals, and not 5-, 7-fold, etc., we recommend the explanations given in another section.

Improper rotations (rotations followed by reflection through a plane perpendicular to the rotational axis) are designated by the order of rotation, with a bar above that number.

The screw axes (or helicoidal axes, ie, symmetry axes involving rotation followed by a translation along the axis) are represented by the order of rotation, with an added subindex that quantifies the translation along the axis. Thus, a screw axis of type 6₂ means that in each of the six rotations an associated translation occurs of 2/6 of the axis of the elementary cell in that direction.

The mirror planes are represented by the letter m.

The glide planes (mirror planes involving reflexion and a translation parallel to the plane) are represented by the letters a, b, c, n or d, depending if the translation associated with the reflection is parallel to the reticular translations (a, b, c), parallel to the diagonal of a reticular plane (n), or parallel to a diagonal of the unit cell (d).

The letters and numbers that are used to represent the symmetry elements also have an equivalence with some graphic symbols.

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Finally, combining the 32 crystal classes (crystallographic point groups) with the 14 Bravais lattices, we find up to 230 different ways to replicate a finite object (motif) in 3-dimensional space. These 230 ways to repeat patterns in space, which are compatible with the 32 crystal classes and with the 14 Bravais lattices, are called space groups, and represent the 230 different ways to fit the Bravais lattices to the symmetry of the objects. The interested reader should also consult the excellent work on the symmetry elements present in the space groups, offered by Margaret Kastner, Timathy Medlock and Kristy Brown  through this link of the Bucknell University.

A wall of bricks showing the most appropriate lattice which best represents both the brick and its symmetry. Note that in this case the point symmetry of the brick and the point symmetry of the reticular point are coincident. The space group, considering the thickness of the brick, is Cmm2.