Short description about dihedrals?

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In summary, dihedrals are groups that have a geometric interpretation as rotations and reflections. In abstract algebra, the elements of a dihedral group are represented by s and r, corresponding to rotations and reflections respectively. The group presentation for D_n is given by \langle s, r| s^n=1, r^2=1 \rangle, where sr=rs^{-1}. However, for n>1, the group is not finite and is in fact the free product of C_n and C_2.
  • #1
vivaitalia1
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Can someone give me a short description about dihedrals? for example what would be the elements of D10? or D4?
 
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  • #2
Well, do you know the geometric interpretation of a dihedral group? The elements are (in the geometric interpretation) rotations and reflections.
 
  • #3
Yes I know the geometric interpretation. Is it the same for abstract algebra?
 
  • #4
Yes, the geometric interpretation is the motivation for dihedral groups. As I said, the elements are analogous to rotations and reflections. In general a group presentation for [itex]D_n[/itex] is

[tex]\langle s, r| s^n = 1, r^2 = 1 \rangle.[/tex]

In that presentation, s corresponds to a rotation and r to a particular reflection.
 
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  • #5
You also need [itex]sr = rs^{-1}[/itex] in that presentation.
 
  • #6
morphism said:
You also need [itex]sr = rs^{-1}[/itex] in that presentation.

Indeed! I should be more careful. The group in my post isn't even finite for n>1. :yuck:
 
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  • #7
It is the free product of C_n and C_2, in fact.
 

1. What are dihedrals?

Dihedrals, also known as torsion angles, are angles between two planes in a molecule. They represent the rotation of one plane with respect to the other, and are often used to describe the conformation of molecules.

2. What is the importance of dihedrals in chemistry?

Dihedrals play a crucial role in understanding the 3D structure of molecules and their properties. They also provide insights into the stability and reactivity of molecules, and are important in predicting and designing molecular structures in drug discovery and material science.

3. How are dihedrals calculated?

Dihedrals can be calculated using the coordinates of the atoms involved in the angle. The formula for calculating dihedrals is based on the dot product of two vectors formed by the atoms. It is usually measured in degrees.

4. What factors affect dihedrals?

The bond types, bond lengths, and electronic structure of the atoms involved can affect dihedrals. Steric hindrance, electrostatic interactions, and hydrogen bonding can also influence dihedrals. Additionally, changes in temperature and solvent can affect the conformation and therefore the dihedrals of a molecule.

5. What is the significance of studying dihedrals in protein structure?

The dihedral angles in proteins, particularly in the backbone, determine the secondary structure and overall 3D shape of the protein. Understanding these angles is crucial in predicting protein folding and stability, as well as in identifying potential binding sites for ligands and substrates.

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