symmetry

Examples of symmetry in the following topics:

• Animal Characterization Based on Body Symmetry

  • Animals can be classified by three types of body plan symmetry: radial symmetry, bilateral symmetry, and asymmetry.
  • In contrast to radial symmetry, which is best suited for stationary or limited-motion lifestyles, bilateral symmetry allows for streamlined and directional motion.
  • Animals in the phylum Echinodermata (such as sea stars, sand dollars, and sea urchins) display radial symmetry as adults, but their larval stages exhibit bilateral symmetry .
  • This is termed secondary radial symmetry.
  • The larvae of echinoderms (sea stars, sand dollars, and sea urchins) have bilateral symmetry as larvae, but develop radial symmetry as full adults.

• Symmetry of Functions

  • They can have symmetry after a reflection.  
  • In the next graph below, quadratic functions have symmetry over a line called the axis of symmetry.  
  • The graph has symmetry over the origin or point $(0,0)$.  
  • This type of symmetry is a translation over an axis.
  • Determine whether or not a given relation shows some form of symmetry

• Chirality and Symmetry

  • Some examples of symmetry elementsare shown below.
  • In these two cases the point of symmetry is colored magenta.
  • The boat conformation of cyclohexane shows an axis of symmetry (labeled C2 here) and two intersecting planes of symmetry (labeled σ).
  • The existence of a reflective symmetry element (a point or plane of symmetry) is sufficient to assure that the object having that element is achiral.
  • (ii) Asymmetry: The absence of all symmetry elements.

• Symmetry and Centricity

  • Think of pitch symmetry in terms of a musical "mirror."
  • Pitch symmetry always implies an axis of symmetry.
  • The pitch-space line shows that it has a different axis of symmetry—around E2.
  • Pitch-class symmetry is very similar to pitch symmetry, but understood in pitch-class space.
  • Mapping this on the pitch-class circle shows the passage's pitch-class symmetry.

• Trigonometric Symmetry Identities

  • The trigonometric symmetry identities are based on principles of even and odd functions that can be observed in their graphs.
  • This symmetry is used to derive certain identities.
  • The following symmetry identities are useful in finding the trigonometric function of a negative value.
  • Cosine and secant are even functions, with symmetry around the $y$-axis.
  • Explain the trigonometric symmetry identities using the graphs of the trigonometric functions

• Body Plans

  • Animal body plans follow set patterns related to symmetry.
  • Asymmetrical animals are those with no pattern or symmetry, such as a sponge.
  • Bilateral symmetry is illustrated in a goat.
  • Animals exhibit different types of body symmetry.
  • The sponge is asymmetrical, the sea anemone has radial symmetry, and the goat has bilateral symmetry.

• Theoretical Models for Pericyclic Reactions

  • The opposite is true for the π*-orbital on the right, which has a mirror plane symmetry of A and a C2 symmetry of S.
  • Such symmetry characteristics play an important role in creating the orbital diagrams used by Woodward and Hoffmann to rationalize pericyclic reactions.
  • The symmetries of the appropriate reactant and product orbitals were matched to determine whether the transformation could proceed without a symmetry imposed conversion of bonding reactant orbitals to antibonding product orbitals.
  • If the correlation diagram indicated that the reaction could occur without encountering such a symmetry-imposed barrier, it was termed symmetry allowed.
  • If a symmetry barrier was present, the reaction was designated symmetry-forbidden.

• Enantiomorphism

  • A regular tetrahedron has six planes of symmetry and seven symmetry axes (four C3 & three C2) and is, of course, achiral.
  • If one of the carbon substituents is different from the other three, the degree of symmetry is lowered to a C3 axis and three planes of symmetry, but the configuration remains achiral.
  • Further substitution may reduce the symmetry even more, but as long as two of the four substituents are the same there is always a plane of symmetry that bisects the angle linking those substituents, so these configurations are also achiral.
  • A carbon atom that is bonded to four different atoms or groups loses all symmetry, and is often referred to as an asymmetric carbon.
  • The former has a plane of symmetry passing through the chlorine atom and bisecting the opposite carbon-carbon bond.

• Cylindrical and Spherical Coordinates

  • Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with a round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, and so on.
  • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.

• Double Integrals in Polar Coordinates

  • When domain has a cylindrical symmetry and the function has several specific characteristics, apply the transformation to polar coordinates.
  • In $R^2$, if the domain has a cylindrical symmetry and the function has several particular characteristics, you can apply the transformation to polar coordinates, which means that the generic points $P(x, y)$ in Cartesian coordinates switch to their respective points in polar coordinates.
  • This is the case because the function has a cylindrical symmetry.
  • In general, the best practice is to use the coordinates that match the built-in symmetry of the function.