radial symmetry

Examples of radial 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.
  • Radial symmetry is the arrangement of body parts around a central axis, like rays on a sun or pieces in a pie.
  • All true animals, except those with radial symmetry, are bilaterally symmetrical.
  • 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.

• Body Plans

  • Animal body plans can have varying degrees of symmetry and can be described as asymmetrical, bilateral, or radial.
  • They can be asymmetrical, radial, or bilateral in form .
  • Radial symmetry describes an animal with an up-and-down orientation: any plane cut along its longitudinal axis through the organism produces equal halves, but not a definite right or left side.
  • Bilateral symmetry is illustrated in a goat.
  • The sponge is asymmetrical, the sea anemone has radial symmetry, and the goat has bilateral symmetry.

• Rhizaria

  • A second subtype of Rhizaria, the radiolarians, exhibit intricate exteriors of glassy silica with radial or bilateral symmetry .

• Classes of Echinoderms

  • Of all echinoderms, the Ophiuroidea may have the strongest tendency toward 5-segment radial (pentaradial) symmetry.
  • Their early larvae have bilateral symmetry, but they develop fivefold symmetry as they mature.
  • Several sea urchins, however, including the sand dollars, are oval in shape, with distinct front and rear ends, giving them a degree of bilateral symmetry.
  • Although the basic echinoderm pattern of fivefold symmetry can be recognized, most crinoids have many more than five arms.
  • Sea cucumbers are the only echinoderms that demonstrate "functional" bilateral symmetry as adults, as they lie horizontally as opposed to the vertical axis of other echinoderms.

• Compositional Balance

  • The three most common types of compositional balance are symmetrical, asymmetrical, and radial.
  • The opposite of symmetry is asymmetry.
  • Asymmetry is defined as the absence of, or a violation of, the principles of symmetry.
  • Radial balance refers to circular elements in compositions.
  • The three common types of balance are symmetric, asymmetric, and radial.

• 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.
  • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.
  • Spherical coordinates ($r$, $\theta$, $\varphi$) as often used in mathematics: radial distance $r$, azimuthal angle $\theta$, and polar angle $\varphi$.
  • The dot is the point with radial distance $\rho = 4$, angular coordinate $\varphi = 130$ degrees, and height $z = 4$.

• Phylum Echinodermata

  • Echinoderms are invertebrates that have pentaradial symmetry, a spiny skin, a water vascular system, and a simple nervous system.
  • Adult echinoderms exhibit pentaradial symmetry and have a calcareous endoskeleton made of ossicles, although the early larval stages of all echinoderms have bilateral symmetry .
  • Echinoderms possess a unique ambulacral or water vascular system, consisting of a central ring canal and radial canals that extend along each arm .
  • The ring canal connects the radial canals (there are five in a pentaradial animal), and the radial canals move water into the ampullae, which have tube feet through which the water moves.
  • The nervous system in these animals is a relatively simple structure with a nerve ring at the center and five radial nerves extending outward along the arms.

• 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.  
  • For example:  Does the function below show symmetry?  
  • Describe the type of symmetry.
  • This type of symmetry is a translation over an axis.

• Wave Equation for the Hydrogen Atom

  • The solution of the Schrödinger equation for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic—it is radially symmetric in space and only depends on the distance to the nucleus.
  • In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wavefunctions must be found.
  • This holds for all problems with rotational symmetry.
  • where R are radial functions and theta ($\theta$) and phi ($\phi$) are spherical harmonic terms.

• 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.