Rotation (geometry) in the context of Map (mathematics)

Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, the face of a human being has a plane of symmetry down its centre, or a pine cone displays a clear symmetrical spiral pattern. Internal features can also show symmetry, for example the tubes in the human body (responsible for transporting gases, nutrients, and waste products) which are cylindrical and have several planes of symmetry.

Biological symmetry can be thought of as a balanced distribution of duplicate body parts or shapes within the body of an organism. Importantly, unlike in mathematics, symmetry in biology is always approximate. For example, plant leaves – while considered symmetrical – rarely match up exactly when folded in half. Symmetry is one class of patterns in nature whereby there is near-repetition of the pattern element, either by reflection or rotation.

In chemistry, a molecule or ion is called chiral (/ˈkaɪrəl/) if it cannot be superposed on its mirror image by any combination of rotations, translations, and some conformational changes. This geometric property is called chirality (/kaɪˈrælɪti/). The terms are derived from Ancient Greek χείρ (cheir) 'hand'; which is the canonical example of an object with this property.

A chiral molecule or ion exists in two stereoisomers that are mirror images of each other, called enantiomers; they are often distinguished as either "right-handed" or "left-handed" by their absolute configuration or some other criterion. The two enantiomers have the same chemical properties, except when reacting with other chiral compounds. They also have the same physical properties, except that they often have opposite optical activities. A homogeneous mixture of the two enantiomers in equal parts, a racemic mixture, differs chemically and physically from the pure enantiomers.

In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.

It is important to note the distinction between rotary reflection and rotary inversion symmetry operations and their associated symmetry elements. Rotary reflections are generally used to describe the symmetry of individual molecules and are defined as a 360°/n rotation about an n-fold rotation axis followed by a reflection over a mirror plane perpendicular to the n-fold rotation axis. Rotoinversions are generally used to describe the symmetry of crystals and are defined as a 360°/n rotation about an n-fold rotation axis followed by an inversion through the origin. Although rotary reflection operations have a rotoinversion analogue and vice versa, rotoreflections and rotoinversions of the same order need not be identical. For example, a 6-fold rotoinversion axis and its associated with symmetry operations are distinct from those resulting from a 6-fold reflection axis.