Metachirality is a stronger form of
chirality
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
.
It applies to objects or systems that are chiral (not identical to their
mirror image
A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
) and where, in addition, their mirror image has a
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
that differs from the symmetry group of the original object or system.
Many familiar chiral objects, like the capital letter 'Z' embedded in the plane, are not metachiral.
The symmetry group of the capital letter 'Z' embedded in the plane consists of the
identity transformation
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
and a rotation over 180˚ (a half turn).
In this case, the mirror image has the same symmetry group.
In particular, asymmetric objects (that only have the identity transformation as symmetry, like a human hand) are not metachiral,
since the mirror image is also asymmetric.
In general, two-dimensional objects and bounded three-dimensional objects are not metachiral.
An example of a metachiral object is an infinite
helical staircase
Stairs are a structure designed to bridge a large vertical distance between lower and higher levels by dividing it into smaller vertical distances. This is achieved as a diagonal series of horizontal platforms called steps which enable passage ...
.
A
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined hel ...
in 3D has a ''handedness'' (either left or right, like
screw thread
A screw thread, often shortened to thread, is a helical structure used to convert between rotational and linear movement or force. A screw thread is a ridge wrapped around a cylinder or cone in the form of a helix, with the former being called ...
), whereby it differs from its mirror image.
An infinite helical staircase, however, does have symmetries:
screw operations, that is, a combination of a
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
and a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
.
The symmetry group of the mirror image of an infinite helical staircase also contains screw operations.
But they are of the opposite handedness and, hence,
the symmetry groups differ.
Note, however, that these symmetry groups are
isomorphic.
Of the 219
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it ...
s, 11 are metachiral.
A nice example of a metachiral spatial structure is the
K4 crystal, also known as Triamond,
and featured in the ''Bamboozle'' mathematical artwork.
Bamboozle: A Mathematical Artwork in MetaForum
/ref>
See also
* Orientation (mathematics)
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space ...
* Stereochemistry
Stereochemistry, a subdiscipline of chemistry, involves the study of the relative spatial arrangement of atoms that form the structure of molecules and their manipulation. The study of stereochemistry focuses on the relationships between stereo ...
* Right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of ...
* Handedness
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjec ...
* Asymmetry
Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstract systems and it may be displayed in pre ...
References
{{reflist
Chirality