Chirality (mathematics)

Last updated August 22, 2024

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral.

Examples

The tetrominos S and Z are enantiomorphs in 2-dimensions
S	Z

Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule.

Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out.^[1]

The J-, L-, S- and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane.

Chirality and symmetry group

A figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. (In Euclidean geometry any isometry can be written as $v\mapsto Av+b$ with an orthogonal matrix $A$ and a vector $b$ . The determinant of $A$ is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving.

A general definition of chirality based on group theory exists.^[2] It does not refer to any orientation concept: an isometry is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.^[3]^[4]

Chirality in two dimensions

The colored necklace in the middle is chiral in two dimensions; the two others are achiral.
This means that as physical necklaces on a table the left and right ones can be rotated into their mirror image while remaining on the table. The one in the middle, however, would have to be picked up and turned in three dimensions. Bracelets33.svg — The colored necklace in the middle is **chiral** in two dimensions; the two others are **achiral**.
This means that as physical necklaces on a table the left and right ones can be rotated into their mirror image while remaining on the table. The one in the middle, however, would have to be picked up and turned in three dimensions.

A scalene triangle does not have mirror symmetries, and hence is a chiral polytope in 2 dimensions. Triangle.Scalene.svg — A scalene triangle does not have mirror symmetries, and hence is a chiral polytope in 2 dimensions.

In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry. (An axis of symmetry of a figure $F$ is a line $L$ , such that $F$ is invariant under the mapping $(x,y)\mapsto (x,-y)$ , when $L$ is chosen to be the $x$ -axis of the coordinate system.) For that reason, a triangle is achiral if it is equilateral or isosceles, and is chiral if it is scalene.

Consider the following pattern:

This figure is chiral, as it is not identical to its mirror image:

But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a frieze group generated by a single glide reflection.

Chirality in three dimensions

In three dimensions, every figure that possesses a mirror plane of symmetry S₁, an inversion center of symmetry S₂, or a higher improper rotation (rotoreflection) S_n axis of symmetry^[5] is achiral. (A plane of symmetry of a figure $F$ is a plane $P$ , such that $F$ is invariant under the mapping $(x,y,z)\mapsto (x,y,-z)$ , when $P$ is chosen to be the $x$ - $y$ -plane of the coordinate system. A center of symmetry of a figure $F$ is a point $C$ , such that $F$ is invariant under the mapping $(x,y,z)\mapsto (-x,-y,-z)$ , when $C$ is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry. An example is the figure

F_{0}=\left\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\right\}

which is invariant under the orientation reversing isometry $(x,y,z)\mapsto (-y,x,-z)$ and thus achiral, but it has neither plane nor center of symmetry. The figure

F_{1}=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}

also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.

Achiral figures can have a center axis.

Knot theory

A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called a chiral knot. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral.

Related Research Articles

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 ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

A mirror image 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 specular reflection off from surfaces of lustrous materials, especially a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3D structures.

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.

In mathematics, the modular group is the projective special linear group $of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.$

In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass.

In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.

In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space $; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n), for inhomogeneous special orthogonal group.$

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections.

A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D).

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

A regular octahedron has 24 rotational symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron.

<span class="mw-page-title-main">Tetrahedral symmetry</span> 3D symmetry group

A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

In knot theory, a Lissajous knot is a knot defined by parametric equations of the form

In geometry, a point reflection is a transformation of affine space in which every point is reflected across a specific fixed point. When dealing with crystal structures and in the physical sciences the terms inversion symmetry, inversion center or centrosymmetric are more commonly used.

Chirality is a property of asymmetry important in several branches of science. The word chirality is derived from the Greek χείρ (kheir), "hand", a familiar chiral object.

In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. More generally, the term motion is a synonym for surjective isometry in metric geometry, including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.

<span class="mw-page-title-main">Symmetry (geometry)</span> Geometrical property

In geometry, an object has symmetry if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.

The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers. When studying these transformations, the dual numbers are often interpreted as representing oriented lines on the plane. The Laguerre transformations map lines to lines, and include in particular all isometries of the plane.

References

↑ Toong, Yock Chai; Wang, Shih Yung (April 1997). "An example of a human topological rubber glove act". Journal of Chemical Education. 74 (4): 403. Bibcode:1997JChEd..74..403T. doi:10.1021/ed074p403.
↑ Petitjean, M. (2020). "Chirality in metric spaces. In memoriam Michel Deza". Optimization Letters. 14 (2): 329–338. doi: 10.1007/s11590-017-1189-7 .
↑ Petitjean, M. (2021). "Chirality in geometric algebra". Mathematics. 9 (13). 1521. doi: 10.3390/math9131521 .
↑ Petitjean, M. (2022). "Chirality in affine spaces and in spacetime". arXiv: 2203.04066 [math-ph].
↑ "2. Symmetry operations and symmetry elements". chemwiki.ucdavis.edu. 3 March 2014. Retrieved 25 March 2016.

External links

Symmetry, Chirality, Symmetry Measures and Chirality Measures: General Definitions
Chiral Polyhedra by Eric W. Weisstein, The Wolfram Demonstrations Project.
Chiral manifold at the Manifold Atlas.

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[1] Toong, Yock Chai; Wang, Shih Yung (April 1997). "An example of a human topological rubber glove act". Journal of Chemical Education. 74 (4): 403. Bibcode:1997JChEd..74..403T. doi:10.1021/ed074p403.

[2] Petitjean, M. (2020). "Chirality in metric spaces. In memoriam Michel Deza". Optimization Letters. 14 (2): 329–338. doi: 10.1007/s11590-017-1189-7 .

[3] Petitjean, M. (2021). "Chirality in geometric algebra". Mathematics. 9 (13). 1521. doi: 10.3390/math9131521 .

[4] Petitjean, M. (2022). "Chirality in affine spaces and in spacetime". arXiv: 2203.04066 [math-ph].

[5] "2. Symmetry operations and symmetry elements". chemwiki.ucdavis.edu. 3 March 2014. Retrieved 25 March 2016.

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