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In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χειρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek εναντιος (enantios) 'opposite' and μορφη (morphe) 'form'. A non-chiral figure is called achiral or amphichiral. For other uses, see Geometry (disambiguation). ...
A mirror image is a mirror based duplicate of a single image. ...
The helix (and by extension a spun string, a screw, a propeller, etc.) and Möbius strip are chiral three-dimensional objects. The J, L, S and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. A helix (pl: helices), from the Greek word Îλικας/Îλιξ, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ...
A Möbius strip made with a piece of paper and tape. ...
A tetromino, also spelled tetramino or tetrimino, is a geometric shape composed of four squares, connected orthogonally. ...
Tetris (Russian: ) is a falling-blocks puzzle video game, released on a large spectrum of platforms. ...
Many other familiar objects exhibit the same chiral symmetry of the human body: gloves, glasses, shoes, legs on a pair of pants, etc. A similar notion of chirality is considered in knot theory, as explained below. Trefoil knot, the simplest non-trivial knot. ...
Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. The left-handed orientation is shown on the left, and the right-handed on the right. ...
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 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.) The symmetry group of an object (e. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
Chirality in three dimensions In three dimensions, every figure which possesses a plane of symmetry or a center of symmetry is achiral. (A plane of symmetry of a figure F is a plane P, such that F is invariant under the mapping , 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 , 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 In 3-dimensional geometry, a plane of symmetry is a 2-dimensional flat dividing surface placed such that things on one side are symmetrical (mirror image) to things on the other side. ...
which is invariant under the orientation reversing isometry and thus achiral, but it has neither plane nor center of symmetry. The figure
also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.
Note also that achiral figures can have a center axis. A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...
Chirality in two 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 , when L is chosen to be the x-axis of the coordinate system.) Consider the following pattern: The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. ...
> > > > > > > > > > > > > > > > > > > > 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. A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. ...
Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...
Knot theory A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called chiral. For example the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral. A trefoil knot. ...
In the mathematical field of knot theory, an amphichiral knot, also called an achiral knot or amphicheiral knot, is an oriented knot equivalent to its mirror image. ...
The unknot, and a knot equivalent to it The unknot is a loop of rope without a knot in it (in knot theory, ropes have no ends; they are loops). ...
In knot theory, a figure-eight knot is the unique knot with a crossing number of four, the smallest possible except for the unknot and trefoil knot. ...
In knot theory, the trefoil knot is the simplest nontrivial knot. ...
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