Euclidean group
Encyclopedia
In mathematics
, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group
of n-dimensional Euclidean space
. Its elements, the isometries
associated with the Euclidean metric
, are called Euclidean moves.
These group
s are among the oldest and most studied, at least in the cases of dimension 2 and 3 — implicitly, long before the concept of group was invented.
for E(n) is
which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translational symmetry
, and the remaining n(n − 1)/2 to rotational symmetry
.
, also called rigid motions; they are the rigid body
moves. These include the translations, and the rotation
s, which together generate E+(n). E+(n) is also called a special Euclidean group, and denoted SE(n).
The others are the indirect isometries. The subgroup E+(n) is of index
2. In other words, the indirect isometries form a single coset
of E+(n). Given any
indirect isometry, for example a given reflection R that reverses orientation, all indirect isometries are given as DR, where D is a direct isometry.
The Euclidean group for SE(3) is used for the kinematics of a rigid body
, in classical mechanics
. A rigid body motion is in effect the same as a curve
in the Euclidean group. Starting with a body B oriented in a certain way at time t = 0, its orientation at any other time is related to the starting orientation by a Euclidean motion, say f(t). Setting t = 0, we have f(0) = I, the identity transformation. This means that the curve will always lie inside E+(3), in fact: starting at the identity transformation I, such a continuous curve can certainly never reach anything other than a direct isometry. This is for simple topological reasons: the determinant
of the transformation cannot jump from +1 to −1.
The Euclidean groups are not only topological group
s, they are Lie group
s, so that calculus
notions can be adapted immediately to this setting.
for n dimensions, and in such a way as to respect the semidirect product
structure of both groups. This gives, a fortiori, two ways of writing down elements in an explicit notation. These are:
Details for the first representation are given in the next section.
In the terms of Felix Klein
's Erlangen programme, we read off from this that Euclidean geometry
, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry
. All affine theorems apply. The extra factor in Euclidean geometry is the notion of distance
, from which angle
can then be deduced.
s.
It has as subgroups the translational
group T, and the orthogonal group
O(n). Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:
where A is an orthogonal matrix
or an orthogonal transformation followed by a translation:
.
T is a normal subgroup
of E(n): for any translation t and any isometry u, we have
again a translation (one can say, through a displacement that is u acting on the displacement of t; a translation does not affect a displacement, so equivalently, the displacement is the result of the linear part of the isometry acting on t).
Together, these facts imply that E(n) is the semidirect product
of O(n) extended by T. In other words O(n) is (in the natural way) also the quotient group
of E(n) by T:
Now SO(n), the special orthogonal group, is a subgroup of O(n), of index
two. Therefore E(n) has a subgroup E+(n), also of index two, consisting of direct isometries. In these cases the determinant of A is 1.
They are represented as a translation followed by a rotation
, rather than a translation followed by some kind of reflection
(in dimensions 2 and 3, these are the familiar reflections in a mirror
line or plane, which may be taken to include the origin
, or in 3D, a rotoreflection
).
We have:
Examples in 3D of combinations:
E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom
:
E(1) - 1:
E(2) - 3:
See also Euclidean plane isometry
.
E(3) - 6:
See also 3D isometries that leave the origin fixed, space group
, involution.
; the translation group is the union of those for all distances.
In 1D, all reflections are in the same class.
In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class.
In 3D:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of n-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. Its elements, the isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
associated with the Euclidean metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
, are called Euclidean moves.
These group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s are among the oldest and most studied, at least in the cases of dimension 2 and 3 — implicitly, long before the concept of group was invented.
Dimensionality
The number of degrees of freedomDegrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
for E(n) is
- n(n + 1)/2,
which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
, and the remaining n(n − 1)/2 to rotational symmetry
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
.
Direct and indirect isometries
There is a subgroup E+(n) of the direct isometries, i.e., isometries preserving orientationOrientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
, also called rigid motions; they are the rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
moves. These include the translations, and the rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s, which together generate E+(n). E+(n) is also called a special Euclidean group, and denoted SE(n).
The others are the indirect isometries. The subgroup E+(n) is of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
2. In other words, the indirect isometries form a single coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
of E+(n). Given any
Universal quantification
In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
indirect isometry, for example a given reflection R that reverses orientation, all indirect isometries are given as DR, where D is a direct isometry.
The Euclidean group for SE(3) is used for the kinematics of a rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
, in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. A rigid body motion is in effect the same as a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
in the Euclidean group. Starting with a body B oriented in a certain way at time t = 0, its orientation at any other time is related to the starting orientation by a Euclidean motion, say f(t). Setting t = 0, we have f(0) = I, the identity transformation. This means that the curve will always lie inside E+(3), in fact: starting at the identity transformation I, such a continuous curve can certainly never reach anything other than a direct isometry. This is for simple topological reasons: the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the transformation cannot jump from +1 to −1.
The Euclidean groups are not only topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
s, they are Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, so that calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
notions can be adapted immediately to this setting.
Relation to the affine group
The Euclidean group E(n) is a subgroup of the affine groupAffine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
for n dimensions, and in such a way as to respect the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
structure of both groups. This gives, a fortiori, two ways of writing down elements in an explicit notation. These are:
- by a pair (A, b), with A an n×n orthogonal matrixOrthogonal matrixIn linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
, and b a real column vector of size n; or - by a single square matrix of size n + 1, as explained for the affine groupAffine groupIn mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
.
Details for the first representation are given in the next section.
In the terms of Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
's Erlangen programme, we read off from this that Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry
Affine geometry
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
. All affine theorems apply. The extra factor in Euclidean geometry is the notion of distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
, from which angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
can then be deduced.
Subgroup structure, matrix and vector representation
The Euclidean group is a subgroup of the group of affine transformationAffine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s.
It has as subgroups the translational
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
group T, and the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
O(n). Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:
where A is an orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
or an orthogonal transformation followed by a translation:
.T is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of E(n): for any translation t and any isometry u, we have
- u−1tu
again a translation (one can say, through a displacement that is u acting on the displacement of t; a translation does not affect a displacement, so equivalently, the displacement is the result of the linear part of the isometry acting on t).
Together, these facts imply that E(n) is the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of O(n) extended by T. In other words O(n) is (in the natural way) also the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of E(n) by T:
- O(n)
E(n) / T.
Now SO(n), the special orthogonal group, is a subgroup of O(n), of index
Index (mathematics)
The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...
two. Therefore E(n) has a subgroup E+(n), also of index two, consisting of direct isometries. In these cases the determinant of A is 1.
They are represented as a translation followed by a rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
, rather than a translation followed by some kind of reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
(in dimensions 2 and 3, these are the familiar reflections in a mirror
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...
line or plane, which may be taken to include the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
, or in 3D, a rotoreflection
Improper rotation
In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...
).
We have:
- SO(n)
E+(n) / T.
Subgroups
Types of subgroups of E(n):- Finite groupFinite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category. - Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically discreteDiscrete spaceIn topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
. E.g. for 1 ≤ m ≤ n a group generated by m translations in independent directions, and possibly a finite point group. This includes latticeLattice (group)In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
s. Examples more general than those are the discrete space groupSpace groupIn mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
s. - Countably infinite groups with arbitrarily small translations, rotations, or combinations. In this case there are points for which the set of images under the isometries is not closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the origin by 1 radian.
- Non-countable groups, where there are points for which the set of images under the isometries is not closed. E.g. in 2D all translations in one direction, and all translations by rational distances in another direction.
- Non-countable groups, where for all points the set of images under the isometries is closed. E.g.
- all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the rotation groupRotation groupIn mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
) - all isometries that keep the origin fixed, or more generally, some point (the orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
) - all direct isometries E+(n)
- the whole Euclidean group E(n)
- one of these groups in an m-dimensional subspace combined with a discrete group of isometries in the orthogonal n-m-dimensional space
- one of these groups in an m-dimensional subspace combined with another one in the orthogonal n-m-dimensional space
- all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the rotation group
Examples in 3D of combinations:
- all rotations about one fixed axis
- ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis
- ditto combined with discrete translation along the axis or with all isometries along the axis
- a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction
- all isometries which are a combination of a rotation about some axis and a proportional translation along the axis; in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold helixHelixA helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...
; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes. - for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
of R3, Dih(R3).
Overview of isometries in up to three dimensions
E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
:
E(1) - 1:
- E+(1):
- identity - 0
- translation - 1
- those not preserving orientation:
- reflection in a point - 1
E(2) - 3:
- E+(2):
- identity - 0
- translation - 2
- rotation about a point - 3
- those not preserving orientation:
- reflection in a line - 2
- reflection in a line combined with translation along that line (glide reflectionGlide reflectionIn geometry, a glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result...
) - 3
See also Euclidean plane isometry
Euclidean plane isometry
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...
.
E(3) - 6:
- E+(3):
- identity - 0
- translation - 3
- rotation about an axis - 5
- rotation about an axis combined with translation along that axis (screw operationScrew theoryScrew theory refers to the algebra and calculus of pairs of vectors, such as forces and moments and angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies....
) - 6
- those not preserving orientation:
- reflection in a plane - 3
- reflection in a plane combined with translation in that plane (glide planeGlide planeIn crystallography, a glide plane is symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged....
operation) - 5 - rotation about an axis by an angle not equal to 180°, combined with reflection in a plane perpendicular to that axis (roto-reflectionImproper rotationIn 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...
) - 6 - inversion in a point - 3
See also 3D isometries that leave the origin fixed, space group
Space group
In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
, involution.
Commuting isometries
For some isometry pairs composition does not depend on order:- two translations
- two rotations or screws about the same axis
- reflection with respect to a plane, and a translation in that plane, a rotation about an axis perpendicular to the plane, or a reflection with respect to a perpendicular plane
- glide reflection with respect to a plane, and a translation in that plane
- inversion in a point and any isometry keeping the point fixed
- rotation by 180° about an axis and reflection in a plane through that axis
- rotation by 180° about an axis and rotation by 180° about a perpendicular axis (results in rotation by 180° about the axis perpendicular to both)
- two rotoreflections about the same axis, with respect to the same plane
- two glide reflections with respect to the same plane
Conjugacy classes
The translations by a given distance in any direction form a conjugacy classConjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
; the translation group is the union of those for all distances.
In 1D, all reflections are in the same class.
In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class.
In 3D:
- Inversions with respect to all points are in the same class.
- Rotations by the same angle are in the same class.
- Rotations about an axis combined with translation along that axis are in the same class if the angle is the same and the translation distance is the same, and in corresponding direction (right-hand or left-hand screw).
- Reflections in a plane are in the same class
- Reflections in a plane combined with translation in that plane by the same distance are in the same class.
- Rotations about an axis by the same angle not equal to 180°, combined with reflection in a plane perpendicular to that axis, are in the same class.
See also
- fixed points of isometry groups in Euclidean spaceFixed points of isometry groups in Euclidean spaceA fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space....
- Euclidean plane isometryEuclidean plane isometryIn 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...
- Poincaré groupPoincaré groupIn physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
- Coordinate rotations and reflectionsCoordinate rotations and reflectionsIn geometry, 2D coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1...
- Reflection through the origin
- Plane of rotationPlane of rotationIn geometry, a plane of rotation is an abstract object used to describe or visualise rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions.Mathematically such...