Euclidean plane isometry
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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a Euclidean plane isometry is an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types:
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
, rotations, reflections, and
glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection ...
s (see below under classification of Euclidean plane isometries). The set of Euclidean plane isometries forms a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
: the
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...
in two dimensions. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections.


Informal discussion

Informally, a Euclidean plane isometry is any way of transforming the plane without "deforming" it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Examples of isometries include: * Shifting the sheet one inch to the right. * Rotating the sheet by ten degrees around some marked point (which remains motionless). * Turning the sheet over to look at it from behind. Notice that if a picture is drawn on one side of the sheet, then after turning the sheet over, we see the
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 ...
of the picture. These are examples of
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 ...
s,
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 ...
s, and
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
s respectively. There is one further type of isometry, called a
glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection ...
(see below under classification of Euclidean plane isometries). However, folding, cutting, or melting the sheet are not considered isometries. Neither are less drastic alterations like bending, stretching, or twisting.


Formal definition

An isometry of the Euclidean plane is a distance-preserving transformation of the plane. That is, it is a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
: M : \textbf^2 \to \textbf^2 such that for any points ''p'' and ''q'' in the plane, :d(p, q) = d(M(p), M(q)), where ''d''(''p'', ''q'') is the usual
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
between ''p'' and ''q''.


Classification

It can be shown that there are four types of Euclidean plane isometries. (Note: the notations for the types of isometries listed below are not completely standardised.)


Reflections

Reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
s, or mirror isometries, denoted by ''F''''c'',''v'', where ''c'' is a point in the plane and ''v'' is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
in R2. (''F'' is for "flip".) have the effect of reflecting the point ''p'' in the line ''L'' that is perpendicular to ''v'' and that passes through ''c''. The line ''L'' is called the reflection axis or the associated mirror. To find a formula for ''F''''c'',''v'', we first use the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
to find the component ''t'' of ''p'' − ''c'' in the ''v'' direction, :t = (p-c) \cdot v = (p_x - c_x)v_x + (p_y - c_y)v_y, :and then we obtain the reflection of ''p'' by subtraction, :F_(p) = p - 2tv. The combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices (i.e. with determinant 1 and −1) forming orthogonal group ''O''(2). In the case of a determinant of −1 we have: ::R_(p) = \begin \cos\theta & \sin\theta \\ \sin\theta & \mathbf \cos\theta \end \begin p_x \\ p_y \end. which is a reflection in the ''x''-axis followed by a rotation by an angle θ, or equivalently, a reflection in a line making an angle of θ/2 with the ''x''-axis. Reflection in a parallel line corresponds to adding a vector perpendicular to it.


Translations

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 ...
s, denoted by ''T''''v'', where ''v'' is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
in R2 have the effect of shifting the plane in the direction of ''v''. That is, for any point ''p'' in the plane, ::T_v(p) = p + v, :or in terms of (''x'', ''y'') coordinates, :: T_v(p) = \begin p_x + v_x \\ p_y + v_y \end. A translation can be seen as a composite of two parallel reflections.


Rotations

Rotations, denoted by ''R''c,θ, where ''c'' is a point in the plane (the centre of rotation), and θ is the angle of rotation. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations. First, a rotation around the origin is given by ::R_(p) = \begin \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end \begin p_x \\ p_y \end. :These matrices are the
orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...
(i.e. each is a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
G whose
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
is its inverse, i.e. GG^T=G^T G=I_2.), with determinant 1 (the other possibility for orthogonal matrices is −1, which gives a mirror image, see below). They form the special
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
SO(2). :A rotation around ''c'' can be accomplished by first translating ''c'' to the origin, then performing the rotation around the origin, and finally translating the origin back to ''c''. That is, ::R_ = T_c \circ R_ \circ T_, :or in other words, ::R_(p) = c + R_(p - c). :Alternatively, a rotation around the origin is performed, followed by a translation: ::R_(p) = c-R_ c + R_(p). A rotation can be seen as a composite of two non-parallel reflections.


Rigid transformations

The set of translations and rotations together form the rigid motions or rigid displacements. This set forms a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under composition, the ''group of rigid motions'', a subgroup of the full group of Euclidean isometries.


Glide reflections

Glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection ...
s, denoted by ''G''''c'',''v'',''w'', where ''c'' is a point in the plane, ''v'' is a unit vector in R2, and ''w'' is non-null a vector perpendicular to ''v'' are a combination of a reflection in the line described by ''c'' and ''v'', followed by a translation along ''w''. That is, ::G_ = T_w \circ F_, :or in other words, ::G_(p) = w + F_(p). :(It is also true that ::G_(p) = F_(p + w); :that is, we obtain the same result if we do the translation and the reflection in the opposite order.) :Alternatively we multiply by an orthogonal matrix with determinant −1 (corresponding to a reflection in a line through the origin), followed by a translation. This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line. The
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
isometry, defined by ''I''(''p'') = ''p'' for all points ''p'' is a special case of a translation, and also a special case of a rotation. It is the only isometry which belongs to more than one of the types described above. In all cases we multiply the position vector by an orthogonal matrix and add a vector; if the determinant is 1 we have a rotation, a translation, or the identity, and if it is −1 we have a glide reflection or a reflection. A "random" isometry, like taking a sheet of paper from a table and randomly laying it back, "
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
" is a rotation or a glide reflection (they have three
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
). This applies regardless of the details of the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, as long as θ and the direction of the added vector are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
and uniformly distributed and the length of the added vector has a continuous distribution. A pure translation and a pure reflection are special cases with only two degrees of freedom, while the identity is even more special, with no degrees of freedom.


Isometries as reflection group

Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a
reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent cop ...
.


Mirror combinations

In the Euclidean plane, we have the following possibilities. *; d_ .html" ;"title="span style="color:gray;">d  ">span style="color:gray;">d  Identity :Two reflections in the same mirror restore each point to its original position. All points are left fixed. Any pair of identical mirrors has the same effect. *; db.html" ;"title="span style="color:gray;">db">span style="color:gray;">dbReflection :As Alice found
through the looking-glass ''Through the Looking-Glass, and What Alice Found There'' (also known as ''Alice Through the Looking-Glass'' or simply ''Through the Looking-Glass'') is a novel published on 27 December 1871 (though indicated as 1872) by Lewis Carroll and the ...
, a single mirror causes left and right hands to switch. (In formal terms, topological orientation is reversed.) Points on the mirror are left fixed. Each mirror has a unique effect. *; dp.html" ;"title="span style="color:gray;">dp">span style="color:gray;">dpRotation :Two distinct intersecting mirrors have a single point in common, which remains fixed. All other points rotate around it by twice the angle between the mirrors. Any two mirrors with the same fixed point and same angle give the same rotation, so long as they are used in the correct order. *; dd.html" ;"title="span style="color:gray;">dd">span style="color:gray;">ddTranslation :Two distinct mirrors that do not intersect must be parallel. Every point moves the same amount, twice the distance between the mirrors, and in the same direction. No points are left fixed. Any two mirrors with the same parallel direction and the same distance apart give the same translation, so long as they are used in the correct order. *; dq.html" ;"title="span style="color:gray;">dq">span style="color:gray;">dqGlide reflection :Three mirrors. If they are all parallel, the effect is the same as a single mirror (slide a pair to cancel the third). Otherwise we can find an equivalent arrangement where two are parallel and the third is perpendicular to them. The effect is a reflection combined with a translation parallel to the mirror. No points are left fixed.


Three mirrors suffice

Adding more mirrors does not add more possibilities (in the plane), because they can always be rearranged to cause cancellation. :Proof. An isometry is completely determined by its effect on three independent (not collinear) points. So suppose ''p''1, ''p''2, ''p''3 map to ''q''1, ''q''2, ''q''3; we can generate a sequence of mirrors to achieve this as follows. If ''p''1 and ''q''1 are distinct, choose their perpendicular bisector as mirror. Now ''p''1 maps to ''q''1; and we will pass all further mirrors through ''q''1, leaving it fixed. Call the images of ''p''2 and ''p''3 under this reflection ''p''2′ and ''p''3′. If ''q''2 is distinct from ''p''2′, bisect the angle at ''q''1 with a new mirror. With ''p''1 and ''p''2 now in place, ''p''3 is at ''p''3′′; and if it is not in place, a final mirror through ''q''1 and ''q''2 will flip it to ''q''3. Thus at most three reflections suffice to reproduce any plane isometry. ∎


Recognition

We can recognize which of these isometries we have according to whether it preserves hands or swaps them, and whether it has at least one fixed point or not, as shown in the following table (omitting the identity).


Group structure

Isometries requiring an odd number of mirrors — reflection and glide reflection — always reverse left and right. The even isometries — identity, rotation, and translation — never do; they correspond to ''rigid motions'', and form a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of the full
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...
of isometries. Neither the full group nor the even subgroup are abelian; for example, reversing the order of composition of two parallel mirrors reverses the direction of the translation they produce. :Proof. The identity is an isometry; nothing changes, so distance cannot change. And if one isometry cannot change distance, neither can two (or three, or more) in succession; thus the composition of two isometries is again an isometry, and the set of isometries is closed under composition. The identity isometry is also an identity for composition, and composition is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
; therefore isometries satisfy the axioms for a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
. For a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, we must also have an inverse for every element. To cancel a reflection, we merely compose it with itself (Reflections are
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
s). And since every isometry can be expressed as a sequence of reflections, its inverse can be expressed as that sequence reversed. Notice that the cancellation of a pair of identical reflections reduces the number of reflections by an even number, preserving the parity of the sequence; also notice that the identity has even parity. Therefore all isometries form a group, and even isometries a subgroup. (Odd isometries do not include the identity, so are not a subgroup). This subgroup is a normal subgroup, because sandwiching an even isometry between two odd ones yields an even isometry. ∎ Since the even subgroup is normal, it is the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
to a
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
, where the quotient is isomorphic to a group consisting of a reflection and the identity. However the full group is not a
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
, but only a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
, of the even subgroup and the quotient group.


Composition

Composition of isometries mixes kinds in assorted ways. We can think of the identity as either two mirrors or none; either way, it has no effect in composition. And two reflections give either a translation or a rotation, or the identity (which is both, in a trivial way). Reflection composed with either of these could cancel down to a single reflection; otherwise it gives the only available three-mirror isometry, a glide reflection. A pair of translations always reduces to a single translation; so the challenging cases involve rotations. We know a rotation composed with either a rotation or a translation must produce an even isometry. Composition with translation produces another rotation (by the same amount, with shifted fixed point), but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point.


Translation, rotation, and orthogonal subgroups

We thus have two new kinds of isometry subgroups: all translations, and rotations sharing a fixed point. Both are subgroups of the even subgroup, within which translations are normal. Because translations are a normal subgroup, we can factor them out leaving the subgroup of isometries with a fixed point, the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
. :Proof. If two rotations share a fixed point, then we can swivel the mirror pair of the second rotation to cancel the inner mirrors of the sequence of four (two and two), leaving just the outer pair. Thus the composition of two rotations with a common fixed point produces a rotation by the sum of the angles about the same fixed point. :If two translations are parallel, we can slide the mirror pair of the second translation to cancel the inner mirror of the sequence of four, much as in the rotation case. Thus the composition of two parallel translations produces a translation by the sum of the distances in the same direction. Now suppose the translations are not parallel, and that the mirror sequence is A1, A2 (the first translation) followed by B1, B2 (the second). Then A2 and B1 must cross, say at ''c''; and, reassociating, we are free to pivot this inner pair around ''c''. If we pivot 90°, an interesting thing happens: now A1 and A2′ intersect at a 90° angle, say at ''p'', and so do B1′ and B2, say at ''q''. Again reassociating, we pivot the first pair around ''p'' to make B2″ pass through ''q'', and pivot the second pair around ''q'' to make A1″ pass through ''p''. The inner mirrors now coincide and cancel, and the outer mirrors are left parallel. Thus the composition of two non-parallel translations also produces a translation. Also, the three pivot points form a triangle whose edges give the head-to-tail rule of
vector addition In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
: 2(''p'' ''c'') + 2(''c'' ''q'') = 2(''p'' ''q''). ∎


Nested group construction

The subgroup structure suggests another way to compose an arbitrary isometry: : Pick a fixed point, and a mirror through it. # If the isometry is odd, use the mirror; otherwise do not. # If necessary, rotate around the fixed point. # If necessary, translate. This works because translations are a normal subgroup of the full group of isometries, with quotient the orthogonal group; and rotations about a fixed point are a normal subgroup of the orthogonal group, with quotient a single reflection.


Discrete subgroups

The subgroups discussed so far are not only infinite, they are also continuous (
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s). Any subgroup containing at least one non-zero translation must be infinite, but subgroups of the orthogonal group can be finite. For example, the
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of a regular
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
consist of rotations by integer multiples of 72° (360° / 5), along with reflections in the five mirrors which perpendicularly bisect the edges. This is a group, D5, with 10 elements. It has a subgroup, C5, of half the size, omitting the reflections. These two groups are members of two families, D''n'' and C''n'', for any ''n'' > 1. Together, these families constitute the rosette groups. Translations do not fold back on themselves, but we can take integer multiples of any finite translation, or sums of multiples of two such independent translations, as a subgroup. These generate the
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
of a periodic
tiling Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester The ...
of the plane. We can also combine these two kinds of discrete groups — the discrete rotations and reflections around a fixed point and the discrete translations — to generate the
frieze group In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetri ...
s and
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformatio ...
s. Curiously, only a few of the fixed-point groups are found to be
compatible Compatibility may refer to: Computing * Backward compatibility, in which newer devices can understand data generated by older devices * Compatibility card, an expansion card for hardware emulation of another device * Compatibility layer, compo ...
with discrete translations. In fact, lattice compatibility imposes such a severe restriction that, up to
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
, we have only 7 distinct frieze groups and 17 distinct wallpaper groups. For example, the pentagon symmetries, D5, are incompatible with a discrete lattice of translations. (Each higher dimension also has only a finite number of such
crystallographic 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 unc ...
s, but the number grows rapidly; for example, 3D has 230 groups and 4D has 4783.)


Isometries in the complex plane

In terms of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, the isometries of the plane are either of the form :\begin\mathbb&\longrightarrow&\mathbb\\ z&\mapsto&a+\omega z\end or of the form :\begin\mathbb&\longrightarrow&\mathbb\\ z&\mapsto&a+\omega\overline z\mbox\end for some complex numbers ''a'' and ω with , ω,  = 1. This is easy to prove: if ''a'' = ''f''(0) and ω = ''f''(1) − ''f''(0) and if one defines :\beging\colon&\mathbb&\longrightarrow&\mathbb\\ &z&\mapsto&\frac\mbox\end then ''g'' is an isometry, ''g''(0) = 0, and ''g''(1) = 1. It is then easy to see that ''g'' is either the identity or the conjugation, and the statement being proved follows from this and from the fact that ''f''(''z'') = ''a'' + ω''g''(''z''). This is obviously related to the previous classification of plane isometries, since: * functions of the type ''z'' → ''a'' + ''z'' are translations; * functions of the type ''z'' → ω''z'' are rotations (when , ω,  = 1); * the conjugation is a reflection. Note that a rotation about complex point ''p'' is obtained by complex arithmetic with :z \mapsto \omega (z - p) + p = \omega z + p(1 - \omega) where the last expression shows the mapping equivalent to rotation at 0 and a translation. Therefore, given direct isometry z \mapsto \omega z + a, one can solve p(1 - \omega) = a to obtain p = a/(1 - \omega) as the center for an equivalent rotation, provided that \omega \ne 1, that is, provided the direct isometry is not a pure translation. As stated by Cederberg, "A direct isometry is either a rotation or a translation.", quote from page 151


See also

*
Beckman–Quarles theorem In geometry, the Beckman–Quarles theorem, named after Frank S. Beckman and Donald A. Quarles Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all ...
, a characterization of isometries as the transformations that preserve unit distances *
Congruence (geometry) In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be ...
*
Coordinate rotations and reflections In geometry, two-dimensional 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 im ...
* Hjelmslev's theorem, the statement that the midpoints of corresponding pairs of points in an isometry of lines are collinear


References

{{Reflist


External links


Plane Isometries
Crystallography Euclidean plane geometry Euclidean symmetries Group theory Articles containing proofs