Изометрии евклидовой плоскости: виды и свойства

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 (see below ). The set of Euclidean plane isometries forms a group under composition: the Euclidean group 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.

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 of the picture. These are examples of translations, rotations, and reflections respectively. There is one further type of isometry, called a glide 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.

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.)

The set of translations and rotations together form the rigid motions or rigid displacements. This set forms a group under composition, the group of rigid motions, a subgroup of the full group of Euclidean isometries.

Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a reflection group.

Adding more mirrors does not add more possibilities (in the plane), because they can always be rearranged to cause cancellation.

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 of the full Euclidean group 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. Since the even subgroup is normal, it is the kernel of a homomorphism to a quotient group, where the quotient is isomorphic to a group consisting of a reflection and the identity. However the full group is not a direct product, but only a semidirect product, of the even subgroup and the quotient group.

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 proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point.

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.

The subgroups discussed so far are not only infinite, they are also continuous (Lie groups). 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 of a regular pentagon 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, Dn and Cn, 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 of a periodic tiling 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 groups and wallpaper groups. Curiously, only a few of the fixed point groups are found to be compatible with discrete translations. In fact, lattice compatibility imposes such a severe restriction that, up to isomorphism, 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 groups, but the number grows rapidly; for example, 3D has 230 groups and 4D has 4783.)