It is the only isometry which belongs to more than one of the types described above. The identity 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. 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. That is, we obtain the same result if we do the translation and the reflection in the opposite order.)Īlternatively we multiply by an orthogonal matrix with determinant −1 (corresponding to a reflection in a line through the origin), followed by a translation. Neither are less drastic alterations like bending, stretching, or twisting.Īn isometry of the Euclidean plane is a distance-preserving transformation of the plane. However, folding, cutting, or melting the sheet are not considered isometries. There is one further type of isometry, called a glide reflection (see below under classification of Euclidean plane isometries). These are examples of translations, rotations, and reflections respectively. 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. Turning the sheet over to look at it from behind. ![]() Rotating the sheet by ten degrees around some marked point (which remains motionless).Shifting the sheet one inch to the right.For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Informally, a Euclidean plane isometry is any way of transforming the plane without "deforming" it. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections. The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. There are four types: translations, rotations, reflections, and glide reflections (see below under Classification § Notes). On the next page, we'll see tiles that DO flip over.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. The cats and the ducks are also "tiles" that translate/slide/glide left or right, up or down, to fill in the picture. ![]() Now take a look at the other pictures on this page. The original line XY is "translated" along the Y axis to make line X 1Y 1. In math class, we'd say that we can move a line along a graph by saying "X=Y" for the original line and "X 1 + 4 = Y 1" for the line that would be 4 boxes above it on a piece of graph paper. So, why do we call it "translation"? Well, we call that movement a "translation" because we "translate" the tile along the X-axis and the Y-axis. This kind of tessellation symmetry- tile repeating- is called Translation and/or Sliding. The tiles in this picture are copies of one another that are simply shifted from one place to another, without tilting or flipping them over or resizing them. The tessellation is made by repeating the tile over and over again, and fitting all the copies of the tile together. This is the basic "tile" shape of the first tessellation on this page.
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