Understanding Translations, Reflections, and Rotations in Geometry
When considering shapes in spaces, it is imperative to address the movements of shapes in various ways. Typically, there are considered to be three ways in which shapes can be moved. These ways are: flipped, slid, or turned. The shape before it is moved in anyway becomes known as the preimage. The preimage becomes the image once the move occurs. Both the preimage and the image are identical in all aspects except for the coordinates within space they exist.
When a shape is flipped over a line, it is known as a reflection. Reflections are best understood when considering a mirror and how objects appear flipped when viewed in the mirror. The line, over which the shape is flipped, becomes known as the line of reflection. In reflections, it is important to remember that the shapes always keep the same size and shape, but the images are opposite of the original image (reversed). In this way, a reflection is a mirror image of the shape provided. Your own reflection as you look into the mirror is an example of this process of reflection.
When a shape is turned, it is called a rotation. A rotation of a shape occurs around one point. The shape is moved the point by any number of degrees. Since it occurs around one point, every rotation has a center and an angle. The rotation can occur in either a clockwise or counterclockwise direction. Therefore, every rotation can be described in terms of the angle by which the rotation occurred. When an object is rotated one hundred eighty degrees or half way around it is referred to as a half turn. When an object is rotated ninety degrees or one fourth of a turn, it is referred to as a quarter turn. In considering rotations, the number of matches the object will make as it completes one rotation is known as the order. There cannot be an order of one, as if the object requires one full rotation to align; it really does not show rotational symmetry.
In order for a tessellation to be considered as having rotational symmetry it must represent the exactly same tessellation as the original after it has been rotated. If the tessellation does in fact show the same result as the original after it has been rotated, then it can be considered to be a tessellation with rotational symmetry.
Shapes can be moved in multiple ways at one time. Objects could be slid and reflected. A glide reflection happens when a reflection is combined with a translation. In the glide reflection, the translation must occur along the line of reflection or mirror line. The glide reflection is the only type of symmetry which involves more than one step.
In order for a tessellation to demonstrate glide reflective symmetry, the tessellation after the glide reflection occurs must show the same result as the original tessellation before the glide reflection. Any tessellation which shows both reflective symmetry and translational symmetry will automatically have glide reflectional symmetry. However, all tessellations which have glide reflective symmetry do not necessarily have reflective symmetry or translational symmetry.
When recording any form of transformation, one will utilize the coordinate system. In the coordinate system, ordered pairs are used to represent points on a graph. When looking at a graph, the horizontal axis is referred to as the x axis. In a set of ordered pairs, the first number in the ordered pair refers to the x point. This tells you how many spaces to move horizontally. The vertical axis is referred to as the y axis. In ordered pairs, the second number refers to the y point. This tells you how many spaces to move vertically. Therefore, the ordered pair (6, 5) would tell you to move horizontally (over) six places and then up (vertically) five places. The space you come to on the graph is the spot represented by the given ordered pair. It is through the use of ordered pairs that one can describe the placement of objects on a plane and later, how they have been transformed.
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