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Lesson 4: September 17, 2024 Difference between Transformation and Isometry The Mathematics of Patterns and Symmetries - A transformation changes the size, shape, or position of a figure and creates...

Lesson 4: September 17, 2024 Difference between Transformation and Isometry The Mathematics of Patterns and Symmetries - A transformation changes the size, shape, or position of a figure and creates a new figure. Pattern and Symmetries - A geometry transformation is either rigid or non-rigid; another word for a rigid transformation Pattern is "isometry". An isometry, such as a rotation, - A pattern is a repeated arrangement of translation, or reflection, does not change the numbers, shapes, colors and so on. The Pattern size or shape of the figure. A dilation is not an can be related to any type of event or object. If isometry since it either shrinks or enlarges a the set of numbers are related to each other in a figure. specific rule, then the rule or manner is called a pattern. Sometimes, patterns are also known as Types of Transformations a sequence. Patterns are finite or infinite in - Translation numbers. - Reflection - A pattern is a regularity in the world, in human- - Rotation made design, or in abstract ideas. - Dilation - As such, the elements of a pattern repeat in a Note: The initial object to be transformed is called predictable manner. pre-image and the transformed object is called - A geometric pattern is a kind of pattern formed of image. geometric shapes and typically repeated like a wallpaper design. Any of the senses may directly Translation observe patterns. - A mathematical term used in geometry to - Patterns are finite or infinite in numbers. describe a function that moves an object a certain distance. The object is not altered in any Symmetries other way. - Symmetry is defined as one shape is exactly like - Example: the other shape when it is moved, rotated, or flipped. - Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Reflection - It is a transformation in which the figure or object Transformation and Symmetries is mirror image of the other. - It is ideal to start with the concept of motif. - Examples: - Any artistic creation starts with a motif. According to Grünbaum and Shephard, a motif is ”any non- empty plane set” (1987). Any object drawn in a plane is a motif. For example, a sketch of fish is a motif. When you repeat the drawing of the fish in the plane not only once, but several times, you have a pattern. - A pattern can be described as “repetitions of a motif in the plane” (Grünbaum and Shephard, Rotation 1987). An isometry is the rotation of a motif in a - It is a transformation that turns a figure about a fixed angle about a fixed point. Each rotation of a fixed point called the center of rotation. figure is an isometry. The image of the basic Rotations can be done clockwise or motif under the additional number of rotation is a counterclockwise. pattern (Renee Scott, 2008). - Examples:  Frieze Patterns - The name Frieze Patterns is attributed to John Conway, an English mathematician who is active in finite theory, knot theory, number theory, combinatorial game theory and coding theory. (Trivia, he died April 11, 2020 due to COVID19.) - Frieze patterns are patterns that repeat in a straight vertical or horizontal line. Frieze patterns are found in architecture, fabrics, and wallpaper borders, just to name a few. There are seven Dilation types of frieze patterns, and we will discuss each - A dilation is a transformation that changes the type and how to identify them. Archaeologists size of a figure. It can become larger or smaller, often use their knowledge of frieze patterns to but the shape remains the same. classify the artifacts that they find. - Examples: Different Types of Frieze Patterns - Hop A pattern which only involves translation. Glided Reflection - Translation and reflection can be combined to yield an effect shown below. - Step It is a combination of translation and reflection shown by the following figure, Conway also called it glide reflection symmetry. Observe and evaluate this image - Slide - The image has translation, reflection and/or The third consists of translation and vertical rotation for the given image but not dilation since reflection symmetries. all triangles have the same size. - Can you identify the type of transformation/s that is/are being used? - Spinning Hop - Spinning Jump It contains translation and rotation (by half turn It contains all symmetries (translation, or rotation at 180° angle) symmetries horizontal and vertical reflection, and rotation) - Spinning Sidle It contains translation, glide, reflection and Wallpaper Groups rotation (by a half-turn or rotation at 180° If translation symmetry is added in a second, angle) symmetries independent direction, one gets wallpaper groups. It turns out there are only 17 different wallpaper groups (again, considering only discrete group) View this link to see all wallpaper groups. https://www2.clarku.edu/faculty/djoyce/wallpaper/ seventeen.html Filipino Arts Filipino indigenous fabrics showcasing patterns and symmetries - Jump It contains translation and horizontal reflection symmetries What is symmetry? Symmetry comes from a Greek word symmetria meaning ‘to measure together' and is widely used in the study of geometry. Mathematically, symmetry means that one shape becomes exactly like another when you move it in some way: turn, flip or slide. For two objects to be symmetrical, they must be the same size and shape, with one object Filipino Arts and Symmetry having a different orientation from the first. There can also be symmetry in one object, such as a face. If you draw a line of symmetry down the center of your face, you can see that the left side is a mirror image of the right side. Not all objects have symmetry; if an object is not symmetrical, it is called asymmetric. Symmetry and Patterns in Medical Science - Symmetry and pattern is widely used in fields of mathematics and arts, it also appears in chemistry and biology. The human anatomy for example is widely known to be symmetrical in nature. And most animals exhibit symmetry. - There are two kinds of symmetry: “Symmetry establishes a ridiculous and wonderful Bilateral symmetry, in which an object has cousinship between objects, phenomena and two sides that are mirror images of each other. theories outwardly unrelated: terrestrial magnetism, Radial symmetry, this is where a center point woman's veils, polarized light, natural selection, the and numerous lines of symmetry could be theory of groups, structure of space, vase designs, drawn. One good example is the spider web. quantum physics, scarabs, flower petals, X-ray Oftentimes in the field of medical science, interference patterns, cell division in sea urchins, symmetry is the basis to detect anomalies in equilibrium positions of crystals, Romanesque human bodies like disfigured parts of the body cathedrals, snowflakes, music, the theory of wherein one part being larger or smaller than relativity…” - Herman Weil its counterpart. - Patterns in nature are visible regularities of form Why do we find symmetries pleasurable to look at? found in the natural world. These patterns occur “I would claim that symmetry represents order, and in different contexts and can sometimes be we crave order in this strange universe we find shaped mathematically. Patterns in living things ourselves in,” - Alan Lightman, physicist are explained by the biological processes of natural and sexual selection. One good example Tessellation of patterns found in nature. A pattern covering a plane by fitting together - Bacterial Population Growth replicas of the same basic shape. The word Under favorable conditions, a growing tessellation comes from Latin word tessera, which bacterial population doubles at regular means a square tablet or die used in gambling. intervals. Growth is by geometric progression: 1,2,4,8 or 2n where n is number of Tessellations have been created by nature and generations. This is called exponential growth. man either by accident or design. Examples range from hexagonal pattern of bees’ Symmetries honeycomb, snakeskin, or a tiled floor to intricate decorations. Different Types of Tessellation - Regular Tessellation - Demi-Regular Tessellation A tessellation made up of congruent regular Is an edge-to-edge tessellation, but the order polygons which have the following properties: or arrangement of polygons at each vertex is - The tessellation must tile a floor (that goes not the same. on forever) with no overlaps or gaps. - The tiles must be the same regular polygons. Tessellations in Nature - Semi-Regular Tessellation Also known as Archimedean Tessellations are regular tessellations of two or more different polygons around a vertex which has the same arrangement of polygons. Fractals - The function which iterates a figure to make it smaller and smaller or bigger and bigger using a scaling factor is called Fractals. Fractals are mathematical constructs characterized by self- similarity. This means that as one examines finer and finer details of the object, the magnified area is seen to be like the original but is not identical to it. - Note: The process of dilation in transformation is somewhat similar but different from fractals as the former is basically the same figure being smaller or larger. The latter is just being categorized as being almost identical to the original. - Fern is one good example of fractals. Each branch of leaf is like one another. Iteration - Iteration means repeating a process over and over. In mathematics, iteration means repeating a function over and over. The Iterative Function System (IFS) is a method for generating fractals involving a large number of calculations of a simple formula. Recursion is a special kind of iteration. With recursion there is given starting information and a rule for how to use it to get new information.