Fractal Geometry, Tessellations and Symmetry Lesson 3 PDF

Fractal Geometry, Tessellations and Symmetry Lesson 3 GED0103 Math in the Modern World Institute of Arts and Sciences FAR EASTERN UNIVERSITY What is Fractal Geometry? What are fractals? Mathematical constructs characterized by self-sim...

Fractal Geometry, Tessellations and Symmetry Lesson 3 GED0103 Math in the Modern World Institute of Arts and Sciences FAR EASTERN UNIVERSITY What is Fractal Geometry? What are fractals? Mathematical constructs characterized by self-similarity. Geometric patterns that is repeated at ever smaller scales to produce irregular shapes and surfaces that can not be represented by classical geometry. Came from the Latin adjective “fractus” or verb “frangere” which means to break. named and popularized by mathematician Benoit Mandelbrot (1924 – 2010). In 1977, Mandelbrot wrote The Fractal Geometry of Nature, and he stated: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel is a straight line.” Some popular fractals are: Sierpinski triangle, Pascal’s triangle, Koch snowflake, fractal trees and Barnsley ferns What are fractals? Purely a wonder phenomenon, They are everywhere ▪ They are present in food, germs, plants, animals, mountains, forests, water and in the sky. They are repetitive, circular, looping recurrent, recursive, iterative and infinite a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity Fractals found in nature  Fractals found in nature Romanesco brocolli brocolli Fractals found in nature Queen Anne’s Lace Ferns Fractals found in nature clouds Mountain ranges Fractals found in nature Fractals found in nature Fractals found in nature FEATURES Common Techniques for Generating Fractals It has a fine structure at Escape-Time arbitrarily small scales. Iterated Function Systems It is too irregular to be easily Random described in traditional Euclidean geometric language. Recursive It is self-similar (at least approximately or stochastically). It has a simple and recursive definition. Example 1: Sierpinski Fractals Sierpinski triangle, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Named after Polish mathematician Waclaw Sierpinski, but appeared as a decorative pattern many centuries prior to his work. The Sierpinski triangle can be constructed as follows: 1. Start with an equilateral triangle (actual size does not matter) 2. Find the midpoint of each side. 3. Connect the midpoints by a straight line. 4. Observe that you created three more triangles one on top and two at the bottom. 5. Repeat the process with all other triangles. Example 1: Sierpinski Fractals 0 1 2 3 4 Iteration Math Notation Number of Triangles (No of times you have drawn a triangle) 0 30 1 1 31 3 2 32 9 3 33 27 4 34 81 Example 1: Sierpinski Fractals has appeared many centuries earlier in artwork, patterns and mosaics. here are some examples of floor tilings from different churches in Rome: Pascal’s Triangle a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. a number pyramid in which every number is the sum of the two numbers above. named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n=0 at the top (the 0th row). Other Fractal shapes What is Tessellations? A tessellation is a special type of tiling (a pattern of geometric shapes that fill a two-dimensional space with no gaps and no overlaps) that repeats forever in all directions. From the word “tessera” in Latin that means a small stone cube. They were used to make up 'tessellata' - the mosaic pictures forming floors and tilings in Roman buildings. The term has become more specialized and is often used to refer to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps. Tessellations figure prominently throughout art and architecture from various time periods throughout history, from the intricate mosaics of Ancient Rome, to the contemporary designs of M.C. Escher. What is Tessellations? While any polygon (a two-dimensional shape with any number of straight sides) can be part of a tessellation, not every polygon can tessellate by themselves In a tessellation, whenever two or more polygons meet at a point (or vertex), the internal angles must add up to 360°. Only three regular polygons(shapes with all sides and angles equal) can form a tessellation by themselves—triangles, squares, and hexagons. Examples: What is Tessellations? What about circles? Circles are a type of oval—a convex, curved shape with no corners. Circles can only tile the plane if the inward curves balance the outward curves, filling in all the gaps. While they can't tessellate on their own, they can be part of a tessellation, but only if you view the triangular gaps between the circles as shapes. Types of Tessellations Regular tessellations are composed of identically sized and shaped regular polygons. Semi-regular tessellations are made from multiple regular polygons. Only eight combinations of regular polygons create semi-regular tessellations. Meanwhile, irregular tessellations (or demiregular or polymorph tessellations) consist of figures that aren't composed of regular polygons that interlock without gaps or overlaps. As you can probably guess, there are an infinite number of figures that form irregular tessellations! Types of Tessellations Regular tessellations are composed of identically sized and shaped regular polygons. Semi-regular tessellations are made from multiple regular polygons. Only eight combinations of regular polygons create semi-regular tessellations. Meanwhile, irregular tessellations (or demiregular or polymorph tessellations) consist of figures that aren't composed of regular polygons that interlock without gaps or overlaps. As you can probably guess, there are an infinite number of figures that form irregular tessellations! Look at a vertex A vertex is just a "corner point". What shapes meet here? Three hexagons meet at this vertex, and a hexagon has 6 sides. So this is called a "6.6.6" tessellation. For a regular tessellation, the pattern is identical at each vertex! Regular Tesselations Triangles The interior angle of each equilateral triangle is 60 degrees. 60 + 60 + 60 + 60 + 60 + 60 = 360 degrees Squares The interior angle of each square is 90 degrees. 90 + 90 + 90 + 90 = 360 degrees Hexagons The interior angle of each Hexagon is 90 degrees. 120 + 120 + 120 = 360 degrees What is Symmetry? Symmetry is a concept that states that when we move one shape, such as turning, flipping, or sliding it, it becomes identical to the other. The word “Symmetry” comes from the Greek word which implies “to measure together”. When two or more parts are identical after a flip, slide or turn. The two objects are claimed to be symmetrical, if they have the identical size and shape with one object having a different orientation from the first. Some great examples of symmetry in nature are starfish, peacocks, turtles, sunflowers, honeycombs, snowflakes, rainbows, etc. Types of Symmetry 1. Translational Symmetry 2. Rotational Symmetry 3. Reflectional Symmetry 4. Glide Reflection Symmetry Types of Symmetry: Translational Symmetry Translational symmetry is where a figure or an image is translated at a set distance in the same direction as the original. (moving or sliding) The spaces between points, angles, sizes, and shapes of the figure will not change. The only thing that changes is its location. You may move it right or left. You may move it up or down. You may move it through a combination of these two, but these are the only possibilities. Types of Symmetry: Rotational Symmetry also known as radial symmetry, is where a shape or an image looks precisely similar to the orotational symmetry by original form or image after some rotation. (rotating or turning) We count the number of turns it takes, also referred to as order, for a shape to look the same. For example, a rectangle has an order of 2, and a five-point star has an order of 5. You can find rotational symmetry naturally in sea stars, jellyfish, and sea anemones. It also appears in human-made objects like airplane propellers, Ferris wheels, dartboards! Types of Symmetry: Reflectional Symmetry also known as reflexive, or mirror symmetry, refers to a shape that can be reflected over a line and still resemble its original shape. This is perhaps the most intuitively understood type of symmetry, observed when one half of a figure is a mirror image of the other half. A classic example is a heart shape or the wings of a butterfly. The line which a reflection takes place over is known as the line of symmetry. It’s also important to note that some shapes can have multiple lines of symmetry. Take a square, for example - you can draw four lines of symmetry on a square—one horizontally across the middle, one vertically down the middle, and two going diagonally each way. Types of Symmetry: Glide Reflection Symmetry best thought of as a hybrid between reflection and translational symmetry. It is a type of symmetry where the figure or image looks precisely the original when it is reflected over a line and then translated at a given distance in a given direction. The footprints trail is one of the best examples of Glide Reflection Symmetry. This type of symmetry involves both processes but in a specific order; reflection over a line and translation along the line. A shape must first be reflected and then translated in any direction for glide reflection to have taken place. As in translational symmetry, glide-reflectional symmetry exists only for infinite patterns. Line of Symmetry The line of symmetry is an imaginary line or axis that passes through the center of any picture, shape or object and it is divided into two identical halves. English Alphabets with vertical line of symmetry: A, H, I, M, O, T, U, V, W, X, Y. English Alphabets with horizontal line of symmetry: B, C, D, E, H, I, K, O, X English Alphabets with no line of symmetry: F, G, J, L, N, P, Q, R, S, Z.