Chapter 19: Symmetry and Patterns Lesson Plan For All Practical Purposes Symmetry and Patterns Fibonacci Numbers and the Golden Ratio Symmetries Preserve the Pattern Rosette, Strip, and Wallpaper Patterns Notation for Patterns Symmetry Groups Fractal Patterns and Chaos Mathematical Literacy in Today’s World, 7th ed. 1 © 2006, W.H. Freeman and Company

Chapter 19: Symmetry and Patterns Symmetry and Patterns In the narrowest sense, symmetry refers to the mirror-image of an object. In the wider sense, symmetry includes notions of balance, similarity, and repetition. Symmetry allows us to appreciate patterns. Patterns that are pleasing exhibit elements of balance, similarity, and repetition. Symmetry often results in beauty through balance or harmonious arrangement. Symmetry is often esthetically pleasing. Symmetry – Equivalence or correspondence of form on opposite sides of a dividing line or plane about a center or an axis. Examples of symmetry and patterns: Crystals are symmetric in both their appearance and their atomic structure. A spiral pattern is an example of a kind of symmetry, such as in sunflowers, pinecones, and a nautilus. 2

Chapter 19: Symmetry and Patterns Fibonacci Numbers and the Golden Ratio The numbers in the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . This sequence begins with the number 1 and 1 again, and the next number is obtained by adding the two preceding numbers. Phyllotaxis – The spiral pattern that occurs in shoots, leaves, or seeds around a plant’s stem. The number of spirals in plants with phyllotaxis always occur according to consecutive Fibonacci numbers. Recursive rule – Sometimes a sequence of numbers is specified by stating the value of the first term and then giving an equation to calculate succeeding terms from preceding ones. Let Fn be the nth Fibonacci number, then F1 = 1 F2 = 2 and Fn + 1 = Fn + Fn − 1 for n  2 Leonardo of Pisa (“Fibonacci”) 3

Chapter 19: Symmetry and Patterns Fibonacci Numbers and the Golden Ratio The golden ratio, which is usually denoted by the Greek letter phi () is: 1 + √ 5 2 The Greeks considered this specific numerical proportion (known as the  = ≈ 1.618034… golden ratio in modern times) essential to beauty and symmetry. Golden Rectangle Rectangle whose ratio of height to width is that of the golden ratio. The esthetic claim of a golden rectangle—one whose length of sides is in the ratio of 1 to  —is the most pleasing of all rectangles. Greeks used straightedge and compass to construct the golden rectangle (as seen in the diagram above). 4

Chapter 19: Symmetry and Patterns Fibonacci Numbers and the Golden Ratio The Geometric Mean The quantity s = √ lw is the geometric mean of l and w. The Greeks were interested in cutting a single line segment of length l into lengths s and w where l = w + s, so that s would be the mean proportional between w and l. Thus, l/s = s/w and s2 = lw and s = √ lw . Surprisingly, the ratio  eventually arises. More generally, the geometric mean of Geometric Mean – The geometric mean of two numbers a and b is √ab . n numbers is the nth root of the product of all n factors: The geometric mean of x1, . . . , xn is n√ x1, . . . , xn Example: The geometric mean of 1, 2, 3, and 4 is: 4√ 1 × 2 × 3 × 4 = 4√ 24 = 24¼ ≈ 2.213 5

Chapter 19: Symmetry and Patterns Symmetries Preserve the Pattern Symmetry is described by symmetry, repetition, and balance. Balance refers to regularity in how the repetitions are arranged. Motif – The individual element of figure of the design. Pattern – How the copies of the motif are arranged. Preserve the pattern – Pattern looks exactly the same, with all the parts appearing in the same places after the motion is applied. Rigid Motion Rigid motion is one that preserves the size and shape of figures. In particular, any two points are the same distance apart after the motion as before. Any rigid motion of the plane must be one of: Reflection (across a line) Rotation (around a point) Translation (in a particular direction) Glide reflection (across a line) Glide reflection of footprints 6

Chapter 19: Symmetry and Patterns Rosette, Strip, and Wallpaper Patterns Symmetries of the Pattern A transformation of a pattern is a symmetry of the pattern if it preserves the pattern. A pattern is like a recipe for repeating a figure (motif) indefinitely. Patterns in the Plane Patterns in a two-dimensional plane can be categorized by the direction of the pattern’s repetition: No direction – the rosette patterns Exactly on direction (and its reverse) – the strip patterns More than one direction – the wallpaper patterns 7

Chapter 19: Symmetry and Patterns Rosette, Strip, and Wallpaper Patterns Rosette Patterns A pattern whose only symmetries are rotations about a single point and reflections through that point. Two classes of rosettes: Cyclic rosettes – without reflection symmetry (example: pinwheel). Dihedral rosettes – those with reflection symmetry (example: flower). Pinwheel – cyclic rosettes Flower with petals – dihedral rosettes 8

Chapter 19: Symmetry and Patterns Rosette, Strip, and Wallpaper Patterns Strip Pattern A pattern that has indefinitely many repetitions in one direction. It offers repetition and translation symmetry along the direction of the strip. Translation symmetry An infinite figure has translation symmetry if it can be translated (slid without turning) along itself without appearing to have changed. There are seven different strip patterns, as shown. 9

Chapter 19: Symmetry and Patterns Rosette, Strip, and Wallpaper Patterns A pattern in the plane that has indefinitely many repetitions in more than one direction. There are exactly 17 wallpaper patterns. Notations are used to describe four categories of symmetry patterns used. The standard abbreviation is written on top, and the full notation is below. 10

Chapter 19: Symmetry and Patterns Notation for Patterns Crystallographers’ notation is commonly used as the standard notation for patterns. Full notation of strip and wallpaper patterns consist of four symbols, described as follows: Notation for strip patterns First symbol indicates horizontal translation. Second symbol describes vertical reflection symmetry. Third symbol describes horizontal or glide reflection symmetry. Fourth symbol describes rotational symmetry. Notation for wallpaper patterns First symbol indicates if all rotation centers lie on the reflection line. Second symbol indicates the amount of rotational symmetry. Third symbol indicates mirror, glide, or no reflection symmetry. Fourth symbol describes the symmetry relative to an axis and an angle to the symmetry axis of the third symbol. 11

Chapter 19: Symmetry and Patterns Symmetry Groups Properties of Symmetries of Patterns If we combine two symmetries by applying first one and then the other, we get another symmetry. There is an identity, or “null,” symmetry that does not move anything, but leaves every point of the pattern exactly where it is. Each symmetry has an inverse, or “opposite,” that undoes it and also preserves the pattern. A rotation is undone by an equal rotation in the opposite direction. A reflection is its own inverse. A translation or glide reflection is undone by another of the same distance in the opposite direction. In applying a number of symmetries one after the other, we may combine consecutive ones without affecting the result. 12

Chapter 19: Symmetry and Patterns Symmetry Groups A group is a collection of elements {A, B, . . . } and an operation ס between pairs of them such that the following properties hold: Closure: The result of one element operating on another is itself an element of the collection (A ס B is in the collection). Identity element: There is a special element 1, called the identity element, such that the result of an operation involving the identity and any element is the same element (1 ס A = A and A ס 1 = A). Inverses: For any element A, there is another element, called its inverse and denoted A−1, such that the result of an operation involving an element and its inverse is the identity element (A ס A−1 = 1 and A−1 ס A = 1). Associativity: The result of several consecutive operations is the same regardless of grouping or parenthesizing, provided the consecutive order of operations is maintained: A ס B ס C = A ס (B ס C) = (A ס B) ס C. Symmetry Group of the Pattern The symmetries that preserve a pattern form the symmetry group of the pattern. 13

Chapter 19: Symmetry and Patterns Fractal Patterns and Chaos Fractals Another example of symmetry in which linear scaling is used. Similarity with changes of scale or showing different “proportions.” Example of similarities with changes of scale is the nested dolls from Russia (“matrioshka”). Each part of one doll (face, arm, and so on) has the same proportion and scaling factor to the corresponding part of a second doll. Chaos Used to describe systems whose behavior every time is inherently unpredictable. Fractal – A pattern that exhibits similarity at ever finer scales. Nested Dolls “Emperor’s Cloak” name of this work of art was produced by iterating a chaos-producing function. 14