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
© 2006, W.H. Freeman and Company
Mathematical Literacy in
Today’s World, 7th ed.
Chapter 19: Symmetry and Patterns
Symmetry and Patterns
 Symmetry and Patterns
Symmetry –
 In the narrowest sense, symmetry refers to
Equivalence or
the mirror-image of an object.
correspondence
 In the wider sense, symmetry includes
of form on
notions of balance, similarity, and repetition.
opposite sides of
 Symmetry allows us to appreciate patterns.
a dividing line or
 Patterns that are pleasing exhibit elements of
plane about a
balance, similarity, and repetition.
center or an axis.
 Symmetry often results in beauty through
balance or harmonious arrangement.
 Symmetry is often esthetically pleasing.
 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.
Chapter 19: Symmetry and Patterns
Fibonacci Numbers and the Golden Ratio
 Fibonacci Numbers
 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”)
Chapter 19: Symmetry and Patterns
Fibonacci Numbers and the Golden Ratio
 The Golden Ratio
 The golden ratio, which is usually
denoted by the Greek letter phi ()
is:
1+ √5
≈ 1.618034…
 =
2
 The Greeks considered this specific
numerical proportion (known as the
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).
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
Geometric Mean –
lengths s and w where l = w + s, so
The geometric
that s would be the mean proportional
mean of two
between w and l. Thus, l/s = s/w and
numbers a and b
s2 = lw and s = √ lw . Surprisingly, the
is √ab .
ratio  eventually arises.
 More generally, the geometric mean of
n numbers is the nth root of the product of all n factors:
The geometric mean of x1, . . . , xn is
n
√ x 1, . . . , x n
 Example: The geometric mean of 1, 2, 3, and 4 is:
4
√ 1 × 2 × 3 × 4 = 4√ 24
= 24¼ ≈ 2.213
Chapter 19: Symmetry and Patterns
Symmetries Preserve the Pattern
 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
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
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
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.
Chapter 19: Symmetry and Patterns
Rosette, Strip, and Wallpaper Patterns
 Wallpaper Pattern
 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.
Chapter 19: Symmetry and Patterns
Notation for 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
1.
2.
3.
4.
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
1.
2.
3.
4.
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.
Chapter 19: Symmetry and Patterns
Symmetry Groups
 Properties of Symmetries of Patterns
1. If we combine two symmetries by applying first one and then the
other, we get another symmetry.
2. There is an identity, or “null,” symmetry that does not move
anything, but leaves every point of the pattern exactly where it is.
3. 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.
4. In applying a number of symmetries one after the other, we may
combine consecutive ones without affecting the result.
Chapter 19: Symmetry and Patterns
Symmetry Groups
 Group
 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 ‫ ס‬B 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 ‫ ס‬A = A and A ‫ ס‬1 = 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 ‫ ס‬A−1 = 1 and A−1 ‫ ס‬A = 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 ‫ ס‬B ‫ ס‬C = A ‫( ס‬B ‫ ס‬C) = (A ‫ ס‬B) ‫ ס‬C.
 Symmetry Group of the Pattern
 The symmetries that preserve a pattern form the symmetry group
of the pattern.
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
chaosproducing
function.