Nature by Numbers: Symmetry

• The natural world exhibits beauty, shapes, and patterns that can be mathematically described.
• Ian Stewart mentions in "Nature by Numbers" that we live in a universe of patterns.

Symmetry

• Exact correspondence of form and configuration exists on opposite sides of a dividing line, plane, or around a center or axis.
• An imaginary line can be drawn across an object, resulting in mirror images.

Rotational Symmetry

• It is called radial symmetry.
• In biology, objects exhibit this when similar parts are arranged around a central axis, maintaining appearance after rotation.

Calculating Angle of Rotation

• To compute the angle of rotation, divide 360 by the number of folds.
• For example, a 5-fold symmetry requires dividing 360 by 5, resulting in 72 degrees. Therefore, rotating a 5-fold shape by 72 degrees will result in the matching appearance as the original position.

Translational Symmetry

• Objects showing symmetry do not change shape or size when moved.
• Movement does not involve reflection or rotation.

Fractals

• They are self-similar, never-ending patterns across scales.
• The image reappears regardless of magnification.

Spirals

• Curved patterns revolve around a central point.

Spots and Stripes

• They are exhibited on animals

Flower Petals

• Considered aesthetically pleasing.
• Vibrant colors and fragrant odors are appealing.
• Flowers have varying petal counts; having 5 petals is most common.

Number Patterns and Sequences (Pascal)

• Refers to Pascal's triangle with numbers arranged in rows.

Fibonacci Sequence

• Numbers often occur in nature.
• Devised during the Middle Ages.
• Leonardo Pisano Bigollo, an Italian mathematician, discovered it.
• Born in 1170 and died in 1240
• Introduced the Arabic number system to Europe.
• "Fibonacci" is a shortened version of the Latin "Filius Bonacci," meaning "The son of Fibonacci."
• The Fibonacci sequence is an integer in the infinite sequence 0, 1, 1, 2, 3, 5, 8, 13...
• It starts with zero or one.
• Subsequent numbers are the sum of the two preceding numbers.
• 2= 1+1
• 3 = 1+2
• 5 = 2+3
• In nature, many plants arrange leaves around the stem according to Fibonacci numbers, such as spirals in sunflowers.

Mathematics as a Language

Objective: To identify symbols, variables, and sentences used in mathematics.

Equalities

• =: Equals, means "is the same as"
• ≅: Congruent, means "same size and shape"
• ≈: Similar, means "same shape"

Inequalities

• <: Is less than

• : Is greater than

• ≤: Is less than or equal to
• ≥: Is greater than or equal to
• ≠: Not equal to

Parts of Speech for Mathematics

• Numbers: They represent quantity. These parallel nouns in English.
• Operation Symbols: Symbols like +, -, ×, ÷ act as connectives, similar to verbs.
• Relation Symbols: Symbols such as <, >, =, ≤, ≥ are used for comparison and act as verbs.
• Grouping Symbols: ( ), [ ], and { } associate groups of numbers and operators.
• Variables: Letters representing quantities act as pronouns.

Mathematical vs. English Language

• Expression: A name given to a mathematical object of interest (e.g., 5, 2+3 are expressions)
• Sentence: States a complete thought (e.g., 3+4=7 is a sentence)
• Numbers have lots of different names for the ame object

Translating Verbal Phrases

• Plus, add, increased by, sum total, more than symbols: +
• Minus, subtract, decreased by, diminished by, difference, subtracted from, less than symbols: -
• Multiply, times, product symbols: x
• Divide, quotient symbols: ÷

Mathematical Language

Objective: To translate English words, phrases, and sentences into mathematical expressions.

Characteristics:

• Precise and can be stated clearly.
• Concise and can be stated briefly.
• Powerful as it expresses complex ideas simply.
• Non-temporal, so it lacks tenses.
• Contains vocabulary and parts of speech.

Arithmetic Sequence

• A sequence where each value increases or decreases by a constant amount.
• Each term is obtained by adding the common difference "d" to the previous term.
• Common difference: d= a2-a1, where a = value.
• nth term: an= a1 + (n − 1)d, an: the last term to solve for.
• Sum of n terms: sn= [a1+ an]

Examples of Calculating the Nth Term and Common Difference

• nth term: an = a1+(n−1)d
• Common difference: d= a– a1

Geometic vs Arithmetic Sequence

• Arithmetic
• In this, the differences between every two consecutive numbers are the same.
• Identified by the first term (a) and the common difference (d).
• Linear relationship between the terms
• Geometric
• In this, the ratios of every two consecutive numbers are the same.
• Identified by the first term (a) and the common ratio (r).
• Exponential relationship between the terms.

Finite Geometric Series

• Formula for the sum of the first n terms: Sn = a1(1-rn)/1-r
• Where:
  • Sn = Sum of the first n terms
  • a1 = First term
  • n = Number of terms
  • r = Common ratio

Illustrative Example

• If Joey doubled savings each week, week savings can be computed by S=a1(1-rn)/1-r

Infinite Geometric Series

• Formula for sum to infinity : s∞=a1/1-r
• Where Sum to infinity
• a1= first term.
• r =common ratio Set theory — is the branch of mathematics that studies sets or the mathematical science of the infinite.

A relation

• A set of ordered pairs.
• A relation can be represented by a set of ordered pairs, a table, a graph or a mapping.

The DOMAIN

• X-coordinates of the ordered pairs.
• ExDomain - {-2, 3, 4, 5}

The RANGE

• Y-coordinates of the ordered pairs.
• ExRange -{-2, 1, 2, 3}

The INVERSE

• Found by switching the coordinates of each ordered pair. {(3, 4), (1, -2), (-2, 5), (2, 3), (-2, -2)}

FUNCTION

• A relation when each element of the domain is paired with exactly one element of the range.
• For every x there is exactly one y.
• The x-coordinate cannot repeat.
• Equations that are functions: {(3, 2), (4, -1), (-3,-2), (9,0)}; {(9, -1), (6, -1), (-9, 2), (-7, -1)}
• Equations that are not functions: {(1, 2), (2, 4), (1, 5)}; {(-9, 2), (-9, 1), (3, 4), (5, -6)}
• Vertical Line test: Test used to decide if a graph is a function
• • If no vertical line can be drawn so that it intersects that graph more than once then graph is a function.
• Function Notation: y is a function, the y is replaced with f(x)

Function

y

-f(x) +7

Evaluating Functions

• If f (x)=4x + 1, find f(-2).
• -Substitute -2 in for x.

Example

- -

• f (x) =3x + 6, find f(x - 1).
• -Substitute (x - 1) in for x.

Examples

• If f (x) = 2x7, find each of the values

Harmonic Sequence

• A sequence where the reciprocals of the terms form an arithmetic sequence. Fibonacci Sequence
• Integer in the infinite sequence
• Simple rule: add the last two to get the next. Set Theory is the branch of mathematics that studies sets or the mathematical science of the infinite.

Set

• defined collection of objects; the objects are called the elements or members.
• J = {5, 11, 19,…}.
• symbol e is used to denote that an object is an element of a set.
• Example: Set A = {1, 2, 3}. We can see that 1, but 5 of A.

Roster Method

• Tabulation method - the set are enumerated and separated by comma. A = { a, e, i, o, u}
• Builder Notation
• describe the elements or members of the set A = {x|x is a collection of vowel letters}

FINITE SET

• whose elements are limited or countable, and the last element can be identified. ExA = {1,2,3,4,5,6,7,8,9}

is a set whose elements are unlimited

• or uncountable, and the last element cannot be specified. An B = {1,2,3,4, 5, 6,7,8...}

Null SET

• Is a unique set with no element A or THE CARDINALLY OF A SET is the elements/members in the set.

Notation U

• that includes all of the elements in a particular discussion.
• Subset

Subset

• We can form what is called a subset.
• is a subset of B if every element of A is an element of B.
  • Notation: ACA
  • For each set A, AA
  • For each set B, B
• proper subset of B if A C Band AB.
• The word "or" is inclusive.
• Example

Venn Diagram

are diagrams that make use of geometric shapes to show relationship between them. The intersection of A and B is A∩B = {x |xe A andre B}

Unions

Unions consist of the combining of all numbers that are present