math

• Mathematics

  A language
• Asymmetric

  An object that is not symmetrical
• Symmetry


  Mathematically, symmetry means that one shape becomes exactly like another shape when you move it in some way like rotate, flip, or slide
• Line or bilateral symmetry


  • Evident in most animals, including humans, where the left and right portions are exactly the same
• Golden ratio


  The ratio of successive Fibonacci numbers approaches the number φ, approximately equal to 1.618
• Fibonacci sequence


  A sequence where each number is the sum of the two preceding ones, starting from 0 and 1
• Given any real number, its square is nonnegative
• Angle of rotation


  The smallest angle that a figure can be rotated while still preserving the original formation
• Radial or rotational symmetry


  • When an object can be rotated by several degrees and still achieve the same appearance as the original position
• Snowflakes are not perfectly symmetric due to the effects of humidity and temperature on the ice crystal as it forms
• Snowflakes


  • Have six-fold symmetry, with an angle of rotation of 60°
• Fold rotational symmetry


  A figure has a rotational symmetry of order n if 1/n of a complete turn leaves the figure unchanged
• Honeycomb tiling


  • The hexagonal formation of a honeycomb enables the bee colony to maximize their storage of honey using the smallest amount of wax
• Hexagonal formations are more optimal in making use of available space compared to square formations
• Packing problems


  Finding the optimum method of filling up a given space such as a cubic or spherical container
• For square packing, the percentage of the square's area covered by circles is approximately 78.54%
• The hexagonal construction of beehives is an example of an optimal geometric pattern that is simple, beautiful, and functionally efficient
• Numbers


  Have many different names in mathematics, similar to synonyms in English
• There are conventions regarding the naming of 'nouns' in mathematics, just as there are in English
• Mathematics meets all the requirements of a language
• For hexagonal packing, the area of each equilateral triangle is √3 cm^2
• Mathematical expression


  A name given to some mathematical object of interest, often a number, set, or function
• Mathematical sentence


  A correct arrangement of mathematical symbols that states a complete thought
• Mathematical sentences have verbs, just as English sentences do
• Characteristics of a language

  • Contains a vocabulary of words or symbols
  • Meaning must be attached to the words or symbols
  • Employs grammar, a set of rules that outline how vocabulary is used
  • Has a syntax that organizes symbols into linear structures or propositions
  • Has a narrative or discourse consisting of strings of syntactic propositions
  • Has a group of people who use and understand the symbols
• Animal prints and patterns


  Distinctive patterns exhibited in the external appearances of animals, believed to be governed by mathematical equations
• Geometric mean

  The nth root of the product of n numbers
• Hexagon with 300 dots - How many dots are there on each side?
• Types of reasoning

  • Inductive
  • Deductive
• The Fibonacci sequence can be defined recursively as F1 = 0, F2 = 1, and Fn = Fn-1 + Fn-2 for n ≥ 3
• Mathematical 'nouns'

  • Numbers or values that are manipulated in a mathematical sentence
  • Variables (symbols or letters that may have one or more possible value)
  • Constants (numbers with fixed value)
• Inductive Reasoning


  A process of reasoning from a specific observation to a general theory of a population. The conclusion formed using inductive reasoning is called a conjecture which may or may not be true.
• Snail shells start as fragile and colorless protoconch, then harden as the snail consumes calcium, and expand proportionately as the snail grows, resulting in a refined spiral structure
• Fibonacci spiral/golden spiral


  A type of logarithmic spiral made up of square blocks arranged in order by Fibonacci numbers
• Fibonacci numbers in nature - Seeds on a sunflower arranged in clockwise or counterclockwise spirals with consecutive Fibonacci numbers of spirals (e.g. 21, 34, 55)
• Other phrases to watch out for

  • 'the quantity' which indicates a grouping symbol
  • 'square', 'cube' and 'raised to the nth power' which indicate exponents
  • 'square root, 'cube root' and 'nth root' that signify radicals
• Sentence


  Something that expresses a complete thought
• Other mathematical symbols

  • Grouping symbols (parentheses)
  • Exponents and radicals
  • Decimal point
  • Percent
• Translating mathematical sentences to verbal phrases

  1. The difference of x and three is equal to the product of two and the square of x

  2. The ratio of six times y and three times z is greater than the product of three, y and z

  3. The product of the sum and difference of a and b is the square of a and the square of b

  4. The cube root of the product of eighteen times x raised to the fifth power is less than or equal to the quotient of y and two
• Open sentences


  Sentences that are true depending upon the number that is chosen for the variable