Application in architecture
Shapes proportioned according to the golden ratio (1:1.618) have long been considered aesthetically pleasant to the eye as it suggests a fine balance between symmetry and asymmetry. The ratio has been used in architecture for centuries.
Famous examples include the Egyptian pyramids, the ancient Greek temple Parthenon and the Notre Dame of Paris. It is still being used in some modern architecture like the United Nations building in New York.
The Neufert, which is an architecture book which is mostly based on the golden ratio, is still an important reference book today.
Application in our lives
The usefulness of mathematics is very subjective. For many it is seen in terms of the arithmetic skills which are required at home , in the office or workplace, some see mathematics as the basis of scientific development and modern technology while some emphasize the increasing use of mathematical techniques as a management tool in commerce and industry.
Medicine and the biological sciences, geography and economics, business and management studies all require the application of math.
Math is also useful in terms of saving money in the bank as people need to be able to calculate and weigh their options in order to gain maximum interest.
Application in nature
The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the number of branches in plants to the breeding habits of animals and insects.
The growth of petals on plants abides to the Fibonacci sequence too. Lilies have 3, wild roses have 5, delphiniums have 8, some daisies have 13 and the list goes on.
Other than the Fibonacci sequence, the golden ratio (1:1.618) applies to nature too. The fins on a dolphin, the body of an ant, bodies and wings of birds and flying insects are all proportionate to the golden ratio. Shells, such as those of snails or sea-life, are also built to such proportions.
Application in human
It has been discovered that the length of the elbow down to the fingertips to the length of the arm of the average human is of the golden ratio (1:1.618).
Other discoveries that involve the golden ratio include the ratio of the length between the navel and feet to one's height, the ratio of the length of mouth to width of nose and the ratio of the length of face to the width of face.
However, do not start measuring faces and calculate the ratio, because it only applies to the "ideal human face" which is determined by scientists and artists.
Our hand follows the Fibonacci sequence too. We have two hands, three sections in our fingers, five fingers on each hand and eight articulated fingers (thumbs are not).
Application in music
Many composers have applied the Fibonacci numbers or the golden ratio in the music they compose, the most famous one being Mozart and the most recent one being Bela Bartok, a Hungarian composer.
It is said that Mozart wrote down mathematical equations beside his music scores.
Mozart also divided most of his piano sonatas into two distinct sections whose lengths reflect the golden ratio, though there is much debate about whether it is done on purpose.
Some musical instruments display the use of Fibonacci numbers and the golden ratio, for example the violin and the piano. The foundation of a scale is based around the third and fifth tones, there are eight notes in an octave and thirteen notes that separate each octave. The pianos keyboard is another obvious example. Within the scale of thirteen keys, there are eight white keys and five black ones, which are divided into groups of two and three. There is also some golden ratio involved in the construction of a violin.
Application in information technology
Information technology refers to the use of computers and software to convert, store, protect, process, transmit, and retrieve information. Computational theory, algorithm analysis, formal methods and data representation are just some computing techniques that require the use of mathematics.
Computational theory and algorithm analysis deals with whether and how efficiently a computer is able to solve problems. Formal methods are mathematically based techniques for the specification, development and verification of software and hardware systems.
Data representation refers to how computers exchange and process information using the ones and zeros of binary, rather than the more inconvenient ten-digit decimal system. Binary basically means composed of two parts. The binary number system was started by Gottfried Leibniz back in 1666. It makes information processing simpler. For a processing system to work there must be at least two symbols therefore binary is the smallest numeral system that is usable. A CPU can only recognize two states, on or off, but from this on-off state, everything works.