Reported by: JOYLIN SOLQUIO PAURILLO

ROSELYN ORBIGOSO DALMAN

FIBONACCI NUMBERS

Phi Ratio - Golden Ratio The Golden Ratio is a special number, represented by

the greek letter "phi", approximately equal to 1.618.

Two quantities are in the golden ratio if the ratio of the sum of the

quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

Other Names: golden section and golden mean. extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, golden number, and mean of Phidias

TIMELINE

Phidias (490-430 BC) made the Parthenon statues that seem to embody the golden ratio.

Plato (427-347 BC), describes five possible regular solids (the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron and icosahedron), some of which are related to the golden ratio.

TIMELINE

Euclid (325-265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio".

Fibonacci (1170-1250) mentioned that the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.

TIMELINE

Luca Pacioli (1445-1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.

Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597.

TIMELINE

Johannes Kepler (1571-1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers, and describes the golden ratio as a "precious jewel“.

Charles Bonnet (1720-1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.

TIMELINE

Martin Ohm (1792-1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.

Mark Barr (20th century) suggests the Greek letter phi , the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.

APPLICATIONS & OBSERVATIONS

The Parthenon's facade as well as elements of its facade and elsewhere are said to be circumscribed by golden rectangles.

Heinrich Agrippa, drew a man over a pentagram inside a circle. This ink drawing was used to show the proportions that became the basic model used by architects for centuries and today.

APPLICATIONS & OBSERVATIONS

Its concept is used in the construction by Marwan Zgheib of the round skyscraper in Abu Dhabi in the U.A.E.

Illustration from Luca Pacioli's De Divina Proportione applies geometric proportions to the human face.

Salvador Dali, used the golden ratio in his masterpiece, The Sacrament of the Last Supper.

APPLICATIONS & OBSERVATIONS

The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water.

Also, many works of Chopin, mainly Etudes (studies) and Nocturnes, are formally based on the golden ratio.

APPLICATIONS & OBSERVATIONS

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves. In these phenomena he saw the golden ratio operating as a universal law.

In 2003, Volkmar Weiss and Harald Weiss analyzed psychometric data and theoretical considerations and concluded that the golden ratio underlies the clock cycle of brain waves.

“Without mathematics

there is no art.”

–Luca Pacioli