MMW chapter 1

Mathematics

organize and systematize ideas about patterns in nature.

utility/beauty

Patterns possess _and_

Pizano or Leonardo of Pisa

introduced fibonacci sequence

Liber Abbaci

“Book of Calculation.”

Guglielmo Bonacci 

Leonardo of Pisa father

golden rectangle

can be broken into squares the size of the next Fibonacci number down and below

Fibonacci spiral

Take a golden rectangle, break it down into smaller squares based from Fibonacci sequence and divide each with an arc.

1 petal

White calla lily contains

3 petals

Trillium contains

2 petals

  Euphorbia contains

5 petals

Columbine contains

8 petals

Bloodroot contains

13 petals

Black-eyed susan contains

21 petals

Sasha daisy contains

34 petals

Field daisies contains

55 and 89 petals

other types of daisies contain

Luca Pacioli

found the relationship between Fibonacci sequence and the golden ratio.

Divine Proportion

The golden ratio was first called as the

section aurea

Latin for Golden Section

Golden rectangle

The Fibonacci numbers can be applied to the proportions of a rectangle, called the

34

A Fibonacci spiral which approximates the golden spiral, using Fibonacci sequence square sizes up to__

Johannes Kepler

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by

Whirling Triangles

A set of_________ were able to draw a logarithmic spiral that will converge at the intersection of the two lines.

Fibonacci numbers

are said to be as one of the nature's numbering systems.

1.6180339887

Golden ratio amount, Also known as Golden Section, or Golden Proportion, or Divine Proportion denoted by Phi (j) =

Vitruvian Man

human body pentagram model of Leonardo da Vinci.

Golden Ratio

shows beauty, balance and harmony in art, architecture & design

Phi

the initial letters of Phidias’.

Mark Barr

(American mathematician), to use Phi in honor of Phidias.

Phidias (490–430 BC)

widely used the golden ratio in his works of sculpture

Parthenon

The exterior dimension of the ___ in Athens, Greece embodies the golden ratio.

“Timaeus” Plato (428-347 BC)

In ____ describes five possible regular solids that relate to the golden ratio.

Tetrahedron/Cube/Octahedron/Icosahedron/Dodecahedron

five possible regular solids that relate to the golden ratio.

Euclid

first to give definition of the golden ratio as “a dividing line in the extreme and mean ratio” in his book the “Elements.”

Leonardo di ser Piero da Vinci (1452–1519)

____ into invention, painting, sculpting, architecture, science, music, mathematics, engineering, literature, anatomy, geology, astronomy, botany, writing, history, and cartography.

The last supper/Vitruvian Man/Monalisa/St. Jerome

Leonardo da Vinci works

Michelangelo di Lodovico Simon (1475–1564)

painter, architect, poet, and engineer from the Renaissance. “greatest living artist of his time”.

Raffaello Sanzio da Urbino (1483–1520)/Raphael

a painter and architect from the Renaissance.

School of Athens/Stairs/Two men/Upper Arch/Holy Family/Crucifixion

works of Raphael

Rembrandt Harmenszoon Van Rijn (1606–1669)

It is said that the golden triangle is applied in his paintings “Self Portrait”.

art, design and architecture

Today the Golden ratio is usually a concept that is applied in ____

George-Pierre Seurat (1859–1891)

was a French post-impressionist painter. His paintings appear to have applied golden ratio to define the horizon, to place point of interest and to create balance.

Salvador Domingo Felipe Jacinto Dali (1904–1989)/ Salvador Dali

framed his paintings using the golden ratio in his masterpiece, “The Sacrament of the Last Supper.”

Great Pyramid of Giza/4700 BC

The _______ built ______ in Ahmes Papyrus of Egypt is with proportion according to a “Golden Ratio.”

756/481

Great Pyramid of Giza length of each side of the base is ___ feet with a height of ___ feet.

1.5717

Great Pyramid of Giza ratio of the base to the height is roughly ___, which is close to the Golden ratio.

Notre Dame/1163 and 1250

____ is a Gothic Cathedral in Paris, built in between ____

Cathedral of Our Lady of Chartes

in Paris, France also exhibits the golden ratio.

Taj Mahal

India used the golden ratio in its construction and was completed in 1648.

United Nations Building

the window configuration reveal golden proportion.

CN Tower

the tallest tower and freestanding structure in the world, contains the golden ratio in its design.

Eiffel Tower

Paris, France, erected in 1889 is an iron lattice.

Symmetry

sense of harmonious and beautiful proportion of balance or an object is invariant to any of various transformations (reflection, rotation or scaling).

bilateral and radial

two main types of symmetry

Bilateral symmetry

symmetry in which the left and right sides of the organism can be divided into approximately mirror image of each other along the midline.

Radial symmetry (or rotational symmetry)

type of symmetry around a fixed point known as the center and it can be classified as either cyclic or dihedral.

Fractal

curve or geometric figure, each part of which has the same statistical character as the whole

Fractals

logarithmic spiral

or growth spiral) is a self-similar spiral curve which often appears in nature.

dendrochronology

Age of the trees can be determined by applying

Application of Calculus

In forensic, clarity the blurred image to clear image.

Optimization (maximize or minimize) surface areas, volumes, profit and cost analysis, projectile motion, etc.

Mathematical Modelling

In medical field, drugs are designed to change the shape or

      motions of a protein by modelling using geometry and

      related areas.

Development of medicine to cure diseases.

Engineering

Numerical analysis: heat, electricity, and magnetism, relativistic mechanics, quantum mechanics, and other theoretical constructs.

Information technology

Computers are invented thru mathematics.

Computer science development includes mathematical          

      principles.

Google use linear algebra to quantify “relevance” with the

      help of mathematics.

Facebook uses graph theory to create links between

      individuals.

Cryptography

Computer security on ATM cards, computer passwords and electronic commerce. 

Archaeology

Surveys and try to find patterns to shed light on past

      human behavior and in carbon dating artifacts. 

Social Sciences

Mathematical modelling on economics, sociology, psychology, and linguistics.

Political Science

Political analysts study voting patterns and the influence of various factors on voting behavior.

Music and Arts

Musical structure

Gothic Architecture

Egyptian Pyramids

Alessandro di Mariano di Vanni Filipepi (1445–1510), known as Sandro Botticelli

The Birth of Venus is one of the world’s famous and appreciated works of art and it was painted between 1482 and 1485.