Golden ratio

1. Using Golden ratio to study geometry

1.1. Golden triangle, pentagon and pentagram

1.1.1. New node

1.2. Scalenity of triangles

1.3. Triangle whose sides form a geometric progression

1.4. Golden triangle, rhombus, and rhombic triacontahedron

2. number

2.1. irrational

2.2. (1+√5)/2

2.3. approximately 1.618

3. History

3.1. definition

3.1.1. Euclid's Book

3.1.1.1. Elements

3.1.2. A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less

3.2. first studied by

3.2.1. Ancient Greek mathematicians

3.3. later studied by

3.3.1. Phidias

3.3.2. Plato

3.3.3. Luca Pacioli

3.3.4. Roger Penrose

4. Application

4.1. Aesthetics

4.1.1. illustrations of regular solids by Leonardo Da Vinci

4.2. Architecture

4.2.1. Parthenon

4.3. Painting

4.3.1. The Sacrament of the Last Supper

4.4. Music

4.4.1. For Ann (rising)

4.4.2. New node