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In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the Platonic solids: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the golden ratio... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the dodecahedron... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.
Alexey Stakhov "Golden" Non-Euclidean Geometry, The: Hilbert's Fourth Problem, "Golden" Dynamical Systems, and the Fine-Structure Constant
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In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the Platonic solids: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the golden ratio... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the dodecahedron... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.
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Alexey Stakhov

"Golden" Non-Euclidean Geometry, The: Hilbert's Fourth Problem, "Golden" Dynamical Systems, and the Fine-Structure Constant

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Life, death, preservation, loss, failure, success, poverty, riches, worthiness, unworthiness, slander, fame, hunger, thirst, cold, heat - these are the alternations of the world, the workings of fate. Day and night they change place before us, and wisdom cannot spy out their source. Therefore, they should not be enough to destroy your harmony; they should not be allowed to enter the storehouse of the spirit. If you can harmonize and delight in them, master them and never be at a loss for joy; if you can do this day and night without break and make it be spring with everything, mingling with all and creating the moment within your own mind - this is what I call being whole in power.

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