Euclid Quote by Alexey Stakhov
““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.””
About This Quote
Source Book: Geometry and the Golden Ratio, Alexey Stakhov, 2005
Euclid introduced the extreme and mean ratio to enable construction of regular polygons, crucial for building Platonic solids and expressing universal harmony.
In simple terms: Euclid used the golden ratio to build shapes like pentagons for Platonic solids.
Key Takeaway
Use geometric ratios to understand and create harmonious structures.
Themes
geometry
mathematics
philosophy
architecture
Mood
curious
analytical
Type
theoretical
educational
When to use this quote
- educational lectures
- architectural design
- mathematical proofs
- philosophical discourse
Key Concepts
golden ratio
Platonic solids
construction
symmetry
Questions to Reflect On
- How does the golden ratio influence modern design?
- Can ratios replace intuition in architecture?
A Different Perspective
The ratio alone does not guarantee aesthetic or functional success without context.
![“Kepler followed Proclus and believed that 'the main goal of Euclid was to build a geometric theory of the so-called Platonic solids.' Kepler was fascinated by Proclus and often quotes him calling him a 'Pythagorean'. [History of Mathematics]” quote by C. Smoryński](https://sms4smile.com/quotes/img-quotes/68c/c-smory-ski-225321_euclid_kepler-followed-proclus-and-believed-tha_2944745.jpg)
![“[T]he 47th proposition in Euclid might now be voted down with as much ease as any proposition in politics; and therefore if Lord Hawkesbury hates the abstract truths of science as much as he hates concrete truth in human affairs, now is his time for getting rid of the multiplication table, and…” quote by Sydney Smith](https://sms4smile.com/quotes/img-quotes/709/sydney-smith-6523_abstract_t-he-47th-proposition-in-euclid-might-no_463551.jpg)
![“As to writing another book on geometry [to replace Euclid] the middle ages would have as soon thought of composing another New Testament.” quote by Augustus De Morgan](https://sms4smile.com/quotes/img-quotes/ed4/augustus-de-morgan-18916_age_as-to-writing-another-book-on-geometry-t_584350.jpg)
