The Divine Proportion

1. The Divine Proportion “Geometry has two great treasures…one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel”. (Johannes Kepler)

2. What is so interesting about the Divine Proportion? • The Divine Proportion, or Golden Section, represented by the Greek letter Φ (phi), is one of those mysterious natural numbers like π (pi) that seem to arise out of the basic structure of the universe. • Φ appears clearly and regularly in the realm of things that grow and unfold in steps, especially living things – but also in art and architecture. • “To the Greeks therefore, and not to the Romans, we are indebted for all that is great, judicious, and distinct in architecture.”

3. Divine Proportion • The Greeks (and others, like Renaissance artists such as Botticelli, Lippi, Michelangelo) may have felt that when a building or artwork was designed to incorporate Φ that it had the purest possible proportions and was the most pleasing to the eye – we will see why in a minute. • …allude to a proper application of the useful rules of architecture, whence a structure will derive figure, strength, and beauty, and whence will result a due proportion and a just correspondence in all its parts.”

4. The Golden Section – A Ratio • The Golden Section is a RATIO – like 2:1. It is also called the “Golden Mean” • If there is a piece of string, and you divide it into a 2:1 ratio, then 1 part is twice as long as the other. • Also, the short part is 1/3 the length of the whole string, and the long part is 2/3 the length. The ratio of the shorter to the longer is 1:2, and that of the longer to the whole is 2:3.

5. So then what is the Golden Section? • But…the Golden Section is a special ratio – where the ratio of the short part to the long part is the same as the long part to the whole.

6. So then what is the Golden Section? • So, “a” is to “b” (a:b) as “b” is to “c” (b:c) • a:b = b:c

7. Some Examples - People

8. Some Examples - Nature

9. Leonardo di ser Piero da Vinci An Old man by Leonardo Da Vinci

10. Leonardo De Vinci The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci

11. Leonardo De Vinci Mona-Lisa by Leonardo Da Vinci

12. Deoxyribonucleic acid (DNA) • The DNA spiral is a Golden Section • The DNA molecule, the program for all life, is based on the golden section.  It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. • 34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates phi, 1.6180339.

13. DNA DNA in the cell appears as a double-stranded helix referred to as B-DNA. This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms to 13 angstroms.

14. So what is this ratio? • The Golden Section/Golden Mean/Divine Proportion is an irregular number – like π, and cannot be expressed fully in decimal form (i.e. π = 3.14128…) • Φ = 1.618033… or (1+5)/2 • Somehow it seems fitting that we cannot represent the root of Sacred Geometry by an ordinary number.

15. So what about Architecture? • The Divine Proportion was used by the Greeks – and is still being used by architects today to design buildings that are aesthetically pleasing. • There is evidence that the Great Pyramid incorporates Φ – in the so-called “King’s Chamber” and also in its overall dimensions.

16. What about the Parthenon?

17. Columns, anyone? • The graceful curves of the Ionic column are designed using the Golden Section.

18. Another Historical Tidbit • Exodus 25:10 – “Have them make a chest of acacia wood = two and a half cubits long, a cubit and a half wide, and a cubit and a half high…” (ratio 2.5:1.5 = 5:3 = 1.666) • Genesis 6:15 – “And this is the fashion that thou shalt make it of: The length of the ark shall be three hundred cubits, the breadth of it 50 cubits, and the height of it 30 cubits…” (50:30 = 5:3 = 1.666) • Φ = 1.618033… or (1+5)/2

19. Anything Else? • It is likely that Virgil’s “Aeneid” and other great works of classical poetry used φ to determine metrical structure of the poem. • Mozart’s sonatas tend to divide in parts exactly at the Golden Section of total time of the work. • In Beethoven’s 5th Symphony the opening motto is repeated at exactly the Φ point through the Symphony (Bar 372) and also at the start of the recapitulation 1-Φ of the way through. • Stradivarius placed the “f” holes in his violins at the Φ point of the body structure.

20. Hmmmm… • It now seems that the Divine Proportion was and still is used by men and women to build beautiful monuments and other works that are pleasing to the senses. • “…so as to compose delightful harmony by a mathematical and proportional arrangement of acute, grave, and mixed sounds.”

21. A little more math for anyone that’s still awake… • The Fibonacci series (1, 1, 2, 3, 5, 8,13, 21…) which describes the growth pattern of a population, is connected to the Golden Mean, because the ratio of any 2 terms tends towards Φ… • For instance.. 2:1 = 2.000, 8:5 =1.600, 13:8 = 1.625, 21:13 = 1.615… • Φ = 1.618033… or (1+5)/2 • Each number in the series is called a “Fibonacci Number”

22. Fibonacci Bunnies • Start with one pair • Mate during first month • One pair born next month and each month thereafter

23. Building a Golden Spiral • Draw 2 squares of 1 unit each, side-by-side • Next draw a 2 unit square, and then a 3, and then a 5, etc • Draw quarter circles in each square, joining them up…

24. Building a Golden Spiral • The spirals increase in distance from the centre by phi every quarter turn…sea shells, snails, ferns, and many other living creatures are built to this specification