The document discusses the golden ratio, attributed to Phidias, and its mathematical properties, particularly in relation to pentagons and five-pointed stars. It also explores the golden ratio's presence in nature, architecture, and art, illustrating examples and proportions involved. The significance of the number five in various contexts is highlighted, including its associations with humanity and symbolism.

1. GOLDEN RATIO
FIDIAS
He is deemed as the
creator and developer
of the idea for the first
time
PARTHENON
http://upload.wikimedia.org/wikipedia/commons/5/51/Kien_truc_symbol.JPG
"Study for Phidias in 'The Apotheosis of Homer' by Ingres,
San Diego Museum of Art" by Wmpearl - Own work.
Licensed under Creative Commons Zero, Public Domain
Dedication via Wikimedia Commons -
http://commons.wikimedia.org/wiki/File:Study_for_Phidias_in
_%27The_Apotheosis_of_Homer%27_by_Ingres,_San_Dieg
o_Museum_of_Art.JPG#mediaviewer/File:Study_for_Phidias
_in_%27The_Apotheosis_of_Homer%27_by_Ingres,_San_Di
ego_Museum_of_Art.JPG

2. D
1.618
EB
EF
E C
F
A B
AD
AB
1.618
AB
AD
0.618
EF
EB
0.618
In every regular pentagon, the side AB is the Golden Ratio of the diagonal AD.
The shape brought about by the diagonals is a five point star or pentacle. In
the center of the pentacle there is a new pentagon upside down, where one
can draw again a pentacle. This process may last for good.

3. AD
AB
AB AD-AB
The Golden Ratio is found when we divide a line segment into two parts so
that:
the longer part divided by the shorter part
is also equal to
the whole length divided by the longer part
and the result of these divisions equals
1.618
1.618
AD
AB
AD-AB
= =

4. AB
AD
AD-AB
AB AD-AB
And, conversely:
the shorter part divided by the longer part
is also equal to
the longer part divided by the whole length
and the result of these divisions equals
0.618
0.618
AD
AB
= =

5. As a consequence:
Assuming AD=1
AB AD-AB
AD
AB=0.618
and
AD-AB=0.382
With the help of a calculator, try to check all the ratios you learned in
the last two slides

6. 5
Number Five is related to human hands, protection and
salvation. It is also the quintessence, that is, which comes
after the four prime elements.
The Five-pointed star represents a human being
Many strongholds have a
pentagonal shape. Do you know
any building in the USA related to
this shape?
The Five-pointed star represents the devil
when put upside down

7. See page for author [Public domain], via Wikimedia Commons
By Claudio.stanco (Own work) [CC-BY-SA-3.0
(http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia
Commons
ARCHITECTURAL EXAMPLES

8. ROGIER VAN DER
WEYDEN
THE DESCENT FROM THE
CROSS (1435)
Roger van der Weyden (1399/1400–1464) [Public domain undefined Public domain], undefined

10. "Diego Velázquez - The Surrender of Breda (Las Lanzas) - WGA24401" by Diego Velázquez - Web Gallery of Art: Image Info about artwork.
Licensed under Public domain via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Diego_Vel%C3%A1zquez_-
_The_Surrender_of_Breda_(Las_Lanzas)_-_WGA24401.jpg#mediaviewer/File:Diego_Vel%C3%A1zquez_-
_The_Surrender_of_Breda_(Las_Lanzas)_-_WGA24401.jpg

11. X= 0.618 a-x=0.382
a=1
y b-y
b
The key becomes the most
relevant part in the picture

12. In order to better analyze the image, these are the main steps to find out the proportion in the
image
1
Point the compass on B and take radius AB. You will get C on the perpendicular to AB from B.
Draw the perpendicular bisector of BC to get M, middle point of BC

13. 2
Point the compass on M and take radius MC. Join A to M and you will get P on the circle.
AP is the golden ratio of AB

14. 3
Point the compass on A and take radius AP, that is, X. Draw the arch PQ and you will have
transferred X over AB. Finally, draw from Q a line perpendicular to AB, you can tell that this line, in
pink color, perfectly fits the first spear of the Spanish army. If you put QB from A, you will get the
whole distribution

15. El Greco [Public domain], via Wikimedia Commons

16. 0.382 0.618
0.382
0.618
El Greco [Public domain], via Wikimedia Commons

17. GOLDEN RATIO IN NATURE

18. THE PINECONE AND THE GOLDEN SPIRAL

19. SEA STARS, VEGETABLES AND GALAXIES

20. FLOWERS
AGRIMONY CARNATION VINCA
PETUNIA PETUNIA AQUILEGIA

21. VEGETABLES GROW FOLLOWING A
GOLDEN SPIRAL LIKE MOVEMENT