Beauty in the Golden Ratio

1
Beauty in the Golden Ratio

• By Elizabeth Leppard

2
What is beauty?

• Def n The quality that gives pleasure to the
  mind or senses and is associated with such
  properties as harmony of form or colour,
  excellence of artistry, truthfulness, and
  originality
• The Greeks said that all beauty is mathematics
• Historically many different numbers have been
  tried in an attempt to describe beauty however,
  only one mathematical relationship has been
  consistently and repeatedly reported to be
  present in beautiful things, and that number is
  Phi (f)

3
What is the Golden Ratio?
1.6180339887 4989484820 4586834365 6381177203
0917980576 2862135448 6227052604 6281890244
9707207204 1893911374 8475408807 5386891752
1266338622 2353693179 3180060766 7263544333
8908659593 9582905638 3226613199 2829026788
0675208766 8925017116 9620703222 1043216269
5486262963 1361443814 9758701220 3408058879
5445474924 6185695364 8644492410 4432077134
4947049565 8467885098 7433944221 2544877066
4780915884 6074998871 2400765217 0575179788
3416625624 9407589069 7040002812 1042762177
1117778053 1531714101 1704666599 1466979873
1761356006 7087480710 1317952368 9427521948
4353056783 0022878569 9782977834 7845878228
9110976250 0302696156 1700250464 3382437764
8610283831 2683303724 2926752631 1653392473
1671112115 8818638513 3162038400 5222165791
2866752946 5490681131 7159934323 5973494985
0904094762 1322298101 7261070596 1164562990
9816290555 2085247903 5240602017 2799747175
3427775927 7862561943 2082750513 1218156285
5122248093 9471234145 1702237358 0577278616
0086883829 5230459264 7878017889 9219902707
7690389532 1968198615 1437803149 9741106926
0886742962 2675756052 3172777520 3536139362
1076738937 6455606060 5922... 1025 digits

• The golden ratio is an irrational number
• It is more commonly written to 3 decimal places
  ie 1.618 and is known as Phi (f)
• Two quantities are said to be in the golden
  ratio, if "the whole (i.e., the sum of the two
  parts) is to the larger part as the larger part
  is to the smaller part", i.e. if
• where a is the larger part and b is the smaller
  part

4
Other names for The Golden Ratio
The Golden Mean
The Phi Ratio
The Fibonacci Ratio
The Golden Section
The Divine Ratio
5
How do we find Phi (f)?

• If we take the ratio
• b (a b) a²
• a² - ba - b² 0
• dividing through by b² gives
• Let
• Since Phi (f) is positive we have

f² - f - 1 0
2f 1 v(1 4)
6
Phi has two unique properties

• If you square phi, you get a number exactly 1
  greater than phi 2.61804...
• Phi 2 Phi 1
• If you divide phi into 1, you get a number
  exactly 1 less than phi 0.61804...
• 1 / Phi Phi 1
• Phi, curiously, can also be expressed all in
  fives as
• 5 .5 .5 .5 Phi

7
The History of Phi (f)

• The golden ratio was first studied by ancient
  mathematicians because of its frequent appearance
  in geometry and may have even been understood and
  used as far back in history as the Egyptians
• More commonly, however, the discovery of the
  golden ratio is ascribed to the ancient Greeks,
  and is usually attributed to Pythagoras (or to
  the Pythagoreans, notably Theodoras) or to
  Hippasus of Metapontum
• The Golden Ratio was known as tau(t) up until the
  beginning of the 20th Century, when an American
  mathematician Mark Barr gave the ratio the name
  phi (f), the first Greek letter in the name of
  Phidias, the Greek sculptor who lived around 490
  to 430 BC. Phidias frequently used the Golden
  Ratio in his art eg the Athena Parthenos

8
The Golden Ratio and Fibonacci numbers

• What are the Fibonacci Numbers?
• Divide each Fibonacci number by the preceding
  number in the sequence
• ? 1/1 1
•   2/1 2
• 3/2 15
• 5/3 1666..
• 8/5 16  
• 13/8 1625
•   21/13 161538...
• fThe limit of the sequence of ratios of
  successive Fibonacci numbers
  1.618033988749894848204586

1 1 2 3 5 8 13 21 34 55 89
9
Fibonacci numbers can be seen in nature

• How fast can rabbits breed in ideal
  circumstances
• Spirals of a pine cone
• The number of petals on a flower
• Shell spirals follow Fibonacci rectangles

10
The Golden Ratio in geometry

• Phi appears in many basic geometry constructions
• Phi appears in 3D geometric solids
• The golden rectangle

11
The Golden Ratio in nature
12
The Golden Ratio and human beauty

• "Beauty is in the phi of the beholder
• A template for human beauty is found in phi and
  the pentagon
• Dr. Stephen Marquardt has analyzed the human face
  from ancient times to the modern day and has
  developed and patented a beauty mask with his
  findings
• This mask uses the pentagon and decagon as its
  foundation, which embody phi in all their
  dimensions
• http//goldennumber.net/images/marquardt2.gif

13
Examples of the beauty mask
14
The Golden Ratio in architecture

• The Parthenon
• The Notre Dame
• The Eden Project

15
The Golden Ratio in art
The Last Supper
Mona Lisa