Golden Ratio

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Golden Ratio
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Golden Section

• The golden section is a line segment sectioned
  into two according to the golden ratio. The total
  length ab is to the longer segment a as a is to
  the shorter segment b.

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Mathematically
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Construct a Golden Rectangle

1. Construct a unit square.
2. Draw a line from the midpoint of one side to an
   opposite corner.
3. Use that line as the radius to draw an arc that
   defines the long dimension of the rectangle.

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Similar Golden Rectangles
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• Mathematician Mark Barr proposed using the first
  letter in the name of Greek sculptor Phidias,
  phi, to symbolize the golden ratio. Usually, the
  lowercase form is used.
• Phidias (490430 BCE) made the Parthenon statues
  that seem to embody the golden ratio.

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Parthenon on the Acropolis
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Great Mosque of Kairouan

• Built by Uqba ibn Nafi from 670 A.D. at the
  founding of the city of Kairouan, Tunisia.

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Da Vincis ArtVitruvian Man
De Divina Proportione
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Mona Lisa St. Jerome
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Heliogabalus

• Heliogabalus is a famous painting of 1888 by the
  Anglo-Dutch academician Sir Lawrence Alma-Tadema
  and based on a probably invented episode in the
  life of the Roman emperor Elagabalus, also known
  as Heliogabalus, (204-222), taken from the
  Augustan History. Elagabalus is portrayed
  attempting to smother his unsuspecting guests in
  rose-petals released from false ceiling panels.
• The canvas measures 213cm by 132 cm, dimensions
  which are almost exactly in the golden ratio,
  11.618.

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Bathers

• Georges Seurat French painter
• 1859-1891 Scene on the Seine
  River

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Piet MondrianDutch painter (1872 1944)
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Composition with Yellow, Blue, and Red
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Golden Ratio Sculpture

• Australian sculptor Andrew Rogers's 50-ton stone
  and gold sculpture, entitled Golden Ratio, is
  installed outdoors in Jerusalem. The height of
  each stack of stones, beginning from either end
  and moving toward the center, is the beginning of
  the Fibonacci sequence 1, 1, 2, 3, 5, 8.

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Ptolemy's theorem

• The golden ratio in a regular pentagon can be
  computed using Ptolemy's theorem.
• b² a² ab yields

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Pentagram

• A pentagram colored to distinguish its
  line segments of different lengths. The four
  lengths are in golden ratio to one another.
• The pentagram includes ten isosceles
  triangles five acute and five obtuse isosceles
  triangles. In all of them, the ratio of the
  longer side to the shorter side is f. The acute
  triangles are golden triangles. The obtuse
  isosceles triangles are a golden gnomon.

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Golden Spiral

• Approximate and true golden spirals. The
  green spiral is made from quarter-circles tangent
  to the interior of each square, while the red
  spiral is a Golden Spiral, a special type of
  logarithmic spiral called a Fibonacci spiral.
  Overlapping portions appear yellow. The length of
  the side of a larger square to the next smaller
  square is in the golden ratio.

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Check these out

• http//goldennumber.net/face.htm
• http//cuip.uchicago.edu/dlnarain/golden/
• http//math.rice.edu/lanius/Geom/golden.html
• http//www.youtube.com/watch?vfmaVqkR0ZXg

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Sources

• http//en.wikipedia.org/wiki/Golden_ratio
• http//mathworld.wolfram.com/GoldenRatio.html
• http//www.geocities.com/jyce3/piet.htm
• http//en.wikipedia.org/wiki/Leonardo_da_Vinci
• http//www.oceansbridge.com/artist-lists/georges-s
  eurat/Bathers-At-Asnieres-1883-84.php