The Golden Ratio

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The Golden Ratio

• Marie Bruley
• Math 21-Merced College

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What is the Golden Ratio?

• Phi
• F

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This is an approximation of its decimal expansion!

• F

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The Golden Ratio In Art

• Throughout the centuries, artists and
  musicians - pursuing aesthetics - have turned to
  natures own tactics and emulated the golden ratio
  in their own creations. There is evidence of this
  from ancient times.

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The Parthenon
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The Parthenons Dimensions is the Golden Rectangle
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The Floor Plan of the Parthenon
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The Pyramids at Giza
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The Use of Phi in the Pyramids
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Michelangelos Use of the Golden Ratio
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Raphaels Use of the Golden Ratio

• Raphael's Crucifixion ... is another well-known
  example. The principal figures outline a golden
  triangle ...

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Other Art

• Norham Castle at Sunrise

• The English romantic artistic Joseph Mallord
  William Turner (1775-1851) is admired for his use
  of color and light.   Norham Castle at Sunrise
• Of particular interest are the geometric
  similarities in his various canvases, with their
  obvious golden subdivisions.

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Modern Art Uses the Golden Ratio

• The more recent search for a grammar of art
  inevitably led to the use of the golden section
  in abstract art. La Parade, painted in the
  characteristic multi-dotted style of the French
  neo-impressionist Seurat (1859-1891), contains
  numerous examples of golden proportions.

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La Parade-By Seurat
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Bathers-By Seurat
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The Golden Ratio in Modern Architecture

• The 20th Century architect Le Corbusier
  (1887-1965)
• He used the Golden Ratio in his architecture
• He believed that humans are comforted by
  mathematics.

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Finding the Golden Ratio Using Fibonacci numbers
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The Ratios of Fibonaccis leads to
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Fibonacci Numbers Method Continued

• This leads to solving

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The Golden Ratio
x units 1 unit
R S T
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The Golden Rectangle
Historical Note The Golden Ratio is also called
the Divine Proportion.
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The Golden Triangle
Again, we have
36º
x x
1 unit
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Websites Regarding the Golden Ratio

• http//ccins.camosun.bc.ca/jbritton/goldslide/jbg
  oldslide.htm
• http//mathworld.wolfram.com/GoldenRatio.html
• http//students.bath.ac.uk/ma1caab/art.html
• http//www.geocities.com/CapeCanaveral/Station/822
  8/

http//www.geocities.com/CapeCanaveral/Station/822
8/
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Acknowledgements!

• Textbooks consulted were
• Excursions in Modern Mathematics by Peter
  Tannenbaum Robert Arnold
• The slides for computing the Golden Ratio were
  taken from Marguerite Smiths presentation
  Fibonacci Numbers and the Golden Mean
• The Golden Ratio in Art slides were taken from
  the website http//ccins.camosun.bc.ca/jbritton
  /goldslide/jbgoldslide.htm