,</math> and <math>r_m = a\frac{\varphi}{2}.</math>

The volume and surface area of the dodecahedron can be expressed in terms of :

</math> and <math>V_d = \frac{5\varphi^3}{6-2\varphi}.</math>

As well as for the icosahedron:

{2}</math> and <math>V_i = \frac{5}{6}(1 + \varphi).</math>

175px|right|thumb|Three golden rectangles touch all of the vertices of a [[regular icosahedron.]]

These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving . The coordinates of the dodecahedron are displayed on the figure to the right, while those of the icosahedron are:

<math display=block>

(0,\pm1,\pm\varphi),\

(\pm1,\pm\varphi,0),\

(\pm\varphi,0,\pm1).

</math>

Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings.

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is , with averages for individual artists ranging from (Goya) to (Bellini).

Music

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,

  • Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is .

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Works cited

  • (Originally titled A Mathematical History of Division in Extreme and Mean Ratio.)

Further reading

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  • Information and activities by a mathematics professor.
  • The Myth That Will Not Go Away , by Keith Devlin, addressing multiple allegations about the use of the golden ratio in culture.
  • Spurious golden spirals collected by Randall Munroe
  • YouTube lecture on Zeno's mice problem and logarithmic spirals