A220674 Decimal expansion of the area of Dürer's approximation of a regular pentagon with each side of unit length.

1, 7, 2, 0, 3, 1, 1, 4, 2, 9, 7, 3, 7, 1, 7, 1, 6, 6, 2, 6, 1, 8, 8, 1, 7, 8, 1, 0, 2, 8, 4, 9, 4, 7, 9, 7, 6, 1, 6, 1, 2, 0, 3, 4, 6, 8, 1, 1, 1, 8, 9, 7, 9, 1, 2, 7, 4, 5, 8, 4, 2, 5, 3, 3, 3, 2, 2, 7, 4, 2, 5, 3, 9, 8, 5, 9, 6, 0, 2, 9, 0, 4, 8, 3, 9, 0, 6, 2, 5, 2, 9, 6, 1, 6, 0, 8, 6, 1, 2, 8

Offset: 1

Author

Wolfdieter Lang, Jan 30 2013

Keywords

• nonn
• cons

Comments

To be read as dimensionless area F_D/r^2 = 1.720311429... where the length of each side is r, which is the radius of each circle in Dürer's construction. See the link Dürer, Zweites Buch, figure 16. Compare this with the regular pentagon with unit side length, which is given in A102771, and is F_5/r^2 = 1.720477400... The relative error is about -0.96*10^{-4}.

The angles in Dürer's pentagon are approximately twice 108.3661201 degrees, twice 107.0378260 degrees and once 109.1921079 degrees. The sum has to be exactly 3*Pi, or 540 degrees, as for any pentagon. For the analytic values see the W. Lang link.

Alonso del Arte pointed out the Hughes reference where this construction is shown on p. 5 and p. 16. See also the historical remarks on p. 17.

In the cut-the-knot link this construction is considered in more detail, and the two interior angles at the bottom of the pentagon are shown to be 108.36612... degrees.

For more references and links see the W. Lang link. The length of the dimensionless diagonals which approximate the golden section are also given there, and the angles of the companion of Dürer's pentagon with the same area are computed there. - Wolfdieter Lang, Feb 14 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Cut-The-Knot, Approximate Construction of Regular Pentagon by A. Dürer
• G. Hughes, The Polygons of Albrecht Durer -1525, arXiv:1205.0080 [math.HO], 2012.
• Wolfdieter Lang, Albrecht Dürer's approximation of a regular 5-gon.
• Wikimedia Commons, Albrecht Dürer, Underweysung der messung ..., 1525, title page.
• Wikisource Albrecht Dürer, Underweysung der messung ..., Zweites Buch, 1525 (in German).

Crossrefs

Cf. A102771 (pentagon area).

Programs

Mathematica

r = Sqrt[7 - 3*Sqrt[3] + 2*(Sqrt[3]-1)* Cos[a]]; area = (2 + Sqrt[3] + r - 2*Cos[a]*(r+2))/4 /. Cos[a] -> (3 - Sqrt[3] - Sqrt[6*Sqrt[3]-4])/4; RealDigits[area, 10, 100] // First (* Jean-François Alcover, Feb 13 2013 *)

Formula

The dimensionless area of Dürer's pentagon is

F_D/r^2 = (1 + 2*x)*y1/2 + x*y2, with x = (a + sqrt(a*(a+8)))/4, a := sqrt(3) - 1, y1 = 1 - sqrt(3)/2 + x, y2 = sqrt(1 - x^2). The approximate values for x, y1 and y2 are 0.8150878978, 0.9490624938, 0.5793373101, respectively. This leads to the approximate value 1.720311430 for F_D/r^2, and the present sequence gives more accurate digits.