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Dürer's constaruction of the Archimedean Spiral

Albrecht Dürer (1471-1528), the famous Renaissance painter, printmaker, engraver, mathematician, and theorist, presented a drawing with a detailed method for the construction of spirals (Schneckenlinien). He published it in 1525 in the first book of his treatise >Underweysung der messung, mit zirckel und richtscheyt< (Teaching of Measuring).
  


 



Dürer Durer
          Archimedean spiral


logarithmic spiral
        ewige linie


Archimedean Logarithmic
Polar equation:
r(φ) = a · φ
 a constant
r(φ) = exp(a·φ)
a constant
Parameter form: x = φ · cos(φ)
y = φ
· sin(φ)
x = exp(a·φ) · cos(φ)
y =
exp(a·φ) · sin(φ)

Property of the Archimedean spiral:
successive turnings have a constant separation distance.

Property of the Logarithmic spiral:
the angle θ between the tangent and radial line at the point (r,φ) is constant, and equal to arctan(1/a).


area
            archimedean spiral

The area of 1 turn of the Archimedean spiral is exactly 1/3 of the circle area.

Web Links

Albrecht Dürer (Wikipedia)

Albrecht Dürer (MacTutor History of Mathematics archive)

Albrecht Dürer: Underweysung der Messung, mit dem Zirckel und Richtscheyt
(Sächsische Landesbibliothek -Staats- und Universitätsbibliothek Dresden)

Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen/Erstes (Wikisource)

Archimedean spiral (Wikipedia)

Logarithmic spiral (Wikipedia)

Spiralen ("Schneckenlinien") bei Albrecht Dürer (TU Freiberg)

Books
Albrecht Dürer: Unterweisung der Messung, Faksimile Neudruck der Ausgabe Nürnberg 1525, Verlag Dr. Alfons Uhl, Nördlingen 2000.

Updated: 2023, Oct 06

© 2011-2023 J. Giesen

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