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1884–1894

Hermes's Suitcase of Göttingen

Computing the 65537-gon

While studying and teaching mathematics in Prussia and Germany, Johann Gustav Hermes (1846–1912) spent 10 years on a 200-page manuscript focusing on the construction of the 65537-gon. After the Second World War, Hermes's manuscripts were moved to the Mathematical Institute in Göttingen, where they can now be viewed in the so-called "suitcase of Göttingen."

Hermes's Suitcase of Göttingen

The 65537 sides in Hermes's computation are of interest because 65537 is the largest known Fermat prime. As such, it corresponds to the largest known "basic" (i.e. not having a number of sides which is a multiple of side counts of smaller constructible polygons) regular polygon that is constructible using classical Greek method involving compass and straightedge. However, despite the Herculean effort in computation and perseverance on the part of Hermes, the assessment of geometer H. S. M. Coxeter that "Hermes wasted ten years of his life on the 65537-gon" reflects the fact that not only do these calculations contain no new mathematics but—especially following the advent of symbolic computer algebra—they can be fully automated.

Artifact format

Suitcase containing 221 large-format pages

Artifact origin

Kaliningrad, Russia

Current artifact location

Georg-August-Universität Göttingen

Timeline

Algebra timeline Babylonian Metric Algebra Problems Tablet Berlin Pythagorean Theorem Papyrus Rhind Papyrus Al-Khwārizmī's Al-Jabr Khayyam's Al-jabr Cardano's Ars Magna Recorde's Whetstone of Witte Faulhaber's Academia Algebrae Wallis's Treatise of Algebra Emerson's Treatise of Algebra Hermes's Suitcase of Göttingen

Interactive Content

Computational Explanation

Other Resources

Additional Reading

  • Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982.
  • Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.
  • Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
  • Hermes, J. "Über die Teilung des Kreises in 65537 gleiche Teile." Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Vol. 3. Göttingen, pp. 170–186, 1894.
  • Krížek, M.; Luca, F. and Somer, L. 17 Lectures on Fermat Numbers: From Number Theory to Geometry. New York: Springer-Verlag, 2001.
  • Trott, M. "cos(2π/257) à la Gauss." § 1.10.2 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 312–321, 2006.
  • Young, J. (Ed.). "Leonard Eugene Dickson Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field. New York: Dover, pp. 352–386, 1955.