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On Models and Visualizations of Some Special Quartic Surfaces

  • Years Ago
  • Jemma Lorenat, Editor
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Notes

  1. [Grossman and Sebline 2015]; to get a glimpse of this exhibition, one can go to this YouTube video:

    https://www.youtube.com/watch?v=a3QqKBWHarA

    or read the review about it by E. Arthur Robinson Jr. in the November 2015 issue of the Notices of the American Mathematical Society (http://www.ams.org/notices/201510/rnoti-p1192.pdf).

  2. For a more detailed account of the relationship between the quartics of Plücker and Kummer, see [Hudson 1905], chapter VI.

  3. Klein and Wenker had already built a version of this model in 1870, and in a letter to Sophus Lie from July 1870 Klein described how he had used it to find the singularities of the asymptotic curves on a Kummer quartic. His findings, including the drawing of these curves that he sent to Lie, were published soon thereafter by Kummer in the Monatsberichte of the Berlin Academy, and then later reprinted [Klein and Lie 1884]. For details, see [Rowe 2018].

References

  1. Apéry, François. Caron’s Wooden Mathematical Objects, online at: http://www.math-art.eu/Documents/pdfs/Cagliari2013/Actes2013-apery.pdf.

  2. Cayley, Arthur. On Plücker’s Models of Certain Quartic Surfaces, Proceedings of the London Mathematical Society, 3(1871): 281–285.

  3. Clebsch, Alfred. Zum Gedächtnis an Julius Plücker, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Band 16(1871): 1–40.

  4. Coolidge, Julian. A History of the Conic Sections and Quadric Surfaces, Oxford, Clarendon Press, 1945.

  5. Fischer, Gerd, Hrsg. Mathematische Modelle, 2 Bde., Berlin: Akademie Verlag, 1986.

  6. Grossman, Wendy A., and Sebline, Edouard, eds. Man Ray. Human Equations: A Journey from Mathematics to Shakespeare, Jerusalem: The Israel Museum, 2015.

  7. Hankins, Thomas. Sir William Rowan Hamilton, Baltimore: Johns Hopkins University Press, 1980.

  8. Hilbert, David, und Cohn-Vossen, Stefan. Anschauliche Geometrie, Berlin: Julius Springer, 1932.

  9. Hudson, Ronald W. H. T. Kummer’s Quartic Surface, Cambridge: Cambridge University Press, 1905; reprinted in 1990.

  10. Klein, Felix. Zur Theorie der Liniencomplexe des ersten und zweiten Grades, Mathematische Annalen 2(1870): 198–228.

  11. Klein, Felix. Ueber die Plückersche Komplexfläche, Mathematische Annalen 7(1874): 208–211.

  12. Klein, Felix. Gesammelte Mathematische Abhandlungen, 3 Bde., Berlin: Julius Springer, 1921, 1922, 1923.

  13. Klein, Felix, and Lie, Sophus. Ueber die Haupttangentencurven der Kummer’schen Fläche vierten Grades mit 16 Knotenpunkten, Mathematische Annalen 23(1884): 198–228.

  14. MacCullagh, James. On the double refraction of light in a crystallized medium, according to the principles of Fresnel, Transactions of the Royal Irish Academy, 16(2)(1830): 6578.

  15. Plücker, Julius. Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement, Erste Abteilung, Leipzig: Teubner, 1868.

  16. Plücker, Julius. Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement, Zweite Abteilung, hrsg. F. Klein, Leipzig: Teubner, 1869.

  17. Rohn, Karl. Drei Modelle der Kummer’schen Fläche, 3 S., Darmstadt: L. Brill, 1877.

  18. Rohn, Karl. Transformation der hyperelliptischen Functionen p = 2 und ihre Bedeutung für die Kummer’sche Fläche, Mathematische Annalen 15(1879): 315–354.

  19. Rowe, David E. Klein, Lie, and their early Work on Quartic Surfaces, Festschrift for Umberto Bottazzini, to appear in 2018.

  20. Schilling, Martin, Hrsg. Katalog mathematischer Modelle fr den höheren mathematischen Unterricht, Leipzig: M. Schilling, 1911.

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  1. Johannes Gutenberg University, Mainz, Germany

    David E. Rowe

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  1. David E. Rowe
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Correspondence to David E. Rowe.

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Years Ago features essays by historians and mathematicians that take us back in time. Whether addressing special topics or general trends, individual mathematicians or “schools” (as in schools of fish), the idea is always the same: to shed new light on the mathematics of the past. Submissions are welcome.

Submissions should be uploaded to http://tmin.edmgr.com or sent directly to Jemma Lorenat, e-mail: jlorenat@pitzer.edu.

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Rowe, D.E. On Models and Visualizations of Some Special Quartic Surfaces. Math Intelligencer 40, 59–67 (2018). https://doi.org/10.1007/s00283-017-9773-3

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