Abstract
Here are three examples of “facts” – but rather more of “results”, of “theorems” – that illustrate the difference between the mathematician and the “person in the street”. They are excellent for explaining the nature of mathematics to a non-professional.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
[B] Berger, M. (1987, 2009).Geometry I, II. Berlin/Heidelberg/New York: Springer
[BG] Berger, M., & Gostiaux, B. (1987). Differential geometry: Manifolds, curves and surfaces. Berlin/Heidelberg/New York: Springer
A’Campo, N. (1980). Sur la première partie du seizième problème de Hilbert. In Séminaire Bourbaki 1978–79: Vol. 770. Springer lecture notes in mathematics (pp. 208–227). Berlin/Heidelberg/New York: Springer
A’Campo, N. (2000a). Generic immersion of curves, knots, monodromy and gordian number. Publications mathm´ atiques de líInstitut des hautes études scientifiques, 770, 208–227
A’Campo, N. (2000b). Planar trees, slalom curves and hyperbolic knots. Publications mathm´ atiques de líInstitut des hautes études scientifiques
Alias, J. (1984). La voie ferrée. Paris: Eyrolles
Angenent, S. (1991). On formation of singularities in the curve shortening flow. Journal of Differential Geometry, 33, 601–633
Angenent, S. (1992). Shrinking doughnuts. In N.G. Lloyd, L.A. Peletier, & J. Serrin (Eds.), Nonlinear diffusion equations and their equilibrium states, 3. Boston: Birkhäuser
Arnold, V. (1978). Mathematical methods of classical mechanics. Berlin/Heidelberg/New York: Springer
Arnold, V. (1994a). Topological invariants of plane curves and caustics. University Lecture Series. Providence, RI: American Mathematical Society
Arnold, V. (1994b). Plane curves, their invariants, perestroikas and classifications. Advances in Soviet Mathematics, 21, 33–91
Arnold, V. (1995). The geometry of spherical curves and the algebra of quaternions. Russian Mathematical Surveys, 50, 1–68
Arnold, V. (1996). Remarks on the extactic points of plane curves. In The Gelfand mathematical seminars (pp. 11–22). Boston: Birkhäuser
Barth, W., & Bauer, T. (1996). Poncelet theorems. Expositiones Mathematicae, 14, 125–144
Benoist, Y. & Hulin, D. (2004). Itération de pliages de quadrilatères. Inventiones Mathematicae, 157, 147–194
Bérard, P., Besson, G., & Gallot, S. (1985). Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy. Inventiones Mathematicae, 80, 295–308
Berger, M. (1972). Enveloppes de droites. Bulletin de l’Association des Professeurs de Mathématiques de l’Enseignement Public, 283, 311–314
Berger, M. (1993). Encounter with a geometer: Eugenio Calabi. In P. de Bartolomeis, F. Tricerri, & E. Vesentini (Eds.), Conference in honour of Eugenio Calabi, manifolds and geometry (pp. 20–60). Pisa: Cambridge University Press
Berger, M. (1999). Riemannian geometry during the second half of the twentieth century. Providence, RI: American Mathematical Society
Berger, M. (2003). A panoramic view of Riemannian geometry. Berlin/Heidelberg/New York: Springer
Blaschke, W. (1949). Kreis und Kugel. New York: Chelsea
Bol, G. (1950). Projektive Differentialgeometrie. Göttingen: Vandenhoeck and Ruprecht
Boy, W. (1903). Curvatura Integra. Mathematische Annalen, 57, 151–184
Brieskorn, E., & Knörrer, H. (1986). Plane algebraic curves. Boston: Birkhäuser
Buchin, S. (1983). Affine differential geometry. New York: Gordon and Breach
Burago, Y., & Zalgaller, V. (1988). Geometric inequalities. Berlin/Heidelberg/New York: Springer
Cartan, E. (1946–1951). Leçons sur la géométrie des espaces de Riemann (2nd ed.). Paris: Gauthier-Villars
Chenciner, A. (2006). Courbes algébriques. Berlin/Heidelberg/New York: Springer
Cheeger, J. (1999). Differentiability of Lipschitz functions on a metric measure space. GAFA, Geometric and Functional Analysis, 9, 428–517
Chmutov, S., & Duzhin, S. (1997). Explicit formulas for Arnold’s generic curve invariants. In The Arnold-Gelfand mathematical seminars (pp. 123–138). Boston: Birkhäuser
Coolidge, J. (1959). A treatise on algebraic plane curves. Dover: Oxford University Press
Darboux, G. (1879). De l’emploi des fonctions elliptiques dans la théorie du quadrilatère plan. Bulletin des sciences mathématiques, 3, 109–128
Darboux, G. (1880). Sur le contact des coniques et des surfaces. Comptes Rendus, AcadÈmie des sciences de Paris, 91, 969–971
De Turck, D., Gluck, H., Pomerleano, D., & Shea Vick, R. (2007). The four vertex theorem and its converse. Notices of the American Mathematical Society, 54, 191–207
Dieudonné, J. (1960). Foundations of modern analysis. New York: Academic press
Dieudonné, J. (1985). History of algebraic geometry. Monterey, CA: Wadsworth
Dillen, F.J.E., Verstraelen, L.C.A. (Eds.). (2000). Handbook of differential geometry. Amsterdam: Elsevier
Dombrowski, P. (1999). Wege in euklidischen Ebenen. Berlin/Heidelberg/New York: Springer
Emch, A. (1900). Illustration of the elliptic integral of the first kind by a certain link work. Annals of Mathematics, 1, 81–92
Emch, A. (1901). An application of elliptic functions to Peaucellier link-work (inversor). Annals of Mathematics, 2, 60–63
Fabricius-Bjerre, F. (1962). On the double tangents of plane closed curves. Mathematica Scandinavica, 11, 113–116
Fischer, G. (1986a). Mathematische modelle. Braunschweig, Germany: Vieweg
Fischer, G. (1986b). Mathematical models: Photograph volume and commentary. Braunschweig, Germany: Vieweg
Fulton, W. (1969). Algebraic curves. New York: Benjamin
Gage, M. (1986b). On an area-preserving evolution equation for plane curves. Contemporary Mathematics, 51, 51–62
Gage, M., & Hamilton, R. (1986a). The heat equation shrinking plane curves. Journal of Differential Geometry, 23, 6996
Giannopoulos, A., & Milman, V. (2001). Euclidean structure in finite dimensional normed spaces. In W.B. Johnson & J. Lindenstrauss (Eds.), Handbook of Geometry of Banach Spaces (Vol. 1). Dordrecht: Kluwer
Gibbons, G., & Rasheed, D. (1995). Nuclear Physics, B454, 185
Gluck, H. (1971). The converse of the four vertex theorem. Líenseignement mathématique, XVII, 295–309
Gluck, H. (1972). The generalized Minkowski problem in differential geometry in the large. Annals of Mathematics, 96, 245–276
Greenberg, M. (1967). Lectures on algebraic topology. New York: Benjamin
Griffiths, P., & Harris, J. (1978). Principles of algebraic geometry. New York: John Wiley
Gromov, M. (1980). Paul Levy’s isoperimetric inequality. Prepublication M/80/320, Institut des Hautes Études Scientifiques. Appears as Appendix C in Gromov (1999)
Gromov, M. (1999). Metric structures for Riemannian and non-Riemannian spaces. Boston: Birkhäuser
Guggenheimer, H. (1963). Differential geometry. New York: McGraw Hill
Guieu, L., Mourre, E. & Ovsienko, V. (1996). Theorem on six vertices of a plane curve via Sturm theory. In Arnold-Gelfand mathematical seminars (pp. 257–266). Boston: Birkhäuser
Harnack, A. (1876). Über Vieltheiligkeit der ebenen algebraischen Curven. Mathematische Annalen, 10, 189–199
Haupt, O., & Künneth, H. (1967). Endliche Ordnung. Berlin/Heidelberg/New York: Springer
Hellgouarch, Y. (2000). Rectificatif à l’article de H. Darmon. Gazette des mathÈmaticiens (Soc. Math. France), 85, 31–32
Hilbert, D., & Cohn-Vossen, S. (1952). Geometry and the imagination. New York: Chelsea
Hofer, H., & Zehnder, E. (1994). Symplectic capacities. Boston: Birkhäuser
Hopf, H. (1935). Über die Drehung der Tangenten und Sehnen ebener Kurven. Compositio Mathematica, 2, 50–62
Humbert, G. (1888). Sur les courbes algébriques planes rectifiables. Journal de Mathématiques Pures et Appliquée, IV, 133–151
Hurwitz, M. (1902). Sur quelques applications géométriques des séries de Fourier. Annales Scientifiques de l’École Normale Supérieure, 19, 357–408
Itenberg, I., & Viro, O. (1996). Patchworking algebraic curves disproves the Ragsdale conjecture. The Mathematical Intelligencer, 18(4), 19–28
Kapovich, M., & Millson, J. (1996). The symplectic geometry of polygons in Euclidean space. Journal of Differential Geometry, 44, 479–513
Kaufman, L. (1994). Knots and physics. Singapore: World Scientific
Kneser, A. (1912). Bermerkungen über die Anzahl der Extreme der Krümmung auf geschlossenene Kurven und über verwandte Fragen in einer nicht-euklidischen Geometrie (pp. 170–180). Leipzig-Berlin: H. Weber Festschrift
Knöthe, H. (1957). Contributions to the theory of convex bodies. Michigan Mathematical Journal, 4, 39–52
Lafontaine, J. (1996). Introduction aux variétés différentielles. Grenoble: Presses Universitaires de Grenoble
Laumon, G. (1976). Degré de la variété duale d’une hypersurface à singularités isolées. Bulletin de la Société MathÈmatique de France, 104, 51–63
Lebesgue, H. (1950). Leçons sur les constructions géométriques (Reprint Jacques Gabay, 1987). Paris: Gauthier-Villars
Lucas, E. (1960). Récréations mathématiques. Paris: Gauthier-Villars
Marchaud, A. (1936). Les surfaces du second ordre en géométrie finie. Journal de Mathématiques Pures et Appliquée, 18, 293–300
Marchaud, A. (1965). Sur les droites de la surface du troisième ordre en géométrie finie. Journal de Mathématiques Pures et Appliquée, 44, 49–69
McDuff, D. (2000). A glimpse into symplectic geometry. In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: Frontiers and perspectives. Providence, RI: American Mathematical Society
McDuff, D., & Salamon, D. (1998). Introduction to symplectic topology. Oxford, UK: Oxford
Mukhopadhyaya, S. (1909). New methods in the geometry of plane arcs. Bulletin of the Calcutta Mathematical Society, 1, 31–37
Osserman, R. (1978). The isoperimetric inequality. Bulletin of the American Mathematical Society, 84, 1182–1238
Osserman, R. (1979). Bonnesen-Fenchel isoperimetric inequalities. The American Mathematical Monthly, 86, 1–29
Osserman, R. (1985). The four-or-more vertex theorem. The American Mathematical Monthly, 92, 332–337
Oxtoby, J. (1980). Measure and category. Berlin/Heidelberg/New York: Springer
Petitot, J., & Tondut, Y. (1998). Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux. Mathématiques, Infomatiques et Sciences Humaines, EHESS, 145, 5–101
Pohl, W. (1975). A theorem of Géométrie Finie. Journal of Differential Geometry, 10, 435–466
Porteous, I. (1994). Geometric differentiation (for the intelligence of curves and surfaces). Cambridge, UK: Cambridge University Press
Porter, T. (1933). A history of the classical isoperimetric problem. In Contributions to the calculus of variations. Chicago: University of Chicago Press
Raphaël, E. (1992). Peut-on faire flotter des troncs d’arbre en apesanteur? Bulletin de la Société Française de Physique, 92, 21
Raphaël, E., di Meglio, J.-M., Berger, M., & Calabi, E. (1992). Convex particles at interfaces. Journal de Physique I, 2, 571–579
Raphaël, E., & Williams, D. (1993). Three-dimensional convex particles at interfaces. Journal of Colloid and Interface Science, 155, 509–511
Rideau, F. (1989). Les systèmes articulés. Pour la Science, 136, 94–101
Risler, J.-J. (1993). Construction d’hypersurfaces réelles (d’après Viro), Séminaire Bourbaki. Astérisque, 216, 69–87
Ronga, F. (1998). Klein’s paper on real flexes vindicated. In B. Jakubczyk, W. Pawlucki, & J. Stasica (Eds.), Singularities symposium – Lojasiewicz 70. Warszawa: Banach Center Publications 44
Santalo, L. (1976). Integral geometry and geometric probability. Reading, MA: Addison-Wesley
Schmidt, E. (1939). Über das isoperimetrische Problem im Raum von n Dimensionen. Mathematische Zeitschrift, 44, 689–788
Schwartz, H. (1884). Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt als jeder andere Körper gleichen Volumens. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, ou OEuvres complètes, Berlin, 1890, 1–13
Sedykh, V. (2000). Discrete variants of the four-vertex theorem (Preprint no. 9615). CERE-MADE, Dauphine: Université Paris IX
Segre, B. (1957). Some properties of differentiable varieties and transformations. Berlin/ Heidelberg/New York: Springer
Segre, B. (1968). Alcune proprieta differenziali in grande delle curve chiuse sghembe. Rendiconti di Matematica, 1, 237–297
Serre, J.-P. (1970). Cours d’arithmétique. Paris: Presses Universitaires de France
Silverman, J. (1986). The arithmetic of elliptic curves. Berlin/Heidelberg/New York: Springer
Silverman, J., & Tate, J. (1992). Rational points on elliptic curves. Berlin/Heidelberg/New York: Springer
Sossinski, A. (1999). Noeuds (Genèse d’une théorie mathématique). Paris: Seuil
Tabachnikov, S. (2002). Dual billiards in the hyperbolic plane. Nonlinearity, 15, 1051–1072
Thom, R. (1962). Sur la théorie des enveloppes. Journal de Mathématiques Pures et Appliquée, 41, 177–192
Thom, R. (1969). Sur les variétés d’ordre fini. In D. Spencer & S. Iyannaga (Eds.), Global analysis. Papers in Honor of K. Kodaira. Princeton, NJ: Princeton University Press
Thorbergsson, G., & Umehara, M. (1999). A unified approach to the four vertex theorems. II. American Mathematical Sociey Translations, 190, 229–252
Thorbergsson, G., & Umehara, M. (2002). Sextactic points on a simple closed curve. Nagoya Mathematical Journal, 167, 55–94
Thorbergsson, G., & Umehara, M. (2004). A global theory of flexes of periodic functions. Nagoya Mathematical Journal, 173, 85–138
Turaev, V. (1994). Quantum invariants of knots and 3-manifods. Berlin: de Gruyter
Uribe-Vargas, R. (2004a). Four-vertex theorems, Sturm theory and Lagrangian singularities. Mathematical Physics, Analysis and Geometry, 7, 223–237
Uribe-Vargas, R. (2004b). On singularities, ”perestroikas” and differential geometry of space curves. L’enseignement mathématique, 50, 69–101
Viro, O. (1990a). Real algebraic plane curves: Constructions with controlled topology. Leningrad Mathematical Journal, 1(5), 1059–1134
Viro, O. (1990b). Progress in the topology of real algebraic varieties over the last six years. Russian Mathematical Surveys, 41, 55–82
Voisin, C. (1996). Symétrie miroir. Paris: Société Mathematique de France
Walker, R. (1950). Algebraic curves. Princeton, NJ: Princeton University Press
Whitney, H. (1937). On regular closed curves in the plane. Compositio Mathematica, 4, 276–284
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Berger, M. (2010). Plane curves. In: Geometry Revealed. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70997-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-70997-8_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70996-1
Online ISBN: 978-3-540-70997-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)