Peano surface

Model of the Peano surface in the Dresden collection

In mathematics, the Peano surface is the graph of the two-variable function

f ( x , y ) = ( 2 x 2 − y ) ( y − x 2 ) . {\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2}).}

It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and minima of functions of two variables.[1][2]

The surface was named the Peano surface (German: Peanosche Fläche) by Georg Scheffers in his 1920 book Lehrbuch der darstellenden Geometrie.[1][3] It has also been called the Peano saddle.[4][5]

Properties

Peano surface and its level curves for level 0 (parabolas, green and purple)

The function f ( x , y ) = ( 2 x 2 − y ) ( y − x 2 ) {\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2})} whose graph is the surface takes positive values between the two parabolas y = x 2 {\displaystyle y=x^{2}} and y = 2 x 2 {\displaystyle y=2x^{2}} , and negative values elsewhere (see diagram). At the origin, the three-dimensional point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} on the surface that corresponds to the intersection point of the two parabolas, the surface has a saddle point.[6] The surface itself has positive Gaussian curvature in some parts and negative curvature in others, separated by another parabola,[4][5] implying that its Gauss map has a Whitney cusp.[5]

Intersection of the Peano surface with a vertical plane. The intersection curve has a local maximum at the origin, to the right of the image, and a global maximum on the left of the image, dipping shallowly between these two points.

Although the surface does not have a local maximum at the origin, its intersection with any vertical plane through the origin (a plane with equation y = m x {\displaystyle y=mx} or x = 0 {\displaystyle x=0} ) is a curve that has a local maximum at the origin,[1] a property described by Earle Raymond Hedrick as "paradoxical".[7] In other words, if a point starts at the origin ( 0 , 0 ) {\displaystyle (0,0)} of the plane, and moves away from the origin along any straight line, the value of ( 2 x 2 − y ) ( y − x 2 ) {\displaystyle (2x^{2}-y)(y-x^{2})} will decrease at the start of the motion. Nevertheless, ( 0 , 0 ) {\displaystyle (0,0)} is not a local maximum of the function, because moving along a parabola such as y = 2 x 2 {\displaystyle y={\sqrt {2}}\,x^{2}} (in diagram: red) will cause the function value to increase.

The Peano surface is a quartic surface.

As a counterexample

In 1886 Joseph Alfred Serret published a textbook[8] with a proposed criteria for the extremal points of a surface given by z = f ( x 0 + h , y 0 + k ) {\displaystyle z=f(x_{0}+h,y_{0}+k)}

"the maximum or the minimum takes place when for the values of h {\displaystyle h} and k {\displaystyle k} for which d 2 f {\displaystyle d^{2}f} and d 3 f {\displaystyle d^{3}f} (third and fourth terms) vanish, d 4 f {\displaystyle d^{4}f} (fifth term) has constantly the sign − , or the sign +."

Here, it is assumed that the linear terms vanish and the Taylor series of f {\displaystyle f} has the form z = f ( x 0 , y 0 ) + Q ( h , k ) + C ( h , k ) + F ( h , k ) + ⋯ {\displaystyle z=f(x_{0},y_{0})+Q(h,k)+C(h,k)+F(h,k)+\cdots } where Q ( h , k ) {\displaystyle Q(h,k)} is a quadratic form like a h 2 + b h k + c k 2 {\displaystyle ah^{2}+bhk+ck^{2}} , C ( h , k ) {\displaystyle C(h,k)} is a cubic form with cubic terms in h {\displaystyle h} and k {\displaystyle k} , and F ( h , k ) {\displaystyle F(h,k)} is a quartic form with a homogeneous quartic polynomial in h {\displaystyle h} and k {\displaystyle k} . Serret proposes that if F ( h , k ) {\displaystyle F(h,k)} has constant sign for all points where Q ( h , k ) = C ( h , k ) = 0 {\displaystyle Q(h,k)=C(h,k)=0} then there is a local maximum or minimum of the surface at ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} .

In his 1884 notes to Angelo Genocchi's Italian textbook on calculus, Calcolo differenziale e principii di calcolo integrale, Peano had already provided different correct conditions for a function to attain a local minimum or local maximum.[1][9] In the 1899 German translation of the same textbook, he provided this surface as a counterexample to Serret's condition. At the point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} , Serret's conditions are met, but this point is a saddle point, not a local maximum.[1][2] A related condition to Serret's was also criticized by Ludwig Scheeffer, who used Peano's surface as a counterexample to it in an 1890 publication, credited to Peano.[6][10]

Models

Models of Peano's surface are included in the Göttingen Collection of Mathematical Models and Instruments at the University of Göttingen,[11] and in the mathematical model collection of TU Dresden (in two different models).[12] The Göttingen model was the first new model added to the collection after World War I, and one of the last added to the collection overall.[6]

References

  1. ^ a b c d e Emch, Arnold (1922). "A model for the Peano Surface". American Mathematical Monthly. 29 (10): 388–391. doi:10.1080/00029890.1922.11986180. JSTOR 2299024. MR 1520111.
  2. ^ a b Genocchi, Angelo (1899). Peano, Giuseppe (ed.). Differentialrechnung und Grundzüge der Integralrechnung (in German). B.G. Teubner. p. 332.
  3. ^ Scheffers, Georg (1920). "427. Die Peanosche Fläche". Lehrbuch der darstellenden Geometrie (in German). Vol. II. pp. 261–263.
  4. ^ a b Krivoshapko, S. N.; Ivanov, V. N. (2015). "Saddle Surfaces". Encyclopedia of Analytical Surfaces. Springer. pp. 561–565. doi:10.1007/978-3-319-11773-7_33. See especially section "Peano Saddle", pp. 562–563.
  5. ^ a b c Francis, George K. (1987). A Topological Picturebook. Springer-Verlag, New York. p. 88. ISBN 0-387-96426-6. MR 0880519.
  6. ^ a b c Fischer, Gerd, ed. (2017). Mathematical Models: From the Collections of Universities and Museums – Photograph Volume and Commentary (2nd ed.). doi:10.1007/978-3-658-18865-8. ISBN 978-3-658-18864-1. See in particular the Foreword (p. xiii) for the history of the Göttingen model, Photo 122 "Penosche Fläsche / Peano Surface" (p. 119), and Chapter 7, Functions, Jürgen Leiterer (R. B. Burckel, trans.), section 1.2, "The Peano Surface (Photo 122)", pp. 202–203, for a review of its mathematics.
  7. ^ Hedrick, E. R. (July 1907). "A peculiar example in minima of surfaces". Annals of Mathematics. Second Series. 8 (4): 172–174. doi:10.2307/1967821. JSTOR 1967821.
  8. ^ Serret, J. A. (1886). Cours de calcul différentiel et intégral. Vol. 1 (3d ed.). Paris. p. 216 – via Internet Archive.{{cite book}}: CS1 maint: location missing publisher (link)
  9. ^ Genocchi, Angelo (1884). "Massimi e minimi delle funzioni di più variabili". In Peano, Giuseppe (ed.). Calcolo differenziale e principii di calcolo integrale (in Italian). Fratelli Bocca. pp. 195–203.
  10. ^ Scheeffer, Ludwig (December 1890). "Theorie der Maxima und Minima einer Function von zwei Variabeln". Mathematische Annalen (in German). 35 (4): 541–576. doi:10.1007/bf02122660. S2CID 122837827. See in particular pp. 545–546.
  11. ^ "Peano Surface". Göttingen Collection of Mathematical Models and Instruments. University of Göttingen. Retrieved 2020-07-13.
  12. ^ Model 39, "Peanosche Fläche, geschichtet" and model 40, "Peanosche Fläche", Mathematische Modelle, TU Dresden, retrieved 2020-07-13

External links