Peano surface explained

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

f(x,y) = ( 2 x ^{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

The function

f(x,y) = ( 2 x ^{2-y)(y-x} ^{2)} whose graph is the surface takes positive values between the two parabolas y = x ^{2} and y =2 x ^{2} , and negative values elsewhere (see diagram). At the origin, the three-dimensional point ( 0 , 0 , 0 ) on the surface that corresponds to the intersection point of the two parabolas, the surface has a saddle point. 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]

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 = mx or x =0 ) is a curve that has a local maximum at the origin,^[1] a property described by Earle Raymond Hedrick as "paradoxical".^[6] In other words, if a point starts at the origin ( 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)} will decrease at the start of the motion. Nevertheless, ( 0 , 0 ) is not a local maximum of the function, because moving along a parabola such as 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^[7] with a proposed criteria for the extremal points of a surface given by

z = f(x _{0+h,y} _{0+k)}

"the maximum or the minimum takes place when for the values of

h and k for which d ^{2f} and d ^{3f} (third and fourth terms) vanish, d ^{4f} (fifth term) has constantly the sign −, or the sign +." Here, it is assumed that the linear terms vanish and the Taylor series of f has the formz = f(x _{0,y} _{0)+Q(h,k)+C(h,k)+F(h,k)+ …} where Q(h,k) is a quadratic form like a h ^{2+b} h k + c k ^{2} ,C(h,k) is a cubic form with cubic terms in h and k ,and F(h,k) is a quartic form with a homogeneous quartic polynomial in h and k .Serret proposes that if F(h,k) has constant sign for all points where Q(h,k) = C(h,k) =0 then there is a local maximum or minimum of the surface at (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] [8]} 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 ) , 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.^[9]

Models

Models of Peano's surface are included in the Göttingen Collection of Mathematical Models and Instruments at the University of Göttingen,^[10] and in the mathematical model collection of TU Dresden (in two different models).^[11] 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.^[12]

Notes and References

1. Emch. Arnold. Arnold Emch. 10.1080/00029890.1922.11986180. 2299024. 10. American Mathematical Monthly. 1520111. 388–391. A model for the Peano Surface. 29. 1922.
2. Book: Genocchi, Angelo. Angelo Genocchi

   . Differentialrechnung und Grundzüge der Integralrechnung. B.G. Teubner. 1899. 332. Angelo Genocchi. Giuseppe. Peano. Giuseppe Peano. German.

3. Book: Scheffers, Georg. Georg Scheffers

   . Lehrbuch der darstellenden Geometrie. 1920. II. Georg Scheffers. 427. Die Peanosche Fläche. https://archive.org/details/lehrbuchderdarst02sche/page/260. 261–263. German.

4. Book: Krivoshapko. S. N.. Ivanov. V. N.. Saddle Surfaces. 10.1007/978-3-319-11773-7_33. 561–565. Springer. Encyclopedia of Analytical Surfaces. 2015. See especially section "Peano Saddle", pp. 562–563.
5. Book: Francis, George K.. 0-387-96426-6. 880519. 88. Springer-Verlag, New York. A Topological Picturebook. A Topological Picturebook. 1987.
6. Hedrick. E. R.. Earle Raymond Hedrick. July 1907. 10.2307/1967821. 4. Annals of Mathematics. 1967821. 172–174. Second Series. A peculiar example in minima of surfaces. 8.
7. Book: Serret, J. A.. Cours de calcul différentiel et intégral. 1. 3d. Paris. 1886. 216. Internet Archive.
8. Book: Genocchi, Angelo. Angelo Genocchi

   . Italian. Calcolo differenziale e principii di calcolo integrale. Angelo Genocchi. Giuseppe. Peano. Giuseppe Peano. Fratelli Bocca. 1884. https://archive.org/details/calcolodifferen00peangoog/page/n234. 195–203. Massimi e minimi delle funzioni di più variabili.

9. Scheeffer. Ludwig. December 1890. 10.1007/bf02122660. 4. Mathematische Annalen. German. 541–576. Theorie der Maxima und Minima einer Function von zwei Variabeln. 35. 122837827 . See in particular pp. 545–546.
10. Web site: Peano Surface. Göttingen Collection of Mathematical Models and Instruments. University of Göttingen. 2020-07-13.
11. http://www.math.tu-dresden.de/modellsammlung/karte.php?ID=314 Model 39, "Peanosche Fläche, geschichtet"
12. Book: Fischer. Gerd. 10.1007/978-3-658-18865-8. 2nd. Mathematical Models: From the Collections of Universities and Museums – Photograph Volume and Commentary. Mathematical Models: From the collections of universities and museums. 2017. 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.