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

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 $f(x,y)=(2x^{2}-y)(y-x^{2})$ whose graph is the surface takes positive values between the two parabolas $y=x^{2}$ and $y=2x^{2}$ , 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.^[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]

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".^[7] 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 $(2x^{2}-y)(y-x^{2})$ 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^[8] with a proposed criterion 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^{2}f

and

d^{3}f

(third and fourth terms) vanish,

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 $f$ has the form $z=f(x_{0},y_{0})+Q(h,k)+C(h,k)+F(h,k)+\cdots$ where $Q(h,k)$ is a quadratic form like $ah^{2}+bhk+ck^{2}$ , $C(h,k)$ 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]^[9] In the 1899 German translation of the same textbook, he provided this surface as a counterexample to Serret's condition. At the point $(0,0,0)$ , 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

^ ^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.
^ ^a ^b Genocchi, Angelo (1899). Peano, Giuseppe (ed.). Differentialrechnung und Grundzüge der Integralrechnung (in German). B.G. Teubner. p. 332.
^ Scheffers, Georg (1920). "427. Die Peanosche Fläche". Lehrbuch der darstellenden Geometrie (in German). Vol. II. pp. 261–263.
^ ^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. ISBN 978-3-319-11772-0. See especially section "Peano Saddle", pp. 562–563.
^ ^a ^b ^c Francis, George K. (1987). A Topological Picturebook. Springer-Verlag, New York. p. 88. ISBN 0-387-96426-6. MR 0880519.
^ ^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.
^ 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.
^ 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)
^ 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.
^ Scheeffer, Ludwig (December 1890). "Theorie der Maxima und Minima einer Function von zwei Variabeln" (PDF). Mathematische Annalen (in German). 35 (4): 541–576. doi:10.1007/bf02122660. S2CID 122837827. See in particular pp. 545–546.
^ "Peano Surface". Göttingen Collection of Mathematical Models and Instruments. University of Göttingen. Retrieved 2020-07-13.
^ Model 39, "Peanosche Fläche, geschichtet" and model 40, "Peanosche Fläche", Mathematische Modelle, TU Dresden, retrieved 2020-07-13