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Peano surface

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

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

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[edit]

The function ${\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2})}$ whose graph is the surface takes positive values between the two parabolas ${\displaystyle y=x^{2}}$ and ${\displaystyle y=2x^{2}}$, and negative values elsewhere (see diagram). At the origin, the three-dimensional point ${\displaystyle (0,0,0)}$ 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]

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 ${\displaystyle y=mx}$or${\displaystyle x=0}$) 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 ${\displaystyle (0,0)}$ of the plane, and moves away from the origin along any straight line, the value of ${\displaystyle (2x^{2}-y)(y-x^{2})}$ will decrease at the start of the motion. Nevertheless, ${\displaystyle (0,0)}$ is not a local maximum of the function, because moving along a parabola such as ${\displaystyle y={\sqrt {2}}\,x^{2}}$ (in diagram: red) will cause the function value to increase.

The Peano surface is a quartic surface.

As a counterexample[edit]

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

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

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

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 ${\displaystyle (0,0,0)}$, 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[edit]

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[edit]

External links[edit]

• Weisstein, Eric W. "Peano Surface". MathWorld.